Lessons 91–100,Investigation 10 Math...Investigation 10 Block: 90-Minute Class Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Lesson 91 Lesson 92 Lesson 93 Lesson 94 Lesson 95 Cumulative Test

602A Saxon Algebra 1

S E C T I O N O V E R V I E W

10

Lesson Planner

Resources and Planner CD

for lesson planning support

Pacing Guide

45-Minute Class

Day 1 Day 2 Day 3 Day 4 Day 5 Day 6

Lesson 91 Lesson 92 Lesson 93 Lesson 94 Lesson 95 Cumulative Test 18


Lesson 96 Lab 9Lesson 97 Lesson 98 Lesson 99 Lesson 100 Cumulative Test 19

Day 13

Investigation 10

Block: 90-Minute Class


Lesson 91Lesson 92

Lesson 93Lesson 94

Lesson 95Cumulative Test 18

Lab 9Lesson 96Lesson 97

Lesson 98Lesson 99

Lesson 100Cumulative Test 19

Day 7

Investigation 10Lesson 101

Lesson New Concepts

91 Solving Absolute-Value Inequalities

92 Simplifying Complex Fractions

93 Dividing Polynomials

94 Solving Multi-Step Absolute-Value Equations

95 Combining Rational Expressions with Unlike Denominators

Cumulative Test 18, Performance Task 18

96 Graphing Quadratic Functions

LAB 9 Graphing Calculator Lab: Graphing Linear Inequalities

97 Graphing Linear Inequalities

98 Solving Quadratic Equations by Factoring

99 Solving Rational Equations

100 Solving Quadratic Equations by Graphing

Cumulative Test 19, Performance Task 19

INV 10 Investigation: Transforming Quadratic Equations

Resources for Teaching

• Student Edition• Teacher’s Edition• Student Edition eBook• Teacher’s Edition eBook • Resources and Planner CD • Solutions Manual• Instructional Masters• Technology Lab Masters• Warm Up and Teaching Transparencies• Instructional Presentations CD• Online activities, tools and homework help www.SaxonMathResources.com

Resources for Practice and Assessment

• Student Edition Practice Workbook• Course Assessments• Standardized Test Practice• College Entrance Exam Practice• Test and Practice Generator CD using

ExamView™

Resources for Differentiated Instruction

• Reteaching Masters• Challenge and Enrichment Masters• Prerequisite Skills Intervention• Adaptations for Saxon Algebra 1• Multilingual Glossary• English Learners Handbook• TI Resources

* For suggestions on how to implement Saxon Math in a block schedule, see the Pacing section at the beginning of the Teacher’s Edition.

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Section Overview 10 602B

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0Le ssons 91–10 0, I nve st igat ion 10

Diff erentiated Instruction

Below Level Advanced Learners

Warm Up ............................. SE pp. 602, 609, 616, 624, 631, 638, 647, 655, 662, 669

Skills Bank ........................... SE pp. 846–883Reteaching Masters.............. Lessons 91–100,

Investigation 10Warm Up Transparencies .... Lessons 91–100Prerequisite Skills ................. Skills 61, 81 Intervention

Challenge ............................ TE pp. 608, 614, 622, 630, 637, 644, 654, 660, 668, 674

Extend the Example ............. TE pp. 605, 611, 620, 627, 629, 634, 641, 650, 657, 660, 665, 672

Extend the Exploration ........ TE pp. 648Extend the Problem ............. TE pp. 608, 613, 614, 622, 636, 642, 652,

653, 654, 666, 667, 668, 674, 675, 676Challenge and Enrichment ... Challenge: 91–100; Enrichment: 93 Masters

English Learners Special Needs

EL Tips ............................... TE pp. 603, 610, 617, 627, 636, 643, 648, 656, 664, 670, 677

Multilingual Glossary .......... Booklet and Online English Learners Handbook

Inclusion Tips ...................... TE pp. 604, 611, 619, 625, 632, 641, 649, 657, 663, 672

Adaptations for Saxon .......... Lessons 91–100, Cumulative Tests 18, 19 Algebra 1

For All Learners

Exploration .......................... SE pp. 648Caution ................................ SE pp. 603, 618, 627, 633,

649, 657, 664, 671Hints .................................... SE pp. 611, 617, 620, 624, 633,

639, 647, 648, 656, 663, 670, 672, 673

Alternate Method ................ TE pp. 605, 640Online Tools

Error Alert ........................... TE pp. 603, 606, 607, 610, 612, 615, 617, 620, 622, 626, 627, 630, 632, 634, 637, 641, 642, 648, 651, 652, 654, 658, 659, 660, 665, 666, 667, 670, 673, 674, 675, 676, 677

SE = Student Edition; TE = Teacher’s Edition

Math Vocabulary

Lesson New Vocabulary Maintained EL Tip in TE

91absolute-value inequality absolute value

absolute-value expressionabsolute

92complex fraction greatest common factor reciprocal

93coeffi cient rational expression

mono-, bi-, poly-

94 absolute value ring

95least common denominatorleast common multiple

cushion

96

standard form of a quadratic equation axis of symmetryvertexy-interceptzero of a function

diver

97

solution of a linear inequality coordinate plane linear equationordered pair

shaded

98 root of an equation quadratic function root

99

rational equationextraneous solution

cross productsleast common denominatorrational expression

extraneous

100

function notationparabola quadratic equation

related

INV 10

parent functionquadratic functiontransformation

parent function

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S E C T I O N O V E R V I E W 1 0

602C Saxon Algebra 1

Math Highlights

Enduring Understandings – The “Big Picture”

After completing Section 10, students will understand:

• How to solve absolute-value equations.

• How to simplify complex fractions and solve rational equations.

• How to divide polynomials.

• How to graph quadratic functions and solve quadratic equations by graphing or factoring.

Essential Questions

• How do solutions of absolute-value inequalities relate to compound inequalities?

• How can the methods of simplifying fractions be applied to simplifying complex fractions?

• How can long division be applied to dividing algebraic expressions?

• How can the methods of adding and subtracting fractions be applied to simplifying complex fractions?

• What are elements of a parabola and how can any quadratic function be graphed using these elements?

• What is the difference between a quadratic function and a quadratic equation?

• How can the solutions of a quadratic equation be found by graphing the related function?

Math Content Strands Math Processes

Polynomials• Lesson 93 Dividing Polynomials

Rational Expressions and Functions• Lesson 92 Simplifying Complex Fractions• Lesson 95 Combining Rational Expressions with Unlike

Denominators• Lesson 99 Solving Rational Equations

Systems of Equations and Inequalities• Lab 9 Graphing Calculator: Graphing Linear

Inequalities• Lesson 97 Graphing Linear Inequalities

Quadratic Equations

• Lesson 96 Graphing Quadratic Functions• Lesson 98 Solving Quadratic Equations by Factoring• Lesson 100 Solving Quadratic Equations by Graphing• Investigation 10 Transforming Quadratic Functions

Absolute-Value Equations and Inequalities• Lesson 91 Solving Absolute-Value Inequalities• Lesson 94 Solving Multi-Step Absolute-Value Equations

Connections in Practice Problems Lessons

Data Analysis 100Geometry 91, 92, 93, 94, 95, 96, 97, 98, 99, 100Measurement 91, 95, 96, 98, 99Probability 92, 93, 97, 99

Reasoning and Communication Lessons

• Analyze 91, 93, 94, 95, 96, 98, 99, Inv. 10• Error analysis 91, 92, 93, 94, 95, 96, 97, 98, 99, 100• Estimate 91• Formulate 94, 99• Generalize 91, 95, 96, 97, 98, 99, 100, Inv. 10• Justify 91, 92, 93, 94, 95, 96, 98, 99• Math Reasoning 91, 92, 94, 95, 96, 98, 99, 100,

Inv. 10• Multiple choice 91, 92, 93, 94, 95, 96, 97, 98, 99, 100• Multi-step 91, 92, 93, 94, 95, 96, 97, 98, 99, 100• Predict Inv. 10• Verify 94, 96, 97, 99, 100• Write 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Graphing Calculator 93, 96, 97, 98, 100, Inv. 10

Connections

In Examples: Archery, Baseball, Carnival, Gardening, Length of a Garden, Painting, Physics, Polling, Speed Walking, Traveling

In Practice problems: Ages, Archeology, Architecture, Art, Band, Banking, Baseball, Basketball, Bike riding, Canoeing, Carpeting, Census, Commuting, Cost, Diving, Exercise, Flooring, Football, Geography, Hardiness zones, Hiking, Horseback riding, House painting, Housekeeping, Industry, Investments, Lawn care, Olympic swimming pool, Pendulums, Physics, Population, Profit, Rate, Refurbishing, Running, Shopping, Skating, Soccer, Space, Swimming, Track, Travel, United States flag, Vertical motion

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Section Overview 10 602D

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Content Trace

LessonWarm Up:

Prerequisite Skills

New Concepts Where PracticedWhere

Assessed

Looking

Forward

91 Lessons 5, 74 Solving Absolute-Value Inequalities

Lessons 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 104, 106, 107

Cumulative Tests 19, 20, 21, 22, 23

Lessons 94, 101, 107

92 Lessons 11, 57, 72, 83

Simplifying Complex Fractions Lessons 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 105, 107, 108


Lessons 95, 99

93 Lessons 38, 43, 53, 75, 83

Dividing Polynomials Lessons 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 106, 107, 108, 109


Lessons 95, 99

94 Lessons 5, 26 Solving Multi-Step Absolute–Value Equations

Lessons 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 107, 108, 109, 110


Lessons 101, 107

95 Lessons 2, 57, 72, 75

Combining Rational Expressions with Unlike Denominators

Lessons 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 108, 110, 111


Lesson 99

96 Lessons 9, 89 Graphing Quadratic Functions Lessons 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 109, 110, 111, 112

Cumulative Test 20, 21

Lessons 98, 100, 102, 110

97 Lessons 49, 50 Graphing Linear Inequalities Lessons 98, 99, 100, 101, 102, 103, 105, 106, 107, 110, 112, 113, 115

Cumulative Test 20, 21, 22

Lesson 109

98 Lessons 72, 75, 83, 89

Solving Quadratic Equations by Factoring

Lessons 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 111, 113, 114

Cumulative Test 20, 21, 22

Lessons 100, 102, 104, 110, 113

99 Lessons 39, 57 Solving Rational Equations Lessons 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 112, 114, 115

Cumulative Test 20, 21, 22, 23

Lessons in other Saxon High School Math programs

100 Lessons 9, 84 Solving Quadratic Equations by Graphing

Lessons 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 113, 115, 116

Cumulative Test 20, 21

Lessons 102, 104, 110, 113

INV 10 N/A Transforming Quadratic Functions

Lessons 101, 103, 106, 112, 113, 114, 116, 119

Cumulative Test 22

Lessons 107, 114, 119

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S E C T I O N O V E R V I E W 1 0

602E Saxon Algebra 1

Ongoing Assessment

Type Feature Intervention *

BEFORE instruction Assess Prior Knowledge

• Diagnostic Test • Prerequisite Skills Intervention

BEFORE the lesson Formative • Warm Up • Skills Bank• Reteaching Masters

DURING the lesson Formative • Lesson Practice• Math Conversations with the Practice

problems

• Additional Examples in TE• Test and Practice Generator (for additional

practice sheets)

AFTER the lesson Formative • Check for Understanding (closure) • Scaffolding Questions in TE

AFTER 5 lessons Summative After Lesson 95• Cumulative Test 18 • Performance Task 18After Lesson 100• Cumulative Test 19• Performance Task 19

• Reteaching Masters• Test and Practice Generator (for additional

tests and practice)

AFTER 20 lessons Summative • Benchmark Tests • Reteaching Masters• Test and Practice Generator (for additional

tests and practice)

* for students not showing progress during the formative stages or scoring below 80% on the summative assessments

Evidence of Learning – What Students Should Know

Because the Saxon philosophy is to provide students with sufficient time to learn and practice each concept, a lesson’s topic will not be tested until at least five lessons after the topic is introduced.

On the Cumulative Tests that are given during this section of ten lessons, students should be able to demonstrate the following competencies:

• Solve multi-step compound inequalities, and absolute-value equations and inequalities.• Graph quadratic functions, and identify the vertex, the maximum, and the minimum of a parabola.• Simplify complex fractions.• Multiply binomials with radicals.• Factor polynomials, and divide polynomials by monomials.• Use the Pythagorean Theorem, and find the distance between two points.• Find the probability of mutually exclusive events.

Test and Practice Generator CD using ExamView™

The Test and Practice Generator is an easy-to-use benchmark and assessment tool that creates unlimited practice and tests in multiple formats and allows you to customize questions or create new ones. A variety of reports are available to track student progress toward mastery of the standards throughout the year.

NorthStar Math offers you real-time benchmarking, trackingand student progress monitoring.Visit www.NorthStarMath.com for more information.

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Section Overview 10 602F

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Assessment Resources

Resources for Diagnosing and Assessing

• Student Edition• Warm Up• Lesson Practice

• Teacher’s Edition• Math Conversations with the Practice problems• Check for Understanding (closure)

• Course Assessments• Diagnostic Test• Cumulative Tests• Performance Tasks • Benchmark Tests

Resources for Test Prep

• Student Edition Practice• Multiple-choice problems• Multiple-step and writing problems• Daily cumulative practice

• Standardized Test Practice

• College Entrance Exam Practice

• Test and Practice Generator CD using ExamViewTM

Resources for Intervention

• Student Edition • Skills Bank

• Teacher’s Edition• Additional Examples• Scaffolding questions

• Prerequisite Skills Intervention• Worksheets

• Reteaching Masters• Lesson instruction and practice sheets

• Test and Practice Generator CD using ExamViewTM

• Lesson practice problems• Additional tests

Cumulative TestsThe assessments in Saxon Math are frequent and consistently placed after every fi ve lessons to offer a regular method of ongoing testing. These cumulative assessments check mastery of concepts from previous lessons.

Performance Tasks The Performance Tasks can be used in conjunction with the Cumulative Tests and are scored using a rubric.

After Lesson 95 After Lesson 100 For use with Performance Tasks

Name ___________________________________________ Date_______________ Class _______________

Cumulative Test

© Saxon. All rights reserved. 89 Saxon Algebra 1

18B

1. (82) Of five puppies in a litter, four have weights of 4.8 lb, 4.9 lb, 4.5 lb, and 4.4 lb. What could be the weight of the fifth puppy if the average weight of all five puppies is to be between 4.2 and 4.8 lb?

2. (84) Use a table to graph the

function f x( ) = 3x 2 .

3. (61) Simplify 10,000,000 using powers

of ten.

4. (74) Solve | x 5 |= 2 .

5. (76) Find the product 5 = 5( )2

.

6. (69) A rectangular patio has a length of

363 feet and a width of 243 feet. What is the patio’s perimeter?

7. (73) Write a compound inequality that represents all real numbers that are less than 0 or greater than 2. Graph the solution.

8. (68) Nelida has a bag of bagels. There are 4 plain bagels, 5 garlic bagels, 3 sesame bagels, and 6 onion bagels. If Nelida picks a bagel at random, what is the probability she will pick a plain or an onion bagel?

9. (86) Find the distance between the points (4, –2) and (6, 4).

10. (87) Factor 4x 2 = 8xy = 3x = 6y .

11. (62) The stem-and-leaf plot below shows scores on a history test. Find the median, mode, range, and relative frequency of 89.

Scores on a History Test

Stem Leaves

6 0, 1, 1

7 2, 2, 4, 5, 5, 5, 7

8 1, 3, 5, 9, 9

9 5

Key 7 | 2 = 72

Name ___________________________________________ Date_______________ Class _______________

Cumulative Test


18A

1. (82) A farmer randomly chose 4 pumpkins out of a patch of 15 pumpkins. Three of the pumpkins weigh 15 lb, 18 lb, and 16 lb. What could be the weight of the fourth pumpkin if the average weight of all 4 pumpkins is to be between 15 and 20 lb?

2. (84) Use a table to graph the

function f x( ) = 2x 2 .

3. (61) Simplify 1,000,000 using powers of

ten.

4. (74) Solve | x 2 |= 9 .

5. (76) Find the product 3 2( )2

.

6. (69) A rectangular rug has a length of

98 feet and a width of 50 feet. What is the rug’s perimeter?

7. (73) Write a compound inequality that represents all real numbers that are less than –1 or greater than 4. Graph the solution.

8. (68) Jed opens a drawer containing T-shirts. There are 5 white shirts, 3 blue shirts, 2 red shirts, and 2 gray shirts. If Jed picks a shirt at random, what is the probability he will pick a white or a blue T-shirt?

9. (86) Find the distance between the points (2, –3) and (5, 6).

10. (87) Factor 3x 2 + 9xy + 4x + 12y .

11. (62) The stem-and-leaf plot below shows ages of students in a cooking class. Find the median, mode, range, and relative frequency of age 35.

Ages of Students in a Cooking Class

Stem Leaves

1 6, 8, 8

2 4, 5, 5, 5, 6, 8

3 0, 1, 2, 5, 5, 5, 5, 8

4 0, 2, 9

Key 2 | 4 = 24

Name ___________________________________________ Date_______________ Class _______________

Cumulative Test


19B

1. (89) A rock is thrown from a height of 25 feet above the ground. The rock starts with a vertical speed of 96 feet per second. Ignoring friction, the equation y = 16t 2 + 96t + 25 gives the height y

as a function of time t. Find the highest point the rock reaches and how long it takes to reach this point.

2. (84) Determine whether the equation below represents a quadratic function.

y + 7x 2 = 5x 4

3. (83) Determine whether the trinomial below is a perfect-square trinomial. If it is, factor the trinomial.

32x 2 48x + 18

4. (86) Find the midpoint of the line segment with endpoints (–8, 5) and (2, 2).

5. (76) Find the product 5 + 6( )2.

6. (78) Identify the asymptotes for the rational function below.

y =2

x + 8+ 4

7. (73) Write a compound inequality that describes the graph below.

8. (68) What is the probability of rolling either a sum of 7 or a sum of 9 using two different number cubes, each numbered 1 to 6?

Factor the polynomials in problems 9–10.

9. (79) 4x3 + 8x 2 + 12x

10. (87) 3y 2 + 20y + 12 + 5y 3

Name ___________________________________________ Date_______________ Class _______________

Cumulative Test


19A

1. (89) A ball is thrown from a height of 10 feet above the ground. The ball starts with a vertical speed of 64 feet per second. Ignoring friction, the equation y = 16t 2 + 64t + 10 gives the height y

as a function of time t. Find the highest point the ball reaches and how long it takes to reach this point.

2. (84) Determine whether the equation below represents a quadratic function.

y + 2x = 3x 2 + 6

3. (83) Determine whether the trinomial below is a perfect-square trinomial. If it is, factor the trinomial.

8x 2 8x + 2

4. (86) Find the midpoint of the line segment with endpoints (6, 1) and (3, –5).

5. (76) Find the product 2 + 7( )2

.

6. (78) Identify the asymptotes for the rational function below.

y =3

x + 7+ 2

7. (73) Write a compound inequality that describes the graph below.

8. (68) What is the probability of rolling either a sum of 5 or a sum of 8 using two different number cubes, each numbered 1 to 6?

Factor the polynomials in problems 9–10.

9. (79) 3x3 + 3x 2 18x

10. (87) 2y 2 + 3y 3 + 9y + 6

© Saxon. All rights reserved. xi Saxon Algebra 1

Student Rubric

StudentEvalutaion

Knowledge andSkil lsUnderstanding

CommunicationandRepresentation

Process andStrategies

4

I understand and can

justify my reasoning in

more than one way.

I explained my work in

detail and justified my

solution. I described

my work in great

detail and illustrated

my thinking.

I selected the most

appropriate strategy

and used the process

to solve the problem.

3I understand the task

and can show that I

understand.

I can explain my

thinking. I can show

my work. .

I selected a strategy

and followed the

process.

2

I have some

understanding of the

task.

I can explain some of

my thinking. I can

show some of my

work.

I selected a strategy

but became confused

on the process.

1

I need help in

understanding the

task.

I need help in

explaining my thinking.

I need help in showing

my work.

I need help in choosing

a strategy.

© Saxon. All rights reserved. xx SSaxon Algebra 1

TTeacher Rubric

CriteriaPerformance

Knowledge andSkil lsUnderstanding

CommunicationandRepresentation

Process andStrategies

4

The student got it!

The student did it in

new ways and

showed how it

worked. The student

knew and understood

what math concepts

to use.

The student clearly

detailed how he/she

solved the problem.

The student included

all the steps to show

his/her thinking. The

student used math

language, symbols,

numbers, graphs

and/or models to

represent his/her

solution.

The student had an

effective and inventive

solution. The student

used big math ideas

to solve the problem.

The student

addressed the

important details. The

student showed other

ways to solve the

problem. The student

checked his/her

answer to make sure

it was correct

3

The student

understood the

problem and had an

appropriate solution.

All parts of the

problem are

addressed.

The student clearly

explained how he/she

solved the problem.

The student used

math language,

symbols, tables,

graphs, and numbers

to explain how he/she

did the problem.

The student had a

correct solution. The

student used a plan

to solve the problem

and selected an

appropriate strategy.

2

The student

understood parts of

the problem. The

student started, but

he/she couldn’t finish.

The student explained

some of what he/she

did. The student tried

to use words,

symbols, tables,

graphs and numbers

to explain how he/she

did the problem.

The student had part

of the solution, but

did not know how to

finish. The student

was not sure if he/she

had the correct

answer. The student

needed help.

1

The student did not

understand the

problem.

The student did not

explain how he/she

solved the problem.

He/she did not use

words, symbols,

tables or graphs to

show how he/she

The student couldn’t

get started. The

student did not know

how to begin.

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Saxon Algebra 1602

Solving Absolute-Value Inequalities

Warm Up

91LESSON

1. Vocabulary An equation with one or more absolute-value expressions is called an . absolute-value equation

Simplify.

2. �8 - 15� 7 3. �-3 + 9� 6

Solve.

4. �x - 4� = 7 x = 11, -3 5. �x + 7� = 2 x = -5, -9

An absolute-value inequality is an inequality with at least one absolute-value expression. The solution to an absolute-value inequality can be written as a compound inequality.

The inequality �x� < 6 describes all real numbers whose distance from 0 is less than 6 units. The solutions are all real numbers between -6 and 6. The solution can be written -6 < x < 6 or as the compound inequality x > -6 AND x < 6.

640 2-2-4-6

The inequality �x� > 6 describes all real numbers whose distance from 0 is greater than 6 units. The solutions are all real numbers less than -6 or greater than 6. The solution can be written as the compound inequality x < -6 OR x > 6.

640 2-2-4-6

Example 1 Solving Absolute-Value Inequalities by Graphing

Solve each inequality by graphing.

a. �x� < 4 b. �x� > 7

SOLUTION

If the absolute value of x is less than 4, then x is less than 4 units from zero on a number line.

40 2-2-4

The graph shows x < 4 AND x > -4. This can also be written -4 < x < 4.

SOLUTION

If the absolute value of x is greater than 7, then x is more than 7 units from zero on a number line.

6 840 2-2-4-6-8

The graph shows x > 7 OR x < -7.

(74)(74)

(5)(5) (5)(5)

(74)(74) (74)(74)

New ConceptsNew Concepts

Math Language

The absolute value of a number is its distance from zero on the number line.

Reading Math

For the inequality -4 < x < 4, you can say, “x is between -4 and 4.”

Math Reasoning

Analyze Why is the word “OR” used here to describe the solution?

Sample: A number cannot be both greater than 7 and less than -7.

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MATH BACKGROUNDLESSON RESOURCES

Student Edition Practice Workbook 91

Reteaching Master 91Adaptations Master 91Challenge and Enrichment

Master C91

In this lesson, the absolute-value inequalities have the form

⎢x - a� > b,

where a and b are constants. When graphed on the number line, the graph is always symmetric about the center point a.

In this case, the inequality can be understood as “the distance from x to a is greater than b.” Students might have an easier time understanding why such graphs on the number line are symmetrical; one can take b steps from a in two directions.

Warm Up1

602 Saxon Algebra 1

91LESSON

Problems 2–5

Remind students that the expression inside the absolute- value bars can be negative, but the absolute value of any expression cannot be negative.

2 New Concepts

In this lesson, students learn to solve absolute-value inequalities.

Discuss the defi nition of an absolute-value inequality. Explain that the solution to an absolute-value inequality can be written as a compound inequality.

Example 1

Remind students that the absolute value of a number is its distance from zero on the number line.

Additional Example 1

Solve each inequality by graphing.

a. ⎢x� < 2.5 -2.5 < x < 2.5;

20 1-1-2

b. ⎢x� > 10 x > 10 OR x < -10;

100 5-5-10

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Lesson 91 603

Example 2 Isolating the Absolute Value to Solve

Solve and graph each inequality.

a. �x� + 7.4 ≤ 9.8

SOLUTION

Begin by isolating the absolute value.

�x� + 7.4 ≤ 9.8

___-7.4 ___-7.4 Subtraction Property of Inequality

�x� ≤ 2.4 Simplify.

Since the absolute value of x is less than or equal to 2.4, it is 2.4 units or less from zero.

0 2 31-2-3 -1

The solution can be written x ≥ -2.4 AND x ≤ 2.4 or -2.4 ≤ x ≤ 2.4.

b. �x�

_ 4 > 2

SOLUTION


�x�

_ 4 > 2

4 · �x�

_ 4 > 2 · 4 Multiplication Property of Inequality

�x� > 8 Simplify.

The absolute value of x is greater than 8, so it is more than 8 units from zero.

1280 4-4-8-12

The solution is x > 8 OR x < -8.

c. -2�x� < -6

SOLUTION


-2�x� < -6

-2�x�

_ -2

> -6 _ -2

Division Property of Inequality

�x� > 3 Simplify.

Since the absolute value of x is greater than 3, it is more than 3 units from zero.

40 2-2-4

The solution is x > 3 OR x < -3. Online Connection

www.SaxonMathResources.com

Caution

Be sure to reverse the direction of the inequality sign if you multiply or divide by a negative number when solving the inequality.


For this lesson, explain the meaning of the word absolute. Say:

“The word absolute means defi nite or fi nal. It can describe something that exists independently and not in relation to other things.”

Connect the meaning of absolute with absolute value by pointing to the defi nition of the absolute value and emphasizing that the distance from n to 0 is not relative to

another value—it is absolute.

Discuss examples of other uses of the word absolute: the absolute-temperature scale Kelvin, absolute power, etc.

ENGLISH LEARNERS

Lesson 91 603

Example 2

Point out that in solving absolute-value inequalities, it is often easier to isolate the absolute value before solving for the variable.

Error Alert When graphing inequalities on the number line, students often will use open and closed circles incorrectly. Remind them to pay attention to what symbol is used in the expression.


Solve and graph each inequality.

a. ⎢x� - 2.2 ≤ 7.8 -10 ≤ x ≤ 10;

100 5-5-10

b. 3⎢x� > 2.7 x > 0.9 OR x < -0.9;

0.5 10-0.5-1

c. -3⎢x⎢ ≤ -9 x ≥ 3 OR x ≤ -3;

0 3-3


Saxon Algebra 1604

Some absolute-value inequalities have variable expressions inside the absolute-value symbols. The expression inside the absolute-value symbols can be positive or negative.

The inequality �x + 1� < 3 represents all numbers whose distance from -1 is less than 3.

0 2-2-4

3 units3 units

The inequality �x + 1� > 3 represents all numbers whose distance from -1 is greater than 3.

0 2-2-4

3 units3 units

Rules for Solving Absolute-Value InequalitiesFor an inequality in the form �K� < a, where K represents a variable expression and a > 0, solve -a < K < a or K > -a AND K < a.

For an inequality in the form �K� > a, where K represents a variable expression and a > 0, solve K < -a OR K > a.

Similar rules are true for �K� ≤ a or �K� ≥ a.

Example 3 Solving Inequalities with Operations Inside

Absolute-Value Symbols

Solve each inequality. Then graph the solution.

a. �x - 5� ≤ 3

SOLUTION

Use the rules for solving absolute-value inequalities to write a compound inequality.

�x - 5� ≤ 3

x - 5 ≥ -3 AND x - 5 ≤ 3 Write the compound inequality.

__ +5 __ +5 __ +5 __ +5 Addition Property of Inequality

x ≥ 2 AND x ≤ 8 Simplify.

Now graph the inequality.

b. �x + 7� > 3

SOLUTION

Use the rules for solving absolute-value inequalities to write a compound inequality.

�x + 7� > 3

x + 7 < -3 OR x + 7 > 3 Write the compound inequality.

__ -7 __ -7 __ -7 __ -7 Subtraction Property of Inequality

x < -10 OR x > -4 Simplify.

Now graph the inequality.

80 4-4-8-12

6 840 2 6 840 2

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INCLUSION

Have students draw a number line from -5 to 5 with every whole number labeled and 1 inch apart. Have them locate the point 2, and with a ruler, have them then mark all the points that are less than 3 inches away from 2.

Lastly, have them show that the absolute-value inequality that represents the graph is ⎢x - 2� < 3 and the solution is -1 < x < 5.

604 Saxon Algebra 1

Example 3

Point out that the variable expression inside the absolute- value bars “shifts” the center of the graph. The expression x + ashifts the center of the graph to -a and the expression x - b shifts the center of the graph to b when a and b are non-negative constants.


Solve each inequality. Then graph the solution.

a. ⎢x + 4� ≤ 5 x ≤ 1 AND x ≥ -9

0 2-2-4-6-8-10

b. ⎢x - 2� > 3 x < -1 OR x > 5

4 620-2


Lesson 91 605

Example 4 Solving Special Cases

Solve each inequality.

a. �x� + 6 ≤ 4

SOLUTION

�x� + 6 ≤ 4

�x� ≤ -2 Subtract 6 from both sides.

This inequality states that a number’s distance from 0 is less than or equal to -2. No distance can be negative. Therefore, there are no solutions to this inequality. The solution is identified as { } or ∅, the empty set.

b. �x� + 6 > 1

SOLUTION

�x� + 6 > 1

�x� > -5 Subtract 6 from both sides.

This inequality states that a number’s distance from 0 is greater than -5. Since all distances (and absolute values) are positive, all numbers on the number line are solutions. The solution is identified as �, the set of all real numbers. It means that the inequality is an identity; it works for all real numbers.

Example 5 Application: Polling

A poll finds that candidate Garcia is favored by 46% of the voters surveyed and Jackson is favored by 44%. The poll has an accuracy of plus or minus 3%.

a. Write an absolute-value inequality to show the true percentage of voters for Garcia.

SOLUTION

Let the true percentage of voters for Garcia be g.

For g to be within 3%, the distance from g to 46 must be less than or equal to 3. The distance is represented by the absolute value of their difference.

�g - 46� ≤ 3

b. Solve the inequality to find the range for the true percentage of voters who support Garcia.

SOLUTION

�g - 46� ≤ 3

-3 ≤ g - 46 ≤ 3 Write the inequality without an absolute value.

43 ≤ g ≤ 49 Add 46 to all 3 parts of the inequality.

The range is between 43% and 49%.

Math Reasoning

Justify According to the poll, Garcia and Jackson are in a “statistical dead heat”— a tie. Explain why the poll would not say that Garcia (46%) is leading Jackson (44%).

Sample: Because the accuracy of the poll is plus or minus 3%, either could actually be ahead. Garcia’s percentage could actually be 3% less, at 43%. Jackson’s percentage could actually be 3% more, at 47%, and Jackson would be leading Garcia.


Instead of writing an absolute value inequality, have students think about the defi nition of accuracy. Since the problem states that the poll has an “accuracy of plus or minus 3%,” have students write the range for the true percentage of voters that support Garcia.

The limits of the range will be 46 ± 3%, thus the range is between 43% and 49%.

ALTERNATE METHOD FOR EXAMPLE 5

Lesson 91 605

Example 4

Remind students that they should always isolate the absolute value fi rst to identify special cases.


Solve each inequality.

a. ⎢x� + 3 < 3{ } or ∅; There is no graph.

b. ⎢x� + 3 > 2 �;

0

Example 5

Explain to students that a measurement, such as a poll, has an uncertainty or a range of values within which the true value lies.

Extend the Example

Let the true percentage of voters for Jackson be j. Write and solve an absolute-value inequality to show that j is within 3% of 44.⎪j - 44⎥ ≤ 341 ≤ j ≤ 47


Jim and Tom went fi shing and wanted to see who caught the biggest fi sh. Unfortunately, the scale they used was not very accurate and was only good to ±2 ounces. Jim’s fi sh weighed in at 5 pounds 5 ounces and Tom’s was 5 pounds 10 ounces.

a. Let the true weight of Jim’s fi sh be j and Tom’s fi sh be t. Write the absolute-value inequalities.⎢j - 85 oz� ≤ 2 oz⎢t - 90 oz� ≤ 2 oz

b. Can one confi dently say that Tom’s fi sh is bigger than Jim’s? yes


Practice Distributed and Integrated

Lesson Practice

a. Solve and graph the inequality � x � < 12. -12 < x < 12

b. Solve and graph the inequality � x � > 19. x < -19 OR x > 19

c. Solve and graph the inequality � x � + 2.8 ≤ 10.4. -7.6 ≤ x ≤ 7.6

d. Solve and graph the inequality �x�

_ -5 < -1. x < -5 OR x > 5

e. Solve and graph the inequality ⎪x - 10⎥ ≤ 12. -2 ≤ x ≤ 22

f. Solve and graph the inequality ⎪x + 12⎥ > 18. x < -30 OR x > 6

g. Solve the inequality � x � + 21 ≤ 14. ∅

h. Solve the inequality � x � + 33 > 24. �

Industry A machine part must be 15 ± 0.2 cm in diameter.

i. Write an inequality to show the range of acceptable diameters.

j. Solve the inequality to find the actual range for the diameters.14.8 ≤ m ≤ 15.2

a.–f. See Additional Answers.a.–f. See Additional Answers. (Ex 1)(Ex 1)

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

(Ex 3)(Ex 3)

(Ex 4)(Ex 4)

(Ex 4)(Ex 4)

(Ex 5)(Ex 5)

i. � m - 15 � ≤ 0.2i. � m - 15 � ≤ 0.2

1. Find the axis of symmetry for the graph of the equation y = - 1 _ 2 x2 + x - 3.

x = 1

Add or subtract.

2. 6rs

_ r2s2 +

18r _ r2s2

6(s + 3)

_rs2

3. b _

2b + 1 -

6 _ b - 4

b2 - 16b - 6__

(2b + 1)(b - 4)

Factor.

4. -4y4 + 8y3 + 5y2 - 10y 5. 3a2 - 27 3(a + 3)(a - 3)-y(4y2 - 5)(y - 2)

Evaluate.

6. 4x2 + 6x - 4 (2x - 1)(2x + 4) 7. 9x2 - 2x + 32 (9x + 16)(x - 2)

*8. Solve and graph the inequality � x � < 96. -96 < x < 96;

*9. Write Explain what � x � ≥ 54 means on a number line. Sample: x can be any value that is 54 or more units from 0.

*10. Justify When solving an absolute-value inequality, a student gets �x� ≥ -5. Justify that any value for x makes this inequality true.

*11. Multiple Choice Which inequality is represented by the graph? B

80 4 12-4-8-12

A �x� < 9 B �x� > 9 C �x� ≤ 9 D �x� < -9

*12. Track A runner finishes a sprint in 8.54 seconds. The timer’s accuracy is plus or minus 0.3 seconds. Solve and graph the inequality �t - 8.54� ≤ 0.3. 8.24 ≤ t ≤ 8.84

8.848.24 8.44 8.64

(89)(89)

(90)(90) (90)(90)

(87)(87) (83)(83)

(75)(75) (75)(75)

(91)(91) 960 48-48-96 960 48-48-96

(91)(91)

(91)(91)10. Sample: No matter what I substitute for x, its absolute value is going to be greater than -5 because absolute value is always positive.

10. Sample: No matter what I substitute for x, its absolute value is going to be greater than -5 because absolute value is always positive.

(91)(91)

(91)(91)

Saxon Algebra 1606

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606 Saxon Algebra 1

Lesson Practice

Problem c

Scaff olding Before graphing the inequality, have students fi rst isolate the absolute-value term and solve it completely before attempting to graph the solution.

Problem d

Error Alert Students may assume that there is no solution to an absolute value inequality when the expression containing the absolute value is related to a negative number. Remind them to isolate the absolute value, then consider whether there is a solution or not.

Check for Understanding

The questions below help assess the concepts taught in this lesson.

“Explain how to solve an absolute-value inequality.” Sample: If the absolute-value term has a coeffi cient, isolate it fi rst. Check for special cases. The expression inside the absolute-value bars can be positive and negative.

“Explain what absolute value means on the graph of an equation on the number line.” Sample: The absolute value of a number n is the distance from n to 0.

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*13. Error Analysis Two students simplified 2c _

c - 6 + 12 _

6 - c . Which student is correct?

Explain the error.

Student A Student B

2c - 12

_ c - 6

2(c - 6)

_ c - 6

= 2

2c + 12 _ c - 6

2(c + 6)

_ c - 6

= -2

*14. Multi-Step A farmer has a rectangular plot of land with an area of x2 + 22x + 72 square meters. He sets aside x2 square meters for grazing and 2x - 8 square meters for a chicken coop. a. Write a simplified expression for the total fraction of the field the farmer has

set aside. x - 2_x + 18

b. Estimate About what percent of the field has the farmer set aside if x = 30? about 60%

15. Geometry Write a simplified expression for the total fraction of the larger rectangle that the triangle and smaller rectangle cover.

2(x - 1) _

5(x + 9)

16. Find the product of ( √ � 4 - 6) 2 . 16

17. Analyze Why is it necessary to understand factoring when dealing with rational expressions? Sample: Factoring makes it easier to simplify complicated expressions.

18. Multi-Step The base of triangle ABC is x2 + y. The height is 4x + 2xy

_ x3 + xy . What is the

area of triangle ABC ? a. Multiply the base of the triangle by its height. 2(2 + y)

b. Multiply the product from part a by 1 _ 2 . 2 + y

*19. Error Analysis Two students tried to find the axis of symmetry for the equation y = 8x + 2x2. Which student is correct? Explain the error.

Student A Student B

x = -b _ 2a

= -2 _

2(8) =

-2 _ 16

= - 1 _ 8

x = -8 _

2(2) =

-8 _ 4 = -2

*20. Space If it were possible to play ball on Jupiter, the function y = -13x2 + 39x would approximate the height of a ball kicked straight up at a velocity of 39 meters per second, where x is time in seconds. Find the maximum height the ball reaches and the time it takes the ball to reach that height. (Hint: Find the time the ball reaches its maximum height first.) 29.25 feet in 1.5 seconds

21. Measurement The coordinates of two landmarks on a city map are A(5, 3) and B(7, 10). Each grid line represents 0.05 miles. Find the distance between landmarks A and B. 0.364 mile

(90)(90)

13. Student A; Sample: Student B multiplied the denominator of one of the expressions by -1, but forgot to multiply the numerator of that expression by -1also.

13. Student A; Sample: Student B multiplied the denominator of one of the expressions by -1, but forgot to multiply the numerator of that expression by -1also.

(90)(90)

2x x - 2

2x 2x + 18

5x

2x

2x x - 2

2x 2x + 18

5x

2x

(90)(90)

(76)(76)

(88)(88)

(88)(88)

(89)(89)19. Student B; Sample: Student A used the wrong values for a and b.The equation in standard form is y = 2 x2 + 8x, so a = 2 and b = 8.

19. Student B; Sample: Student A used the wrong values for a and b.The equation in standard form is y = 2 x2 + 8x, so a = 2 and b = 8.

(89)(89)

3 43 4

(86)(86)

Lesson 91 607

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Lesson 91 607

Practice3

Math ConversationsDiscussion to strengthen understanding

Problem 16

Error AlertA common mistake students make when multiplying expressions is not simplifying each expression fi rst. Write the statement:

( √ � 4 - 6) 2 = ( √ � 4 - 6) ( √ � 4 - 6)

= 4 - 12 √ � 4 + 36

= 40 - 12 √ � 4

= 40 - 12(2)

= 16

Compare with the statement:

( √ � 4 - 6) 2 = (2 - 6)2 = (-4)2

= 16

Remind students that it is often benefi cial to simplify the expressions before multiplying them.

Problem 18

Guide the students by asking them the following questions.

“How can x3 + xy be factored?” Sample: Put x outside of the parentheses: x(x2 + y)

“How can the expression

(x2 + y) · 4x + 2xy

________ x(x2 + y) be

simplifi ed?” Sample: Divide out x and x2 + y from both the numerator and denominator: 4 + 2y.

“What is the formula for the area of a triangle?” Sample: one half times the base times the height


Saxon Algebra 1608

22. Archeology Archeologists use coordinate grids to record locations of artifacts. Jonah recorded that he found one old coin at (41, 37), and a second old coin at (5, 2). Each unit on his grid represents 0.25 feet. How far apart were the coins? Round your answer to the nearest tenth of a foot. 12.6 feet

23. Find the length t to the nearest tenth. 5.3

5

t√3

24. House Painting A house painter leans a 34-foot ladder against a house with the bottom of the ladder 7 feet from the base of the house. Will the top of the ladder touch the house above or below a windowsill that is 33 feet off the ground? above; 342 - 72 > 3 32

25. Graph the function y = 4x2.

26. Generalize What is the factored form of a 2m + 2 a m b n + b 2n ? ( am + bn) 2

27. Shopping Roger has $40 to buy CDs. The CDs cost $5 each. He will definitely buy at least 3 CDs. How many CDs can Roger buy? Use inequalities to solve the problems. x ≥ 3 and 5x ≤ 40; 3 ≤ x ≤ 8

28. Multi-Step A summer school program has a budget of $1000 to buy T-shirts. Twenty free T-shirts will be received when they place their order. The number of T-shirts y that the program can get is given by y = 1000

_ x + 20, where x is the price per T-shirt. a. What is the horizontal asymptote of this rational function? y = 20

b. What is the vertical asymptote? x = 0

c. If the price per T-shirt is $10, how many T-shirts can the program receive? 120

*29. Suppose the area of a rectangle is represented by the expression 4x2 + 9x + 2. Find possible expressions for the length and width of the rectangle. width =

(4x + 1), length = (x + 2) or width = (x + 2) and length = (4x + 1)

30. Simplify the expression 6 √ � 8 · √ � 5 . 12√ �� 10

(86)(86)

(85)(85)

(85)(85)

(84)(84)25.

x

y

8

12

16

4

2 4-2-4

25.

x

y

8

12

16

4

2 4-2-4

(83)(83)

(82)(82)

(78)(78)

(Inv 9)(Inv 9)

(76)(76)


LOOKING FORWARD

Solving absolute-value inequalities prepares students for

• Lesson 94 Solving Multi-Step Absolute-Value Equations

• Lesson 101 Solving Multi-Step Absolute-Value Inequalities

• Lesson 107 Graphing Absolute-Value Functions

There are also compound absolute-value inequalities. Consider the inequalities:

⎢x� < 6 AND ⎢x - 2� > 6

Have students solve the compound absolute-value inequalities. Sample: The solution will be the overlap of the two graphs on a number line, -6 < x < -4

CHALLENGE

608 Saxon Algebra 1

Problem 27

Extend the Problem

“If the CDs cost $3 each at the used CD store, how many CDs can Roger buy?” 13 or fewer


Lesson 92 609

Warm Up

92LESSON

1. Vocabulary For any nonzero real number n, the of the number is 1

_ n . reciprocal

Identify the LCM.

2. 4x - 16 and x - 4 4(x - 4) 3. 18x2 and 9x 18x2

Factor.

4. x2 - 4x - 77 (x + 7)(x - 11) 5. 18x2 + 12x + 2 2(3x + 1)(3x + 1)

A complex fraction is a fraction that contains one or more fractions in the numerator or the denominator.

Complex FractionsThere are two ways to write a fraction divided by a fraction.

a _

b _

c _

d =

a _ b ÷

c _ d , when b ≠ 0, c ≠ 0, and d ≠ 0.

A complex fraction can be written as a fraction divided by a fraction. The rules for dividing fractions can be applied to simplify complex fractions.

Example 1 Simplifying by Dividing

Simplify a _ x _

b _ a + x

.

SOLUTION

a _ x _

b _ a + x

= a _ x ÷

b _ a + x Write using a division symbol.

= a _ x ·

a + x _ b Multiply by the reciprocal.

= a(a + x)

_ xb

Multiply.

(11)(11)

(57)(57) (57)(57)

(72)(72) (83)(83)


Simplifying Complex Fractions

Math Reasoning

Write Explain why the complex fraction

a _ x _ b

_ a + x can be

written as a_x ÷ b_a + x .

Sample: The fraction bar means divide, so the two expressions are equal.


LESSON RESOURCES



Master C92

The key to simplifying complex fractions is to remember that a fraction is just a way to write a series of division expressions. Take the complex fraction:

a __ x ____

b _____ a + y

It is the same as the statement: “a divided by x is divided by b divided by a + y.”

MATH BACKGROUND

Warm Up1

92LESSON

Lesson 92 609

Problems 2 and 3

Point out to students that factoring the expressions should always precede identifying the LCM.

2 New Concepts

In this lesson, students learn to simplify complex fractions.

Discuss the defi nition of a complex fraction.

Example 1

Remind students that dividing by a fraction is the same as multiplying by its reciprocal.


Simplify b __ y ____

y _____ a + y

. b(a + y) _______

y2


Saxon Algebra 1610

The product of a number and its reciprocal is 1. To eliminate the fraction in the denominator of a complex fraction, multiply the numerator and the denominator by the reciprocal of the denominator.

Example 2 Simplifying Using the Reciprocal of the Denominator

Simplify am

_ n _ x _ mn

.

SOLUTION

am

_ n _ x _ mn

= am _ n ·

mn _ x _

x _ mn · mn _ x

Multiply by the reciprocal of the denominator.

= am _ n ·

mn _ x _

x _ mn · mn _ x

Divide out common factors.

= am2

_ x _1

Multiply.

= am2_

x Simplify.

Example 3 Factoring to Simplify

Simplify

3x _

6x + 12 _

9 _

x + 2 .

SOLUTION

3x _

6x + 12 _

9 _

x + 2

= 3x _

6x + 12 ÷

9 _ x + 2

Write using a division symbol.

= 3x _

6(x + 2) ·

x + 2 _ 9 Factor out the GCF and multiply by the reciprocal.

= 3

1 x _

6(x + 2) ·

(x + 2) _

9 3 Divide out common factors.

= x_18

Simplify. Online Connection



For this lesson, explain the meaning of the word reciprocal. Say:

“The reciprocal of something is done for another in return for something else.”

Connect the meaning of reciprocal with the fraction 1 __ x and its reciprocal x by emphasizing that the denominator and numerator swapped places.

Discuss examples of other uses of the word reciprocal. For example, when we treat people nicely, we can expect reciprocal comments or treatment.

Also mention that reciprocate is the verb form of the word. Have students come up with things that they do now and would like their friends to reciprocate, e.g., share their snacks.

ENGLISH LEARNERS

610 Saxon Algebra 1

Example 2

Students learn to use the reciprocal of the denominator to simplify complex fractions.


Simplify bt __ as ___ r __ st

. bt2 ___ ar

Error Alert A common mistake that students make when simplifying complex fractions is to divide out factors that are on the very top and very bottom of the fraction. Write the statements:

ab ___ x

___ y ___

bx ≠

a __ x __

y __ x

Remind students to write the fractions using a division symbol.

Example 3

Emphasize to students that factoring all the expressions before multiplying and dividing will make the problem easier to solve.


Simplify x

2 - x _____ 9 _____

x - 1 _____ 3x . x

2 __

3


Lesson 92 611

Example 4 Combining Fractions to Simplify

Simplify 1 _ x _

1 - 1 _ x

.

SOLUTION

1 _ x _

1 - 1 _ x

= 1 _ x _

x - 1

_ x Subtract in the denominator.

= 1 _ x ÷

x - 1 _ x Write using a division symbol.

= 1 _ x ·

x _ x - 1

Multiply by the reciprocal.

= 1 _ x ·

x _ x - 1


= 1 _

x - 1 Simplify.

Example 5 Application: Speed Walking

It took Max 3x2 - 12x _

3x minutes to speed walk to the gym that was

5x - 20 _

x3 miles away. Find his rate in miles per minute.

SOLUTION

r = d _ t Solve for r.

r =

5x - 20 _

x3 __

3x2 - 12x

_ 3x

Evaluate for d and t.

= 5x - 20 _

x3 ÷

3x2 - 12x _ 3x

Write using a division symbol.

= 5x - 20 _

x3 ·

3x _ 3x2 - 12x

Multiply by the reciprocal.

= 5(x - 4)

_ x3

· 3x _

3x(x - 4) Factor out any GCFs.

= 5 _ x3

miles per minute Divide out common factors and simplify.

The expression 5 _ x3 represents Max’s speed-walking rate in miles per minute.

Hint

Find the LCD of 1 and 1 _ x to subtract in the denominator.

Hint

Use the formula d = rt.


Use the following strategy with students who have diffi culty with complex operations. Have students write the expression below.

1 __ 2

____ 3 __ 4

Then have them write the same expression replacing the fractions with their decimal equivalents.

Students should immediately see the order of operations:

0.5 ____ 0.75

= 0.50

____ 0.75

= 2 __ 3

Remind students that the method to simplify a complex fraction is the same.

INCLUSION

Lesson 92 611

Example 4

Students learn to combine fractions to simplify complex fractions.


Simplify 1- 1 ______

x + 1 ________

x ___ 2x . 2x ______

(x + 1)

TEACHER TIPEmphasize that the long fraction bar between the numerator and denominator of a complex fraction indicates division and should be the last operation. The expressions in the numerator and denominator need to be simplifi ed or evaluated separately fi rst.

Example 5

Students learn to fi nd a complex fraction that describes the rate of a person’s walking speed.

Extend the Example

If it takes Max 25 minutes to walk to the grocery store that is 1 mile away, what is x ? x = 5


On her bike, Emily can make it to her school that is x miles away in 3 + 4x minutes. If Jason’s house is 1 __ x miles away from school, how long will it take Emily to get there from her school? 3 + 4x

______ x 2 minutes



Saxon Algebra 1612

Lesson Practice

Simplify.

a. x

_ 4 _

3(x - 3)

_ x

x2_12(x - 3)

b. b _

cd _

2b

_ c 1_

2d

c.

4x2

_ x - 3

_

x _

3x - 9 12x d.

1 _ m + 5 _

2 _ m - x _ m

1 + 5m_2 - x

e. It took Ariel 5x2 - 45x _ 5x

minutes to walk to school that was 3x - 27

_ x3

miles away. Find her rate in miles per minute. 3_x3 miles per minute

(Ex 1)(Ex 1) (Ex 2)(Ex 2)

(Ex 3)(Ex 3) (Ex 4)(Ex 4)

(Ex 5)(Ex 5)

Find the product or quotient.

1. 15x4

_ x - 4

· x2 - 10x + 24 __

3x3 + 12x2

5x2(x - 6) _

(x + 4)

2. x2 + 12x + 36 __

x2 - 36 ÷

1 _ x - 6

x + 6

Solve.

3. -3(r - 2) > -2(-6) r < -2

4. y _

4 +

1 _ 2

< 2 _ 3 y < 2_

3

Simplify.

*5.

5x _

10x + 20 _

15 _

x + 2 x_

30 6. 8 √ � 9 • 2 √ � 5 48 √ � 5

*7. Write Under what conditions is a rational expression undefined? Sample: when the denominator equals zero

*8. Justify Give an example to show a _ b _

c _

d =

a _ b ·

d _ c = ad _ bc

. See Additional Answers.

*9. Skating It took Jim 15 __

x2 + 2x - 3 minutes to skate to the park that was

2x _

8x - 8 + x

_ 4x + 12

miles away. Find his rate in miles per minute. x2 + x_

30 miles

per minute

(88)(88)

(88)(88)

(77)(77)

(77)(77)

(92)(92) (76)(76)

(92)(92)

(92)(92)

(92)(92)

SM_A1_NL_SBK_L092.indd Page 612 1/12/08 3:55:35 PM epg1 /Volumes/ju102/HCAC044/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L09

612 Saxon Algebra 1

Lesson Practice

Problem d

Scaff olding Before writing the quotient using a division symbol, remind students to combine the fractions fi rst.

Problem e

Error Alert Students will commonly calculate the rate incorrectly by dividing the time by the distance. Remind them that since speed has units of miles per minutes, it is “miles divided by minutes.”



“What is another way to write out

the fraction a _ x

_ b _ y ?” Sample: a __ x ÷ b __ y

“What can be done to a complex fraction to make the denominator equal to 1?” Sample: Multiply the numerator and denominators by the reciprocal of the denominator.

Practice3


Problem 6

Before multiplying the expressions together, remind students to check for perfect squares.


Lesson 92 613

*10. Multiple Choice Two fractions have a denominator of x2 + 6x + 9 and x2 - 9. What is the least common denominator? DA x2 + 9 B x2 + 6x + 9

C 2x2 + 9 D (x + 3)2(x - 3)

11. Solve and graph the inequality ⎢x� < 65.

*12. Error Analysis Two students solve the inequality ⎢x - 15⎥ < -4. Which student is correct? Explain the error.

Student A Student B

⎢x - 15⎢ < -4-4 < x - 15 < 4

11 < x < 19

⎢x - 15⎢ < -4no solution

*13. Geometry A triangle has sides measuring 16 inches and 23 inches. The triangle inequality states that the length of the third side must be greater than 7 inches and less than 39 inches. Write this inequality and graph it.

*14. Multi-Step The grades on a math test were all within the range of 80 points plus or minus 15 points.

a. Write an absolute -value inequality to show the range of the grades. ⎪x - 80⎥ ≤ 15

b. Solve the inequality to find the actual range of the grades. 65 ≤ x ≤ 95

*15. Error Analysis Two students simplified x + 2 _

2x - 5 - 3 - x

_ 2x - 5

. Which student is correct? Explain the error.

Student A Student B

x + 2 - 3 + x __

2x - 5 =

2x - 1 _ 2x - 5

x + 2 - 3 - x __

2x - 5 =

-1 _ 2x - 5

*16. Canoeing Vanya paddled a canoe upstream for 4 miles. He then turned the canoe around and paddled downstream for 3 miles. The current flowed at a rate of c miles per hour. Write a simplified expression to represent his total canoeing time if he kept a constant paddling rate of 6 miles per hour. c + 42__

(6 - c)(6 + c)

17. Multiple Choice Which equation’s graph has a maximum? BA y = -5 + x2 B y = -x2 + 5x

C y = x2 + 5 D y = 5x2 - 1

18. Write Describe 2 ways to find the axis of symmetry for a parabola.

19. Justify Give an example of a quadratic function and an example of a function that is not quadratic. Explain why each function is or is not quadratic. See Additional Answers.

(92)(92)

(91)(91)-65 < x < 65;

0 32.5 65-32.5-65-65 < x < 65;

0 32.5 65-32.5-65

(91)(91)

Student B; Sample: Student A did not realize that an absolute value can never be less than -4because absolute value is always positive.

Student B; Sample: Student A did not realize that an absolute value can never be less than -4because absolute value is always positive.

(91)(91)

7 < s < 39; 393123157

7 < s < 39; 393123157

(91)(91)

(90)(90)

15. Student A; Sample: Student B did not fully distribute the negative sign through the numerator of the second expression.

15. Student A; Sample: Student B did not fully distribute the negative sign through the numerator of the second expression.

(90)(90)

(89)(89)

(89)(89)

18. Sample: One way is to use the zeros of the function. The axis of symmetry goes through the zero when there is 1 zero because the zero is contained in the vertex. It goes through the average of the 2 zeros when there are 2 zeros. The second way is to use the formula x = - b_

2a. This is the

only way to find the axis of symmetry when the function has no zeros.

18. Sample: One way is to use the zeros of the function. The axis of symmetry goes through the zero when there is 1 zero because the zero is contained in the vertex. It goes through the average of the 2 zeros when there are 2 zeros. The second way is to use the formula x = - b_

2a. This is the

only way to find the axis of symmetry when the function has no zeros.

(84)(84)


Lesson 92 613

Problem 14

Guide students by asking them the following questions.

“What math operation could the phrase plus or minus indicate?” Sample: absolute value

“What operation could the word range indicate?” Sample: distance, difference

Extend the Problem

Translate the inequality into words. The distance between a number and 80 is less than or equal to 15.


Saxon Algebra 1614

20. Find the length a to the nearest tenth. 10.5

a17

20

21. Subtract 5f + 6

_ f 2 + 7f - 8

- f + 10

_ f 2 + 7f - 8

. 4_f + 8

22. Let A = (-5, 3), B = (0, 7), C = (12, 7), and D = (7, 3). Use the distance formula to determine whether ABCD is a parallelogram. yes

23. Geography Ithaca, New York is almost directly west of Oneonta, New York and directly north of Athens, Pennsylvania. The three cities form a triangle that is nearly a right triangle. Use the distance formula to estimate the distance from Athens to Oneonta. Each unit on the grid represents 5 miles. about 83 miles

x

y

5

5

Ithaca, NY

New YorkPennsylvania

Oneonta, NY

Athens, PA

24. Error Analysis Two students factor the polynomial (3x2 + 6) - (4x3 + 8x). Which student is correct? Explain the error.

Student A Student B

3(x2 + 2) - 4x(x2 - 2) 3(x2 + 2) - 4x(x2 + 2)(3 - 4x)(x2 - 4) (3 - 4x)(x2 + 2)

25. Probability A bag has 2x + 1 red marbles, 3x blue marbles, and x + 2 green marbles. What is the probability of picking a red marble? 1_3

26. Multi-Step A toolbox is 2 feet high. Its volume is represented by the expression 2x2 - 8x + 6.

a. Factor the expression completely. 2(x - 3)(x - 1)

b. Identify the expressions that represent the length and width of the toolbox. Sample: The length is (x - 1) and the width is (x - 3).

27. Hardiness Zones The state of Kansas falls into USDA hardiness zones 5b and 6a. This means that plants in these zones must be able to tolerate an average minimum temperature range greater than or equal to -15°F, and less than -5°F. Write an inequality to represent the temperature range of the hardiness zones in Kansas. -15 ≤ t < -5

(85)(85)

(90)(90)

(86)(86)

(86)(86)

(87)(87)

Student B; Sample: Student A did not distribute the negative sign correctly.

Student B; Sample: Student A did not distribute the negative sign correctly.

(38)(38)

(79)(79)

(82)(82)


Simplify:

x

2 - 9 ______ x __________

x2 + 5x + 6

___________ x2 x(x - 3)

_______ x + 2

CHALLENGE

614 Saxon Algebra 1

Problem 25

Extend the Problem

What is the probability of picking a red marble if there are 30 blue marbles in the bag? 1 __

3

Does the probability of picking a red marble depend on the value of x ? No, the probability is independent of x.


Lesson 92 615

28. Multi-Step A rectangular playing field has a perimeter of 8x feet. Its length is 2 feet greater than 2x.a. Write expressions for the length and width of the rectangle.

b. Write an expression for the area of the field. 4x2 - 4

c. The playing field is in a city park. The park is a large square with a side length of x2. Write and factor an expression for the area of the park that does not include the playing field. x4 - 4x2 + 4 = (x2 - 2)2

29. Vertical Motion The height h of an object t seconds after it begins to fall is given by the equation h = -16t2 + vt + s, where v is the initial velocity and s is the initial height. When an object falls, its initial velocity is zero. Write an equation for the height of an object t seconds after it begins to fall from 14,400 feet. Then factor the expression representing the height. h = -16t2 + 14,400; 16(30 + t)(30 - t)

30. Given the equation r = kst _ p , use the terms “jointly proportional to” and “inversely

proportional to” to describe the relation in the equation such that k is the constant of variation. In the equation, r is jointly proportional to s and t, and inversely proportional to p.

(83)(83)

length: 2x + 2, width: 2x - 2length: 2x + 2, width: 2x - 2

(83)(83)

(Inv 8)(Inv 8)


Solving complex fractions prepares students for

• Lesson 95 Combining Rational Expressions with Unlike Denominators

• Lesson 99 Solving Rational Equations

LOOKING FORWARD

Lesson 92 615

Problem 30

Error Alert Students can sometimes confuse the terms jointly and inversely. Remind them that if the value of a variable increases as the value of a related variable decreases, then the variables are inversely proportional.


Saxon Algebra 1616

Warm Up

93LESSON

1. Vocabulary A is a monomial or the sum or difference of monomials. polynomial

Divide.

2. 72x2 - 8x _

8x 9x - 1 3. 15x2 - 3x

_ 5x - 1

3x

Factor.

4. 2x2 + x - 3 (2x + 3)(x - 1) 5. 25x2 - 9 (5x + 3)(5x - 3)

A polynomial division problem can be written as a rational expression. To divide a polynomial by a monomial, divide each term in the numerator by the denominator.

Example 1 Dividing a Polynomial by a Monomial

Divide (8x3 + 12x2 + 4x) ÷ 4x.

SOLUTION

(8x3 + 12x2 + 4x) ÷ 4x

= (8x3 + 12x2 + 4x)

__ 4x

Write as a rational expression.

= 8x3

_ 4x

+ 12x2

_ 4x

+ 4x _ 4x

Divide each term by the denominator.

= 2 8x3 2

_ 14x

+ 312x2 1

_ 1 4x

+ 4x _ 4x


= 2x2 + 3x + 1 Simplify.

Division of a polynomial by a binomial is similar to division of whole numbers.

Words Numbers Polynomials

Step 1

Factor the numerator and denominator, if possible.

128 _ 4 =

32 · 4 _ 4

x2 + 4x + 3 __ x + 1

= (x + 3)(x + 1)

__ x + 1

Step 2 Divide out any common factors.

32 · 4 _

4

(x + 3)(x + 1) __

x + 1

Step 3 Simplify. 32 x + 3

(53)(53)

(38)(38) (43)(43)

(75)(75) (83)(83)


Dividing Polynomials

Online Connection



LESSON RESOURCESLESSON RESOURCES



Master C93, E93

The equation

(ax2 + bx + c) ÷ (x + d ) = x + e can be written as a rational equation in the form:

a x 2 + bx + c ___________

x + d = x + e

It can also be written in long-division form:

x + d

x + e

� �� a x 2 + bx + c .

Regardless of the form, the quotient times the divisor is always equal to the dividend. This is a good way for students to learn to check their work.

MATH BACKGROUND

Warm Up1

616 Saxon Algebra 1

93LESSON

Problem 5

Remind students that the difference of squares a2x2 - b2 = (ax + b)(ax - b).

2 New Concepts

In this lesson, students learn to divide polynomials.

Discuss the defi nition of a polynomial. Explain that division of polynomials is similar to division of whole numbers.

Example 1

Point out to students that a polynomial is the sum or difference of monomials, and that dividing a polynomial by a monomial can be done by dividing each term by the monomial.


Divide (6x4 + 4x3 - 2x2) ÷ 2x. 3x3 + 2x2 - x


Lesson 93 617

Example 2 Dividing a Polynomial by a Binomial

Divide each expression.

a. (x2 - 6x + 9) ÷ (x - 3)

SOLUTION

(x2 - 6x + 9) ÷ (x - 3)

= x2 - 6x + 9__

x - 3Write as a rational expression.

= (x - 3)(x - 3)__

x - 3Factor the numerator.

= (x - 3)(x - 3)__

(x - 3)Divide out common factors.

= x - 3 Simplify.

b. (x2 - 5x + 6) ÷ (2 - x)

SOLUTION

(x2 - 5x + 6) ÷ (2 - x)

= x2 - 5x + 6__

2 - xWrite as a rational expression.

= (x - 3)(x - 2)

__ 2 - x

Factor the numerator.

= (x - 3)(x - 2)__

-x + 2Write the denominator in descending order.

= (x - 3)(x - 2)

__ -1(x - 2)

Factor out a -1 in the denominator.

= (x - 3)(x - 2)

__ -1(x - 2)


= -x + 3 Simplify.

Just as with whole numbers, long division can also be used to divide polynomials.

27

21 � �� 567

x + 2

x + 1 � �� x2 + 3x + 2

__ -42 _____-(x2 + x)

147 2x + 2

___ -147 _____ -(2x + 2)

0 0

Hint

When the signs of the binomials, one in the numerator and one in the denominator, are opposites, factor out -1.


ENGLISH LEARNERS

For this lesson, explain the meanings of the prefi xes mono-, bi-, and poly-. Say:

“Mono- means one, alone, or single, bi- means two, and poly- means many or much.”

Connect the meaning of monomials and polynomials by the defi nition of polynomials. Emphasize that a polynomial is “an expression with one or multiple monomials.”

Discuss examples of words that use mono-, bi-, and poly- and relate the use of the prefi xes to their defi nitions, e.g. polygon, monopoly, bicycle, polymer, and monorail.

Lesson 93 617

Example 2

Remind students to factor the numerator and denominator, if possible.



a. (x2 + 5x - 14) ÷ (x + 7) x - 2

b. (x2 + 5x -14) ÷ (2 - x) -(x + 7)

Error Alert A common mistake when dividing by a polynomial is to divide the numerator by each term in the denominator. Write the statement:

x ___________ x 2 + 3x - 4

≠ x ___ x 2

+ x ___ 3x

- x __ 4

Remind students that only terms in the numerator can be divided by the denominator.


Saxon Algebra 1618

Example 3 Dividing a Polynomial Using Long Division

Divide using long division.

(-25x + 3x2 + 8) ÷ (x - 8)

SOLUTION

(-25x + 3x2 + 8) ÷ (x - 8)

x - 8 � �� (3x2 - 25x + 8) Write in long-division form with expressions in standard form.

3x

x - 8 � �� 3x2 - 25x + 8x Divide the first term of the dividend by the

first term of the divisor to find the first term of the quotient.

3x

x - 8 � �� 3x2 - 25x + 8

Multiply the first term of the quotient by the binomial divisor. Write the product under the dividend. Align like terms. _____ 3x2 - 24x

3x

x - 8 � �� 3x2 - 25x + 8

Subtract the product from the dividend. Then bring down the next term in the dividend.

______ -(3x2 - 24x)

-x + 8

3x - 1

x - 8 � �� 3x2 - 25x + 8

_____ -3x2 + 24x

-x + 8 Repeat the steps to find each term of the quotient. _____ -(-x + 8)

0 The remainder is 0.

The quotient is (3x - 1) remainder 0.

Check Multiply the quotient and the divisor.

(3x - 1)(x - 8)

= 3x2 - 24x - x + 8

= 3x2 - 25x + 8

The divisor is not always a factor of the dividend. When it is not, the remainder will not be 0. The remainder can be written as a rational expression using the divisor as the denominator.

Example 4 Long Division with a Remainder


(2x2 - 9 - 7x) ÷ (-4 + x)

Caution

Be sure to put the divisor and dividend in descending order before dividing.


618 Saxon Algebra 1

Example 3

Emphasize to students that long division can be used to divide polynomials.



(13x2 + 5x - 42) ÷ (x + 2) 13x - 21

Example 4

Polynomial long division may have remainders. Students can multiply the divisor by the quotient and then add the remainder to that product to check their work.



(4x2 + 7x + 17) ÷ (x - 3) 4x + 19 + 74 _____

x - 3

TEACHER TIPEmphasize that when using long division to divide polynomials, the remainder is a rational expression using the divisor as the denominator.


Lesson 93 619

SOLUTION

(2x2 - 9 - 7x) ÷ (-4 + x)

x - 4 � �� 2x2 - 7x - 9 Write in long-division form with expressions in standard form.

2x

x - 4 � �� 2x2 - 7x - 9

Divide the first term of the dividend by the first term of the divisor to find the first term of the quotient.

2x

x - 4 � �� 2x2 - 7x - 9

Multiply the first term of the quotient by the binomial divisor. Write the product under the dividend. Align like terms. ____ 2x2 - 8x

2x

x - 4 � �� 2x2 - 7x - 9

Subtract the product from the dividend. Then bring down the next term in the dividend.

______ -(2x2 - 8x)

x - 9

2x + 1

x - 4 � �� 2x2 - 7x - 9

Repeat the steps to find each term of the quotient.

______ -(2x2 - 8x)

x - 9

____ -(x - 4)

-5

The quotient is 2x + 1 - 5 _

x - 4 .

Example 5 Dividing a Polynomial with a Zero Coefficient

Divide (-2x + 5 + 3x3) ÷ (-3 + x).

SOLUTION

(-2x + 5 + 3x3) ÷ (-3 + x)

(3x3 - 2x + 5) ÷ (x - 3) Write each polynomial in standard form.

x - 3 � �� 3x3 + 0x2 - 2x + 5 Write in long division form. Use 0x2 as a placeholder for the x2-term.

3x2 + 9x + 25

x - 3 � �� 3x3 + 0x2 - 2x + 5

3x3 ÷ x = 3x2

______ -(3x3 - 9x2) Mulitply 3x2(x - 3). Then subtract.

9x2 - 2x Bring down -2x. 9x2 ÷ x = 9x

______ -(9x2 - 27x) Multiply 9x(x - 3). Then subtract.

25x + 5 Bring down 5. 25x ÷ x = 25

______ -(25x - 75) Multiply 25(x - 3). Then subtract.

80 The remainder is 80.

The quotient is 3x2 + 9x + 25 + 80 _

x - 3 .


INCLUSION

Using long division, have students divide the trinomial x2 - 3x - 18 by the binomial x - 6 on grid paper to help them keep their work properly aligned.

Once the students have the answer, have them multiply the divisor by their answer to check that the product of the two expressons is equal to the dividend.

Next, have the students write a rational expression using the same polynomials.

x2 - 3x - 18 ____________ x - 6

Have them factor and divide out like factors to fi nd the quotient.

Point out that either method, fi nding the quotient using long division or using a rational expression and factoring, should result in the same quotient.

Lesson 93 619

Example 5

Point out to students that when using long division to divide polynomials, it is very important to write the polynomial in standard form and to use 0 as the placeholder for any missing terms.


Divide (6x2 - 7 + 3x3) ÷ (4 + x). 3x2 - 6x + 24 - 103 ______

x + 4



Saxon Algebra 1620

Example 6 Application: Length of a Garden

Jim wants to find the length of the rectangular garden outside his office. The area is (x2 - 11x + 30) square feet. The width is (x - 6) feet. What is the length of the garden?

SOLUTION

l = A _ w Solve for l.

= x2 - 11x + 30 __

(x - 6) Evaluate for A and w.

= (x - 6)(x - 5)

__ (x - 6)


= (x - 6) (x - 5)

__ (x - 6)


= (x - 5) Simplify.

The length of the garden is (x - 5) feet.

Lesson Practice


a. (7x4 + 7x3 - 84x2) ÷ 7x2 x2 + x - 12

b. (x2 - 10x + 25) ÷ (x - 5) x - 5

c. (3x2 - 14x - 5) ÷ (5 - x) -3x - 1


d. (8x2 + x3 - 20x) ÷ (x - 2) x2 + 10x

e. (-3x2 + 6x3 + x - 33) ÷ (-2 + x) 6x2 + 9x + 19 + 5_x - 2

f. (6x + 5x3 - 8) ÷ (x - 4). 5x2 + 20x + 86 + 336_x - 4

g. Carlos wants to find the width of his rectangular deck. The area is (x2 - 10x + 24) square feet and the length is (x - 4) feet. What is the width? (x-6) feet

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

(Ex 4)(Ex 4)

(Ex 5)(Ex 5)

(Ex 6)(Ex 6)

1. Find the distance between (-3, 2) and (9, -3). Give the answer in simplest radical form. 13

2. Solve 5 _

16 y +

3 _ 8 ≥

1 _ 2

, and graph the solution. y ≥ 2_5

; 10-1

Factor.

3. 2x2 + 12x + 16 2(x + 4)(x + 2) 4. 3x3 - 5x2 - 9x + 15 (3x - 5)(x2 - 3)

(86)(86)

(77)(77)

(79)(79) (87)(87)

Hint

To find the length, solve the formula for the area of a rectangle, A = lw, for the length.


620 Saxon Algebra 1

Example 6

In this example, division of polynomials is used to solve a word problem.

Extend the Example

“What is the domain of x ? How do you know ?” x > 6; Since x - 6 is the width and x - 5 is the length of the garden, both have to be greater than zero for there to be a garden.

“If Jim added a fence around his garden and that added another foot to both the width and length, what is the new area?” (x - 5)(x - 4) = x2 - 9x + 20 square feet


Jillian wants to fi nd the radius of the circular fountain in the park. The area is 16πx2 + 24πx + 9π square feet. What is the radius of the fountain? 4x + 3 feet

Lesson Practice

Problem b

Scaff olding Have students write a rational expression, and then have them factor each term completely and divide out common factors.

Problem f

Error Alert Students may forget to include the divisor as the denominator of the remainder. Remind them that the remainder is a fractional part of the divisor.


Lesson 93 621

*5. Find the quotient: 4x3 + 42x2 - 2x __ 2x

. (2x2 + 21x - 1)

6. Find the axis of symmetry for the graph of the equation y = x2 - 2x. x = 1

Simplify.

7.

7x4

_ 4x + 18

_

3x2

_ 6x + 27

7x2_2

*8. 1 _ x3 _

1 _ x3 +

1 _ x3

1_2

*9. Write Explain how to check that (5x + 6) is the correct quotient of (15x2 + 13x - 6) ÷ (3x - 1). Sample: Multiply the divisor by the quotient. The product should equal the dividend.

*10. Justify Show that the quotient of (x2 - 4) ÷ (x + 2) can be found using two different methods.

*11. Swimming The city has decided to open a new public pool. The area of the new rectangular pool is (x2 - 16x + 63) square feet and the width is (x - 7) feet. What is the length? (x - 9) feet

*12. Multiple Choice Simplify x3 - 7x + 3x2 - 21 __

x + 3 . A

A x2 - 7 B x3 - 7

C -3 D x2 - 7

_ 2x

*13. Error Analysis Students were asked to simplify 6x2 - 6x _ 8x2 + 8x

_

3x - 3

_ 4x2 + 4x

. Which student is correct?

Explain the error.

Student A

6x2 - 6x _ 8x2 + 8x

· 4x2 + 4x _ 3x - 3

= 6x(x - 1)

_ 8x(x + 1)

· 4x(x + 1)

_ 3(x - 1)

= x

Student B

6x2 - 6 _

8x2 + 8x ·

4x2 + 4x _ 3x - 3

= 6x(x - 1)

_ 8x(x + 1)

· 4x(x + 1)

_ 3(x - 1)

= 24x2

_ 24x

14. Multi-Step Brent rode his scooter 8x2 - 48x _

24x5 minutes to get to baseball practice that was 7x - 42

_ 4x2 miles away.

a. Find his rate in miles per minute. 21x2_

4 miles per minute

b. If the rate is divided by 1 _ x , what is the new rate? 21x3_

4 miles per minute

*15. Geometry The area of a parallelogram is m + n _

5 square inches and the height is

m2 + n2

_ 15

inches. What is the length of the base? 3(m + n)

_m2 + n2 inches

16. Solve and graph the inequality ⎢x� > 84. x < -84 OR x > 84

(93)(93)

(89)(89)

(92)(92) (92)(92)

(93)(93)

(93)(93)Method 1:

x2 - 4_

x + 2 =

(x - 2)(x + 2) _

(x + 2)

= (x - 2)

Method 2: x - 2 x + 2� �� x2 + 0x - 4

____-x2 - 2x

-2x - 4

____ +2x + 4

0

10.10. Method 1:

x2 - 4_

x + 2 =

(x - 2)(x + 2) _

(x + 2)

= (x - 2)

Method 2: x - 2 x + 2� �� x2 + 0x - 4

____-x2 - 2x

-2x - 4

____ +2x + 4

0

10.10.

(93)(93)

(93)(93)

(92)(92) Student A; Sample: Student B did not write solution in simplest form.Student A; Sample: Student B did not write solution in simplest form.

(92)(92)

(92)(92)

(91)(91) 0 42 84-42-84 0 42 84-42-84


Lesson 93 621



“Explain some of the fi rst steps that should be taken before dividing a polynomial.” Sample: Check if the polynomial can be factored, and write the polynomial in standard form.

“When doing long division, why are placeholders necessary for the missing terms in the polynomial ?” Sample: Different places in the polynomial represent different powers of the variable.

Practice3


Problem 7

Before students simplify the complex fractions, encourage them to factor the polynomials.


Saxon Algebra 1622

*17. Census The census of England and Wales has a margin of error of ±104,000people. The 2001 census found the population to be 52,041,916. Write an absolute-value inequality to express the possible range of the population. Then solve the inequality to find the actual range for the population. ⎢x - 52,041,916� ≤ 104,000; 51,937,916 ≤ x ≤ 52,145,916

18. Error Analysis Two students solve the inequality ⎢x + 11� > -15. Which student is correct? Explain the error.

Student A

⎢x + 11� > -15x can be any real number.

Student B

⎢x + 11� > -15x = Ø

19. Multiple Choice When simplifying 1 _ x2 - 5x - 50

+ 1 _ 2x - 20

, what is the numerator? D

A 1 B 2

C x + 5 D x + 7

*20. Analyze Explain what can be done so that 1 _ 3 - r

and 5r _

r - 3 have like denominators.

21. Probability Mr. Brunetti writes quadratic equations on pieces of paper and puts them in a hat, and then tells his students each to choose two at random. After each student picks, his or her two papers go back in the hat. The functions are y = 4x2 - 3x + 7, y = -7 + x2, y = -2x + 6x2, y = 0.5x2 + 1.1, and y = - 1 _

2 x2 + 7x + 5. What is the probability of a student choosing two functions

that have a minimum? 3_5

22. Error Analysis Two students divide the following rational expression. Which student is correct? Explain the error.

Student A

m2

_ 6m2

÷ (m2 + 2)

m2

_ 6m2

· 1 _

m2 + 2 =

1 _ 6(m2 + 2)

Student B

m2

_ 6m2

÷ (m2 + 2)

m2

_ 6m2

· m2 + 2 _

1 =

m2 + 2 _ 6

23. Cost A bakery sells specialty rolls by the dozen. The first dozen costs 6b + b2 dollars. Each dozen thereafter costs 4b + b2 dollars. If Marcello buys 4 dozen rolls, how much does he pay? 2b(9 + 2b) dollars

24. Find the vertical and horizontal asymptotes and graph y = 4 _ x + 2

. x = -2; y = 0

25. Flooring Theo is installing new kitchen tiles. The design on the tile includes a square within a square. The smaller square has a side length of s centimeters. The expression 4s2 + 12s + 9 describes the area of the entire tile. What is the difference between the length of the tile and the length of the square within the tile? s + 3 centimeters

(91)(91)

(91)(91) Student A; Sample: Student B did not realize that all absolute values are greater than -15.Student A; Sample: Student B did not realize that all absolute values are greater than -15.

(90)(90)

(90)(90)20. Sample: Multiply either of the expressions by -1_

-1 because

-1(3 - r) =

-3 + r = r - 3, or -1(r - 3) =

-r + 3 = 3 - r.

20. Sample: Multiply either of the expressions by -1_

-1 because

-1(3 - r) =

-3 + r = r - 3, or -1(r - 3) =

-r + 3 = 3 - r.

(89)(89)

(88)(88) Student A; Sample: Student B did not multiply by the reciprocal of the rational expression.Student A; Sample: Student B did not multiply by the reciprocal of the rational expression.

(87)(87)

(78)(78)

24.

x

y

O4

4

-4

-8

8

824.

x

y

O4

4

-4

-8

8

8

(83)(83)


CHALLENGE

Polynomial division can include multiple variables. Divide:

2 x 2 + 3xy + y 2

_____________ x + y

Have students check their answers by multiplying the divisor and quotient. (2x + y)

622 Saxon Algebra 1

Problem 19


“How can the denominator of the fi rst term be factored ?” (x - 10)(x + 5)

“How can the denominator of the second term be factored ?” 2(x - 10)

Problem 23

Extend the Problem

“If Marcello had 6b2 + 28b dollars, how many dozen rolls could he buy?” 6 dozen

“If Marcello bought 6 dozen rolls, how much money would he have left?” 2b dollars

Problem 25

Error Alert A common mistake with word problems is to solve the wrong problem. Remind students that the question asks for the difference between the lengths and not the lengths of the two tiles.


Lesson 93 623

26. Multi-Step Tony is sketching the view from the top of a 256-foot-tall observation tower and accidentally drops his pencil. a. Use the formula h = -16t2 + 256 to make a table of values showing the height h

of the pencil, 1, 2, and 3 seconds after it is dropped. (1, 240), (2, 192), (3, 112)

b. Graph the function.

c. About how long does it take for the pencil to hit the ground? about 4 seconds

27. Pendulums The time it takes a pendulum to swing back and forth depends on its length. The formula l = 2.45 t

2

_ π

2 approximates this relationship. Graph the function using 3.14 for π. Use the graph to estimate the time it takes a pendulum that is 1 meter long to swing back and forth. about 2 sec

28. Write Explain how to determine whether a triangle with side lengths 5, 7, and 10 is a right triangle.

29. Multi-Step A bag of marbles contains 3 red, 5 blue, 2 purple, and 4 clear marbles.a. Make a graph that represents the frequency distribution.

b. What is the probability of drawing a red or a clear marble? 7_14

= 1_2

30. Given the following table, find the value of the constant of variation and complete the missing values in the table given that y varies directly with x and inversely with z. k = 3

y x z

1 1 3

3 2 2

6 4 2

9 6 2

2 8 12

(84)(84)

26 b.

x

y

180

240

120

60

1 2 3 4O

26 b.

x

y

180

240

120

60

1 2 3 4O

(84)(84)

27.

x

y

3

4

2

1

1 2 3 4O

27.

x

y

3

4

2

1

1 2 3 4O

(85)(85)28. Sample: Substitute 5 for a, 7 for b, and 10 for c in the equation a2 + b2 = c2

and simplify the equation. If the equation is true, then the triangle is a right triangle. If the equation is false, then the triangle is not a right triangle.

28. Sample: Substitute 5 for a, 7 for b, and 10 for c in the equation a2 + b2 = c2

and simplify the equation. If the equation is true, then the triangle is a right triangle. If the equation is false, then the triangle is not a right triangle.

(80)(80)

29.

543210 Red Blue Purple Clear

Marbles

Colors

29.

543210 Red Blue Purple Clear

Marbles

Colors

(Inv 8)(Inv 8)

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LOOKING FORWARD

Dividing polynomials prepares students for

• Lesson 95 Combining Rational Expressions with Unlike Denominators

• Lesson 99 Solving Rational Equations

Lesson 93 623

Problem 27

If students graph this function without a graphing calculator, tell them to use multiples of π for t to make the calculations easier. For example, let t = 3 · 3.14. Then (3 · 3.14)2 = 32 · (3.14)2 and the (3.14)2 will divide out.

Problem 30

A review of the defi nitions of direct and inverse variation may be helpful.


Saxon Algebra 1624

Warm Up

94LESSON

1. Vocabulary The of a number n is the distance from n to 0 on a number line. absolute value

Simplify.

2. ⎪-9⎥ - 5 4 3. ⎪12 - 23⎥ 11

Solve.

4. 6x - 7 = 11 x = 3 5. 11x + 8 = 41 x = 3

To solve an absolute-value equation, begin by isolating the absolute value. Then use the definition of absolute value to write the absolute-value equation as two equations. Solve each equation, and write the solution set. There are often two answers to an absolute-value equation. The solutions can be graphed on a number line by placing a closed circle at each value in the solution set.

Example 1 Solving Equations with Two Operations

Solve each equation. Then graph the solution.

a. ⎪x⎥

_ 5 + 3 = 18

SOLUTION

First isolate the absolute value. Write the equation so that the absolute value is on one side of the equation by itself.

⎪x⎥

_ 5 + 3 = 18

__ -3 = __ -3 Subtraction Property of Equality

⎪x⎥

_ 5 = 15 Simplify.

5 · ⎪x⎥

_ 5 = 5 · 15 Multiplication Property of Equality

⎪x⎥ = 75 Simplify.

x = 75 or x = -75 Write as two equations without an absolute value.

The solution set is {-75, 75}.

Graph the solution on a number line.

0 25 50 75-25-50-75

(5)(5)

(5)(5) (5)(5)

(26)(26) (26)(26)


Solving Multi-Step Absolute-Value Equations

Online Connection


Math Language

The absolute value of a number n is the distance from 0 to n on a number line. The absolute value of 0 is 0.

Hint

Isolate the absolute value using inverse operations.

SM_A1_NL_SBK_L094.indd Page 624 1/8/08 11:30:00 AM user /Volumes/ju102-1/HCAC044/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L09




Master C94

Solving absolute-value equations involves many of the same steps as solving multi-step equations in order to isolate the absolute value. Then, the defi nition of absolute value is applied to fi nd the solution set.

It is critical to understand that the absolute value of a number is its distance from zero. This distance can be to the left or right of 0. Even though the numbers are negative in one direction, the distance is still positive.

Beware of equations which simplify to an absolute value equal to a negative number. For these equations, there are no solutions because absolute value is never negative.

Emphasize to students that the absolute value expression must be isolated. The expression’s distance from zero is given on the opposite side of the equal sign. If the absolute value has other operators on the same side with it, then the distance of the quantity in the absolute value symbol from zero is not clear.

Warm Up1

624 Saxon Algebra 1

94LESSON

Problem 2

Remind students that the absolute value of a number is its distance from zero. Distance is always positive.

2 New Concepts

In this lesson, students learn to solve multi-step absolute-value equations and to graph the solutions.

Example 1

Students solve equations involving two operations.



a. ⎢x�

____ 2 - 1 = 14 {-30, 30};

3020100-30 -20 -10

b. -3⎢x� - 10 = -5 Ø; There is no graph.

TEACHER TIPExplain to students that the absolute-value symbol is a grouping symbol. Like parentheses, it must be isolated before it can be “undone” in an equation.


Lesson 94 625

b. 4⎪x⎥ - 9 = 15

SOLUTION

First isolate the absolute value.

4⎪x⎥ - 9 = 15

4⎪x⎥ = 24 Add 9 to both sides.

⎪x⎥ = 6 Divide both sides by 4.

x = 6 and x = -6 Write as two equations without an absolute value.


Graph the solution on a number line.

640 2-2-4-6

Example 2 Solving Equations with More than Two Operations

Solve each equation.

a. 5⎪x⎥

_ 2 + 4 = 4

SOLUTION

5⎪x⎥ _

2 + 4 = 4

5⎪x⎥

_ 2 = 0 Subtract 4 from both sides.

5⎪x⎥ = 0 Multiply both sides by 2.

⎪x⎥ = 0 Divide both sides by 5.

Since the absolute value is equal to zero, there is only one solution. The solution set is {0}.

b. 2⎪x⎥

_ 6 + 3 = 1

SOLUTION

2⎪x⎥ _

6 + 3 = 1

2⎪x⎥

_ 6 = -2 Subtract 3 from both sides.

2⎪x⎥ = -12 Multiply both sides by 6.

⎪x⎥ = -6 Divide both sides by 2.

By the definition of absolute value, we know that there are no solutions to this equation. The absolute value is never negative. The solution set is empty. You can write this as {} or Ø.

Math Reasoning

Analyze Why can the absolute value never be negative?

Sample: A distance can only be represented by 0 or a positive number.


INCLUSION

Use the following strategy with students who have diffi culty following multiple steps. The steps to solving these equations should be clearly identifi ed.

1. Isolate the absolute value.

2. Set up two equations to show the defi nition of absolute value.

3. Solve the two equations.

Having students label their work with the steps will lead them through the steps in the appropriate order. Have a bulletin board or poster modeling these steps with an example problem for students to see as they work.

Lesson 94 625

Example 2

Students learn to solve equations with more than two operations.



a. 5⎢x�

____ 2 + 4 = 19 { -6, 6}

b. 2⎢x�

____ 3 + 9 = 11 { -3, 3}


Saxon Algebra 1626

Example 3 Solving Equations with Operations Inside the

Absolute-Value Symbols


a. ⎪2x⎥ + 9 = 15

SOLUTION

⎪2x⎥ + 9 = 15

⎪2x⎥ = 6 Subtract 9 from both sides.

Write the equation as two equations without the absolute value. Then solve both equations.

2x = 6 or 2x = -6

x = 3 Divide both sides by 2. x = -3

The solution set is {3, -3}.

b. 6⎪x + 3⎥ - 8 = 10

SOLUTION

6⎪x + 3⎥ - 8 = 10

6⎪x + 3⎥ = 18 Add 8 to both sides.

⎪x + 3⎥ = 3 Divide both sides by 6.

Write the equation as two equations without the absolute value and solve.

x + 3 = 3 or x + 3 = -3

x = 0 Subtract 3 from both sides. x = -6


c. 5 ⎪ x

_ 3 - 2⎥ = 15

SOLUTION

5 ⎪ x

_ 3 - 2⎥ = 15

⎪ x

_ 3 - 2⎥ = 3 Divide both sides by 5.

Write the equation as two equations without the absolute value and solve.

x

_ 3 - 2 = 3 or

x _

3 - 2 = -3

x

_ 3 = 5 Add 2 to both sides.

x _

3 = -1

x = 15 Multiply both sides by 3. x = -3


Math Reasoning

Write Why do you often have to solve two equations without absolute value to find the solution to one equation with absolute value?

Sample: The defi nition of absolute value creates two possibilities for each absolute-value sign.


626 Saxon Algebra 1

Example 3

Students learn to solve equations with operations inside the absolute-value symbols.



a. ⎢5x� -10 = 25 {-7, 7}

b. 2 ⎪x - 4⎥ + 10 = 30 {-6, 14}

c. 3 ⎪ x __ 2 - 1⎥ = 18 {-10, 14}

Error Alert Students may try to isolate the variable without isolating the absolute value fi rst. Be sure to emphasize that once the absolute value is isolated, the defi nition of absolute value must be applied.


Lesson 94 627

Example 4 Application: Archery

A 60-centimeter indoor archery target has several rings around a circular center. If the average diameter of a ring is d, and an arrow landing in that ring scores p points, the inner and outer diameters of the ring are given by the absolute-value equation ⎪d + 6p - 63⎥ = 3. Find the inner and outer diameters of the 8-point ring.

SOLUTION

⎪d + 6p - 63⎥ = 3

⎪d + 6 · 8 - 63⎥ = 3 Substitute 8 for p.

⎪d + 48 - 63⎥ = 3 Multiply.

⎪d - 15⎥ = 3 Subtract.

Write the absolute-value equation as two equations and solve.

d - 15 = 3 or d - 15 = -3

d = 18 Add 15 to both sides. d = 12

The inner diameter of the ring is 12 centimeters and the outer diameter of the ring is 18 centimeters.

Lesson Practice


a. ⎪x⎥_7 + 10 = 18 {56, -56}; 0 28 56-28-56

b. 3⎪x⎥ - 11 = 10 {7, -7}; 0 3.5 7-3.5-7


c.4⎪x⎥_

9+ 23 = 11 Ø

d.⎪x⎥ + 3_

2- 2 = 1 {3, -3}

e. ⎪7x⎥ + 2 = 37 {5, -5}

f. 5 ⎪x + 1⎥ - 2 = 23 {4, -6}

g. 9 ⎪ x_2

- 1⎥ = 45 {12, -8}

h. Investments A factory produces items that cost $5 to make. The factory would like to invest $100 plus or minus $10 in the first batch. Use the equation ⎪5x - 100⎥ = 10 to find the least and greatest number of items the factory can produce. 18 items, 22 items

8 pt.

inner diameterouter diameter

8 pt.

inner diameterouter diameter

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

(Ex 3)(Ex 3)

(Ex 3)(Ex 3)

(Ex 4)(Ex 4)

Caution

The absolute-value bars act as grouping symbols. Be sure to use the order of operations to simplify any expression within them.


For Example 4, explain the meaning of the word ring. Tape a picture on the chalkboard and draw a circle around it. Say:

“This picture has a ring drawn around it. A ring is a circular line that surrounds an object. Circus acts are performed inside a large ring.”

Ask volunteers to give examples of other types of rings. Sample: rings around a planet; a wedding ring

ENGLISH LEARNERS

Lesson 94 627

Example 4

Students use absolute-value equations to solve an archery problem.

Extend the Example

Find the inner and outer diameters of the 6-point ring. 24 cm and 30 cm


A family wants to spend $200 plus or minus $50 on a campsite for their entire trip. The campsite costs $25 per night. Use the equation ⎪200 - 25x⎥ = 50 to fi nd the greatest and least number of nights the family can camp. 6 and 10 nights

Lesson Practice

Problem c

Error Alert Students need to stop when they isolate the absolute value and fi nd that it equals a negative number. There is no need to go any further. There is no solution.

Problem f

Scaff olding Have students isolate the absolute value. Then apply the defi nition of absolute value.



“What are the steps for solving a multi-step absolute-value equation?” Sample: Isolate the absolute value. Apply the defi nition of absolute value. Set up two equations, and then solve them to fi nd two solutions.

“What happens if the absolute value is equal to a negative number?” Sample: There is no solution.



Saxon Algebra 1628

Add or subtract.

1. m2 _

m - 4 - 16 _

m - 4 m + 4 2.

-66 __ w2 - w - 30

+ w _ w - 6

w + 11_w + 5

*3. Write Explain why some absolute-value equations have no solutions.

*4. Multiple Choice The solution set {-12, 60} correctly solves which absolute-value equation? D

A 6 ⎪ x

_ 4 - 1⎥ = 42 B -2 ⎪

x _

4 - 1⎥ = 16

C 8 ⎪ x

_ 3 - 2⎥ = 48 D -5 ⎪

x _

6 - 4⎥ = -30

5. Refurbishing Rudy has x + y junk cars in his lot. He fixes them up and sells

each car for $400 + $100x

_ y . If he sells 30% of them, how much profit will he make?

6. Physics The function y = -16x2 + 80x models the height of a droplet of water from an in-ground sprinkler x seconds after it shoots straight up from ground level. Explain how you know when the droplet will hit the ground.

Factor.

7. 2a2 + 8ab + 6a + 24b 2(a + 3)(a + 4b)

8. zx10 - 4zx9 - 21zx8 zx8(x - 7)(x + 3)

9. Find the product of b - 4

_ b + 9 · (b2 + 11b + 18). (b - 4)(b + 2)

*10. Error Analysis Students were asked to simplify (15x4 + 4x2 + 3x3) ÷ (x - 6). Which student set their problem up correctly? Explain the error. Student B; Sample: Student A did not put the dividend in descending order or insert a placeholder.

Student A

(x - 6) � �� (15x4 + 4x2 + 3x3)

Student B

(x - 6) � �� (15x4 + 3x3 + 4x2 + 0x + 0)

Simplify.

*11. -1 _ 10x - 10

_

x5

_ 10x2 - 10

1 - x_

x5

12.

2x _

3x + 12 __

6x2

__ x2 + 8x + 16

x + 4_9x

(90)(90) (90)(90)

(94)(94)

3. Sample: An absolute value cannot be negative, so any absolute-valueequation that sets an absolute value equal to a negative number has no solution.

3. Sample: An absolute value cannot be negative, so any absolute-valueequation that sets an absolute value equal to a negative number has no solution.

(94)(94)

(88)(88)

5.30(x2 + 4x + 4y + xy)

___y dollars5.

30(x2 + 4x + 4y + xy) ___

y dollars

(89)(89)6. Sample: The droplet is at ground level when y = 0. This first occurs at 0 seconds, before the water has shot out from the sprinkler. The maximum value occurs at x = - b_

2a= - 80_

-32= 2.5.

Because of symmetry, the droplet is at ground level 2.5 seconds before and after its maximum point, so it hits the ground 5 seconds after it shoots up.

6. Sample: The droplet is at ground level when y = 0. This first occurs at 0 seconds, before the water has shot out from the sprinkler. The maximum value occurs at x = - b_

2a= - 80_

-32= 2.5.

Because of symmetry, the droplet is at ground level 2.5 seconds before and after its maximum point, so it hits the ground 5 seconds after it shoots up.

(87)(87)

(79)(79)

(88)(88)

(93)(93)

(92)(92)

(92)(92)


628 Saxon Algebra 1

Practice3


Problem 1

Combine the numerators, factor, and then cancel common factors in the numerator and denominator.

Problem 6

Remind students that when the droplet hits the ground, its height is 0.

Problem 7

Since the expression has 4 unlike terms, the students should factor by grouping. Factor pairs of terms rather than all four terms.


Lesson 94 629

*13. Canoeing A canoe rental company charges $10 for the canoe and an additional charge per person. There are 4 people going on the trip and they have planned on spending a total of $50. They hope that the total cost is within $20 of the planned spending total. What is the minimum and maximum they can be charged per person? $5, $15

Find the quotient.

14. (1 + 4x4 - 10x2) ÷ (x + 2) 4x3 - 8x2 + 6x - 12 + 25_x + 2

15. 25x3 + 20x2 - 5x

__ 5x (5x - 1)(x + 1)

*16. Solve the equation ⎪x⎥

_ 11

+ 9 = 15 and graph the solution.

*17. Justify Show that the solution set to the equation ⎪x⎥

_ -3 + 1 = 5 is Ø.

*18. Multi-Step Marty measured the area of his rectangular classroom. He determined that the area is (-2x2 + x3 - 98 - 49x) square feet. The length is (x + 6) feet. a. What is the width? (x2 - 8x - 1 - 92_

x + 6 ) feet

b. If the area is (x2 - 36) square feet, what is the width? (x - 6) feet

19. Geometry The area of a triangle is (10y2 + 6y) square centimeters. The base is (5y + 3) centimeters. What is the height? 4y centimeters

*20. Error Analysis Students were asked to simplify 4x

_ 8x + 16

_ 12

_ x + 2

. Which student is correct? Explain the error. Student B; Sample: Student A did not multiply by the reciprocal.

Student A

4x _ 8x + 16

· 12 _

x + 2

= 4x _

8(x + 2) ·

12 _ x + 2

= 6x __

(x + 2)(x + 2)

Student B

4x _ 8x + 16

· x + 2 _

12

= 4x _

8(x + 2) ·

x + 2 _ 12

= x _ 24

*21. Commuting It took Taylor 1 __ x2 + 3x - 40

minutes to get to work, which was

x2

_ 6x + 48

miles away. Find his rate in miles per minute. x3 - 5x2_

6x miles per minute

22. Verify Show that 0 is a solution to the inequality ⎪x - 14⎥ < 30. Sample:⎪0 - 14⎥ = ⎪-14⎥ = 14 and 14 < 30

23. Multiple Choice Which inequality is represented by the graph? C

20100-10-20

A ⎪x⎥ ≤ - 21 B ⎪x⎥ < 21

C ⎪x⎥ ≤ 21 D ⎪x⎥ > 21

24. Analyze When a line segment is horizontal, which expression under the radical in the distance formula is 0: (x2 - x1)

2 or (y2 - y1)2? (y2 - y1)2

(94)(94)

(93)(93)

(93)(93)

(94)(94){-66, 66};

660 33-33-66{-66 366}

660 33-33-66

(94)(94) 17.3 3 33 3 3 3 3g ⎪x⎥

-3=3 3T 33 3 3

-33 3g ⎪x⎥ =

- 3 33v 33 3 g v 3

3 3 3

17.3 3 33 3 3 3 3g ⎪x⎥

-3=3 3T 33 3 3

-33 3g ⎪x⎥ =

- 3 33v 33 3 g v 3

3 3 3

(93)(93)

(93)(93)

(92)(92)

(92)(92)

(91)(91)

(91)(91)

(86)(86)


Lesson 94 629

Problem 15

Extend the Problem

Multiply the dividend by the divisor to verify that the solution is equal to the quotient.

Problem 17

Isolate the absolute value. Note that it is equal to a negative number.

Problem 19

The formula for the area of a triangle is A = 1 __

2 bh.

Problem 22

Substitute 0 into the equation and see if it makes the inequality true.


Saxon Algebra 1630

25. Multi-Step Ornella hiked 8 miles on easy trails and 3 miles on difficult trails. Her hiking rate on the easy trails was 2.5 times faster than her rate on the difficult trails. a. Write a simplified rational expression for Ornella’s total hiking time.

b. Find Ornella’s hiking time if her hiking rate on the difficult trails was 2 miles per hour. 3.1 hours

26. Find the vertical and horizontal asymptotes and graph y = 1 _ x - 2

- 4.

27. Multi-Step Phone Plan A costs $12 per month for local calls and $0.06 per minute for long-distance calls. Phone Plan B costs $15 per month for local calls plus $0.04 per minute for long-distance calls. How many minutes of long-distance calls would Jenna have to make for Plan B to cost less than Plan A? a. Formulate Write an inequality to answer. 12 + 0.06m > 15 + 0.04m

b. Solve the inequality and answer the question. m > 150; 150 minutes

c. Graph the solution. 20015050 100

28. United States Flag An official American flag should have a length that is 1.9 times its width. The area of an official American flag can be found by the function y = 1.9x2. Graph the function. Then use the graph to approximate the width of a flag that has an area of 47.5 square feet.

29. Multi-Step A rectangular garden that is 25 feet wide has a diagonal length that is 50 feet long. a. Find the length of the garden in simplest radical form. 25√ � 3 ft

b. Find the perimeter of the garden to the nearest tenth of a foot. 136.6 ft

30. The volume of a sphere is V = 4 _ 3 πr3. Describe in words the relationship between

the volume of a sphere and its radius. Identify the constant of variation. Thevolume of a sphere is directly proportional to the cube of its radius. The constant of variation is equal to 4_

3π.

(90)(90)

8_2.5r

+ 3_r = 15.5_

2.5r8_

2.5r+ 3_

r = 15.5_2.5r

(78)(78)

xy

4-2

-2

-6

26. x = 2; y = -4

xy

4-2

-2

-6

26. x = 2; y = -4

(81)(81)

(84)(84)

28.

x

y

30

40

20

10

1 2 3 4O

; about 5 feet28.

x

y

30

40

20

10

1 2 3 4O

; about 5 feet

(85)(85)

(Inv 8)(Inv 8)


CHALLENGE LOOKING FORWARD

Write a multi-step absolute-value equation that has no solution. Sample: ⎪ x __

4 - 2⎥ + 15 = 12

Solving multi-step absolute-value equations prepares students for

• Lesson 101 Solving Multi-Step Absolute-Value Inequalities


630 Saxon Algebra 1

Problem 30

Error Alert When fi nding the constant of variation, students may think that π is a variable. Remind students that π is a constant.


Lesson 95 631

Combining Rational Expressions with

Unlike Denominators

Warm Up

95LESSON

1. Vocabulary One of two or more numbers or expressions that are multiplied to get a product is called a (n) . factor

Find the LCM.

2. 8x4y and 12x3y2 24x4y2

3. (9x - 27) and (4x - 12) 36(x - 3)

Factor.

4. x2 + 4x - 21 (x + 7)(x - 3)

5. 10x2 + 13x - 3 (5x - 1)(2x + 3)

The steps for adding rational expressions are the same as for adding numerical fractions. Fractions with unlike denominators cannot be added unless you first find their least common denominator.

Example 1 Finding a Common Denominator

Find the least common denominator (LCD) for each expression.

a. 3 _

(x + 3) -

9 __ (x2 - 2x - 15)

SOLUTION

3 _ (x + 3)

- 9 __

(x2 - 2x - 15)

= 3 _

(x + 3) -

9 __ (x + 3)(x - 5)

Factor each denominator, if possible.

To find the LCD of (x + 3) and (x + 3)(x - 5), use every factor of each denominator the greatest number of times it is a factor of either denominator. Each denominator has a factor of (x + 3). One denominator also has a factor of (x - 5). The product of these two factors is the LCD.

LCD = (x + 3)(x - 5)

b. 2x _

4x2 - 196 +

12x __ x2 + x - 56

SOLUTION

2x _ 4x2 - 196

+ 12x __

x2 + x - 56

= 2x __

4(x - 7)(x + 7) +

12x __ (x - 7)(x + 8)

Factor each denominator completely.

LCD = 4(x - 7)(x + 7)(x + 8)

(2)(2)

(57)(57)

(57)(57)

(72)(72)

(75)(75)


Online Connection


Math Language

A least common

denominator (LCD) is the least common multiple (LCM) of the denominators.


Working with rational expressions can test the students’ understanding of fractions. If the algorithm for adding and subtracting fractions is simply memorized, then adding and subtracting rational expressions can be challenging. Rational expressions can be treated as fractions. In order to add or subtract rational expressions, a common denominator must be found. Any common denominator will work, but the LCD is the most effi cient. Keeping the terms simpler will help eliminate error.

Once a common denominator is found, each term must be multiplied by a factor equal to 1 so that equivalent fractions with a common denominator are found. Combine the fractions while keeping the denominator in factored form. After simplifying the numerator, it is important to factor the numerator to divide out factors that form quotients equal to one.

LESSON RESOURCES



Master C95

MATH BACKGROUND

Warm Up1

95LESSON

Lesson 95 631

Problem 3

Remind students to factor fi rst and then to fi nd the LCM.

2 New Concepts

In this lesson, students combine rational expressions with unlike denominators.

Example 1

Students fi nd a common denominator for rational expressions.


Find the least common denominator (LCD) for each expression.

a. 5 ______ x - 2

+ 6 __________ x2 - x - 2

(x - 2)(x + 1)

b. 3x ____________ x 2 + 8x + 15

+ 4x _________ 5 x 2 - 500

5(x + 3)(x + 5)(x + 10)(x - 10)


Saxon Algebra 1632

Example 2 Using Equivalent Fractions to Add with Unlike

Denominators

Add 6x2

_ x2 - 16

+ x - 1 _ 2x - 8

.

SOLUTION Factor each denominator.

6x2

_ x2 - 16

+ x - 1 _ 2x - 8

= 6x2

__ (x - 4)(x + 4)

+ x - 1 _

2(x - 4)

LCD = 2(x - 4)(x + 4)

Write an equivalent fraction for each addend with the LCD as a denominator.

6x2

__ (x - 4)(x + 4)

· 2 _ 2 =

2(6x2) __

2(x - 4)(x + 4)

Multiply the numerator and denominator of the first fraction by 2.

x - 1 _ 2(x - 4)

· x + 4 _ x + 4

= (x - 1)(x + 4)

__ 2(x - 4)(x + 4)

Multiply the numerator and denominator of the second fraction by x + 4_

x + 4.

2(6x2) __

2(x - 4)(x + 4) +

(x - 1)(x + 4) __

2(x - 4)(x + 4)

Write the sum of the equivalent fractions.

= 2(6x2) + (x - 1)(x + 4)

___ 2(x - 4)(x + 4)

Add.

= 12x2 + x2 + 3x - 4 __

2(x - 4)(x + 4) Expand the numerator.

= 13x2 + 3x - 4 __

2(x - 4)(x + 4) Combine like terms in the

numerator.

Example 3 Using Equivalent Fractions to Subtract with

Unlike Denominators

Subtract 4x2

_ 9x - 27

- 2x - 5 _ x2 - 9

.

SOLUTION Factor each denominator.

4x2

_ 9x - 27

- 2x - 5 _ x2 - 9

= 4x2

_ 9(x - 3)

- 2x - 5 __

(x - 3)(x + 3)

LCD = 9(x - 3)(x + 3)

4x2(x + 3)

__ 9(x - 3)(x + 3)

- 9(2x - 5)

__ 9(x - 3)(x + 3)

Write the difference of the equivalent fractions.

= 4x2(x + 3) - 9(2x - 5)

___ 9(x - 3)(x + 3)

Subtract.

= 4x3 + 12x2 - 18x + 45 __

9(x - 3)(x + 3) Expand the numerator.

There are no like terms, so the difference is 4x3 + 12x2 - 18x + 45

__ 9(x - 3)(x + 3) .

Math Reasoning

Analyze What does it mean to write an equivalent fraction?

Sample: It means to multiply the numerator and denominatorof a fraction by the same factor resulting in a fraction that is equal to the original fraction.


To help students combine rational expressions, set up boxes. After students factor the denominators and determine the LCD for Example 3, have them draw unit factor boxes next to each term.

(4 x2)

________ 9(x - 3)

- (2x - 5)

_____________ (x - 3)(x + 3)

Then have the students compare the denominator of the terms to the LCD.

Say, “Since the fi rst term is missing a factor of (x + 3), multiply by the unit

factor (x + 3)

______

(x + 3)

.” Have students determine

what to multiply the second term by. After the expression has been setup with common denominators it will be ready to simplify.

(x + 3)

________ (x + 3)

(4x2)

___________ 3 · 3(x - 3)

- 9 __ 9 (2x - 5)

_____________ (x - 3)(x + 3)

INCLUSION

632 Saxon Algebra 1

Example 2

Students learn to use equivalent fractions to add rational expressions with unlike denominators.


Add 3 x 2 _____ x 2 - 9

+ x + 2 ______

4x - 12 .

13 x 2 + 5x + 6 _____________ 4(x - 3)(x + 3)

Example 3

Students learn to use equivalent fractions to subtract rational expressions with unlike denominators.


Subtract 2x _______ 4x + 16

- 3x - 4 _______ x 2 - 16

.

x2 - 10x + 8 _____________ 2(x - 4)(x + 4)

Error Alert Students may forget to distribute the negative sign to each term in the numerator of the rational expression being subtracted. Putting parentheses around the terms in the numerator will help students to remember that they must subtract each term.


Lesson 95 633

Example 4 Adding and Subtracting with Unlike Denominators

a. Add -35 _

56 - 7x +

-6x - 2 __ x2 - 6x - 16

.

SOLUTION

-35 _ 56 - 7x

+ -6x - 2 __

x2 - 6x - 16

= -35 _

-7(x - 8) +

-6x - 2 __ (x - 8)(x + 2)

Factor each denominator.

LCD = -7(x - 8)(x + 2) Find the LCD.

-35(x + 2) __

-7(x - 8)(x + 2) +

-7(-6x - 2) __

-7(x - 8)(x + 2) Write each fraction as an

equivalent fraction.

= -35(x + 2) + (-7)(-6x - 2)

___ -7(x - 8)(x + 2)

Add.

= 7x - 56 __

-7(x - 8)(x + 2) Expand the numerator and

collect like terms.

= 7(x - 8)

__ -7(x - 8)(x + 2)


= - 1 _

x + 2 Divide out common factors.

b. Subtract x - 5 _ 2x - 6

- x - 7 _

4x - 12 .

SOLUTION

x - 5 _ 2x - 6

- x - 7 _

4x - 12

= x - 5 _

2(x - 3) -

x - 7 _ 4(x - 3)

Factor each denominator.

LCD = 4(x - 3) Find the LCD.

2(x - 5) _

4(x - 3) -

(x - 7) _

4(x - 3) Write each fraction as an equivalent fraction.

= 2(x - 5) - 1(x - 7)

__ 4(x - 3)

Subtract.

= 2x - 10 - x + 7 __

4(x - 3) Expand the numerator.

= x - 3_

4(x - 3)Collect like terms.

= 1 _ 4 Divide out common factors.

Caution

Before factoring, write expressions in descending order.

Hint

Always check to see if the numerator of the sum can be factored and if a common factor can be divided out.


Lesson 95 633

Example 4

Students add and subtract rational expressions with unlike denominators. Then the sum or difference is simplifi ed by dividing out like factors.


a. Add -24 _______ 32 - 8x

+ 4x + 1 ___________

x2 - 7x + 12 .

7x - 8 ______________ (x - 4)(x - 3)

b. Subtract x + 4 _______ 3x + 9

- x + 9 ________

5x + 15 .

2x - 7 __________ 15(x + 3)



Saxon Algebra 1634

Example 5 Application: Traveling

A pilot’s single-engine aircraft flies at 230 kph if there is no wind. The pilot plans a round trip to a city that is 400 kilometers away. If there is a tailwind

of w kilometers per hour, the time for the outbound flight is 400

_ 230 + w hours. The time for the return flight with a headwind of w kilometers per hour is 400

_ 230 - w hours. What is the total time for the round trip?

SOLUTION

400 _ 230 + w

+ 400 _

230 - w Add to find the total time.

LCD = (230 + w)(230 - w) Find the LCD.

400(230 - w)

__ (230 + w)(230 - w)

+ 400(230 + w)

__ (230 + w)(230 - w)

Write equivalent fractions.

= 184,000

__ (230 + w)(230 - w)

hours Expand the numerator and collect like terms.

The round trip takes 184,000

__ (230 + w)(230 - w)

hours.

Lesson Practice

Find the LCD for each expression.

a. 5x _

5x - 45 -

44 _ x2 - 81

LCD = 5(x - 9)(x + 9)

b. 3x _

x + 4 -

12 __ x2 + 2x - 8

LCD = (x + 4)(x - 2)

c. Add 3x2

_ x2 - 25

+ x - 1 _

4x - 20 .

13x2 + 4x - 5__4(x - 5)(x + 5)

d. Subtract 2x2

_ 6x - 24

- 3x - 4 _ x2 - 16

. x3 + 4x2 - 9x + 12__

3(x - 4)(x + 4)

e. Add x - 1 _ x2 - 1

+ 2 _

5x + 5 . 7_

5(x + 1)

f. Subtract 2 _ x2 - 36

- 1 _

x2 + 6x . 1_

x(x - 6)

g. Trenton hiked 4x

_ x2 - 64 miles on Saturday and 12

_ 7x - 56 miles on Sunday. How many miles did he hike altogether? 40x + 96__

7(x - 8)(x + 8) miles

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

(Ex 4)(Ex 4)

(Ex 4)(Ex 4)

(Ex 5)(Ex 5)

Factor completely.

1. 3x3 - 9x2 - 30x 3x(x + 2)(x - 5) 2. 8x3y2 + 4x2y - 12xy3

3. 32x3 - 24x4 + 4x5 4x3(x - 2)(x - 4) 4. mn3 - 10mn2 + 24mn

(Inv 9)(Inv 9) (Inv 9)(Inv 9)4xy(2x2y + x - 3y2)4xy(2x2y + x - 3y2)

(79)(79) (79)(79)mn(n - 6)(n - 4)mn(n - 6)(n - 4)


634 Saxon Algebra 1

Example 5

Students add rational expressions to solve a rate problem.

Extend the Example

About how many hours does the trip take if the tailwind is 10 kilometers per hour? about 3.485 hours


Sarah biked 7x ______ x 2 - 49 kilometers

on Saturday. She biked 15 ______ 5x + 35

kilometers on Sunday. How many kilometers did she bike altogether? 10x - 21 ____________

(x + 7)(x - 7) kilometers

Lesson Practice

Problem a

Scaff olding Factor both denominators completely. Then have students fi nd the LCM of the denominator.

Problem f

Error Alert Students may not recognize x as a factor in the LCD. Remind them that every factor must be in the LCD.



“How do you fi nd the LCD?” Sample: Factor each denominator. Find the LCM of the denominators. To fi nd the LCM of the denominators, use every factor of each denominator the greatest number of times it is a factor of either denominator.

“When adding rational expressions with unlike denominators, what do you do after you fi nd the LCD?” Sample: Find equivalent fractions with their denominators equal to the LCD.


Lesson 95 635

Find the quotient.

5. 4m

_ 17r

÷ 12m2

_ 5r

5_51m 6. (x2 - 16x + 64) ÷ (x - 8) (x - 8)

7. Find the zeros of the function shown. no zeros

x

y

4

6

2 4-4 -2

*8. Find the LCD of 4 _ x + 4

- 8 __

x2 + 6x + 8 . LCD = (x + 4)(x + 2)

*9. Add 2x _

2x2 - 128 +

5 __ x2 - 7x - 8

. x2 + 6x + 40__

(x - 8)(x + 8)(x + 1)

*10. Solve the equation 9⎢x� - 22 = 14 and graph the solution. {-4, 4};

11. Error Analysis Students were asked to subtract 6x2

_ x2 + 6x - 16

- 3 _

8x - 16 . Which student

is correct? Explain the error. Student A; Sample: Student B didn’t distribute the negative all the way through the second numerator.

Student A

6x2

__ x2 + 6x - 16

- 3 _

8x - 16

6x2

__ (x + 8)(x - 2)

- 3 _

8(x - 2)

6x2(8) __

8(x + 8)(x - 2) -

3(x + 8) __

8(x - 2)(x + 8)

48x2 - 3x - 24

__ 8(x + 8)(x - 2)

Student B

6x2

__ x2 + 6x - 16

- 3 _

8x - 16

6x2

__ (x + 8)(x - 2)

- 3 _

8(x - 2)

6x2(8) __

8(x + 8)(x - 2) -

3(x + 8) __

8(x - 2)(x + 8)

48x2 - 3x + 24

__ 8(x + 8)(x - 2)

*12. Generalize Explain how to find the LCD of two algebraic rational expressions.

*13. Running Michele is training for a marathon. She ran 3x2

_ x2 - 100 miles on Monday and

x - 1

_ 2x - 20 miles on Tuesday. How many miles did she run in all?

*14. Multi-Step The girls’ track team sprinted 2x

_ 4x2 - 196 meters Thursday and 12x

_ x2 + x - 56 meters Friday.

a. What was the total distance that the track team sprinted?

b. If their rate was 2x

_ x + 8 meters per minute, how much time did it take them to sprint on Thursday and Friday? 25x + 176__

4(x - 7)(x + 7) minutes

(88)(88) (93)(93)

(89)(89)

(95)(95)

(95)(95)

(94)(94) 40-4 40-4

(95)(95)

(95)(95)

12. Sample: Factor each denominator. The LCD must contain each factor of each denominator and use each factor the greatest number of times it occurs in either denominator.

12. Sample: Factor each denominator. The LCD must contain each factor of each denominator and use each factor the greatest number of times it occurs in either denominator.

(95)(95)

7x2 + 9x - 10__

2(x + 10)(x - 10) miles

7x2 + 9x - 10__2(x + 10)(x - 10)

miles

(95)(95)

14a. 25x2 + 176x___2(x - 7)(x + 7)(x + 8)

meters14a. 25x2 + 176x___2(x - 7)(x + 7)(x + 8)

meters

SM_A1_NLB_SBK_L095.indd Page 635 4/14/08 7:15:43 PM elhi1 /Volumes/ju108/HCAC057/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L09

Lesson 95 635

Practice3


Problem 5

To divide by a fraction is to multiply by its reciprocal. Before multiplying, divide out factors common to the numerator and denominator, if any.

Problem 9


“What is the LCD of the fractions?” 2(x - 8)(x + 8)(x +1) or (x - 8)(x + 8)(x + 1)

“How do you change the fractions so that they share the LCD?” Sample: Multiply the numerator and denominator of each fraction by its missing factors of the LCD.

“Once the fractions share the LCD, what is the next step?” Sample: Add the fractions.


Saxon Algebra 1636

*15. Error Analysis Two students solve the equation 6⎪x + 3⎢ - 8 = -2. Which student is correct? Explain the error.

Student A

6⎪x + 3⎢ - 8 = -2no solutionAbsolute values cannot equal a negative number.

Student B

6⎢x + 3� - 8 = -26⎢x + 3� = 6⎢x + 3� = 1x + 3 = 1 and x + 3 = -1 x = -2 and x = -4

16. Geometry The perimeter of a square must be 34 inches plus or minus 2 inches. What is the longest and the shortest length each side can be? 8 inches, 9 inches

*17. Multi-Step A student budgets $35 for lunch and rides each week. He gives his friend $5 for gas and then pays for 5 lunches a week. He has a $2 cushion in his budget, meaning that he can spend $2 more or less than he budgeted. a. Write an absolute-value equation for the minimum and maximum he can spend

on each lunch. ⎢5 + 5x - 35� = 2

b. What is the maximum and the minimum he can spend on each lunch? $5.60,$6.40

18. Error Analysis Students were asked to simplify x2 + 10x + 24

_ x . Which student is correct?

Explain the error. Student B; Sample: Student A canceled terms that were not common factors.

Student A

(x + 6)(x + 4)

__ x = (x + 6)(4)

Student B

(x + 6)(x + 4)

__ x

*19. Carpeting The rectangular public library is getting new carpet. The area of the room is (x3 - 18x2 + 81x) square feet. The length of the room is (x - 9) feet. What is the width? ( x2 - 9x) feet

*20. Justify Give an example to show a _ b · c _

d = ac

_ bd

.

21. Multiple Choice Simplify: x2 - 9

_ x2 - 5x + 6

_ x

2 + 5x + 6

_ x2 - 4 . A

A 1 B (x - 3)2

_ (x - 2)2 C -1 D

(x - 3) _

(x - 2)

22. Solve and graph the inequality ⎢x� > 17. x < -17 or x > 17;

23. Measurement When measuring something in centimeters, the accuracy is within 0.1 centimeter. A board is measured to be 15.6 centimeters. The accuracy of the measurement can be represented by the absolute-value inequality ⎢x - 15.6� ≤ 0.1. Solve the inequality. 15.5 ≤ x ≤ 15.7

(94)(94)15. Student B; Sample:Student A did not isolate the absolute value and assumed that because the equation was equal to a negative number, the absolute value would be equal to a negative number.

15. Student B; Sample:Student A did not isolate the absolute value and assumed that because the equation was equal to a negative number, the absolute value would be equal to a negative number.(94)(94)

(94)(94)

(93)(93)

(93)(93)

(92)(92)Sample: 4_

2 · 3_

9 = 2 · 1_

3 = 2_

3 and 4 · 3_

2 · 9 = 12_

18 = 2_

3Sample: 4_

2 · 3_

9 = 2 · 1_

3 = 2_

3 and 4 · 3_

2 · 9 = 12_

18 = 2_

3

(92)(92)

(91)(91) 200 10-10-20 200 10-10-20

3 43 4

(91)(91)


In Problem 17, explain the meaning of the word cushion. Say:

“Cushioning provides or gives security or support to ensure safety. For example, the couch has cushions which provide comfort when we sit on it.”

Explain to students that not only pillows can provide cushioning.

Say:

“Buying extra amounts of ingredients to bake a cake provides a cushion in case the cake doesn’t come out correctly the fi rst time.”

Ask volunteers to give examples of other types of cushioning. Sample: bringing extra money to buy something when the price is estimated

ENGLISH LEARNERS

636 Saxon Algebra 1

Problem 16

Extend the Problem

What is the greatest and least area the square can have? 64 square inches and 81 square inches

Problem 18

Remind students that when a term is within parentheses, it cannot be separated. In order to cancel these factors, they must both be identical.

Problem 19

Factor the expression that describes area. Then divide by the length to get the width.

Problem 22

Extend the Problem

Solve and graph ⎢x� ≤ 17. -17 ≤ x ≤ 17

170-17


Lesson 95 637

24. Multi-Step Reggie has made a stew that is at a temperature of 100°F, and he either wants to heat it up to 165°F or cool it down to 45°F. To heat up the stew will take at most 45 minutes plus 1 minute for each degree the temperature is raised. To cool down the stew will take at least 10 minutes plus 2 minutes for each degree the temperature is lowered. How much time does Reggie need to allow for changing the food temperature either up or down? a. How many degrees up and down does the stew need to go?

b. How much time will it take to cool down or heat up the stew?

c. Is it faster to heat up or cool down the stew? To heat up the stew is faster.

25. Olympic Swimming Pool An Olympic swimming pool must have a width of 25 meters and a length of 50 meters. Find the length of a diagonal of an Olympic swimming pool to the nearest tenth of a meter. 55.9 meters

26. Multi-Step A graphing calculator screen is 128 pixels wide and 240 pixels long. The “pixel coordinates” of three points are shown.

128

0 240P(61, 43)

Q(124, 106)

R(155, 35)

a. Find the distance in pixels between each pair of points.

b. List the line segments in order from shortest to longest. −−−

QR,−−

PQ,−−

PR

27. Write Create a polynomial in which each term has a common factor of 4a, and then factor the expression. Sample: 8 a2b + 4a3 + 12ab + 16ab2; 4a(2ab + a2 + 3b + 4b2)

28. Hiking In celebration of getting to the top of Beacon Rock in southern Washington, Renata throws her hat up and off the top of Beacon Rock. The height of the hat x seconds after the throw (in meters) can be approximated by the function y = -5x2 + 10x + 260. After how many seconds will the hat be at its maximum height? What is this height? 1 second; 265 feet

29. Commuting Mr. Shakour’s round trip commute to and from work totals 30 miles. Because of traffic, his speed on the way home is 5 miles per hour less than what it is on the way to work. Write a simplified expression to represent Mr. Shakour’s total commuting time. 15(2x - 5)

_x(x - 5)

hours

30. A teacher randomly picks a shirt and a skirt from her closet to wear to school. Use the table to find the theoretical probability of the teacher choosing an outfit with a blue shirt and khaki skirt. 1_

10

Ski

rts

ShirtsRed Blue Blue White White

Khaki KR KB KB KW KWNavy NR NB NB NW NWBlack BR BB BB BW BWWhite WR WB WB WW WW

(82)(82)

65 degrees up or 55 degrees down65 degrees up or 55 degrees downheat: t ≤ 110 minutes; cool: t ≥ 120 minutesheat: t ≤ 110 minutes; cool: t ≥ 120 minutes

(85)(85)

(86)(86)

PQ = 89 pixels, QR = 77 pixels, PR = 94 pixelsPQ = 89 pixels, QR = 77 pixels, PR = 94 pixels

(87)(87)

(89)(89)

(90)(90)

(80)(80)


Find the LCD of 3 _________ mz 2 - mz

- 6 ________ pz 2 - p 2 z

.

mpz(z - 1)(z - p)

Combining rational expressions with unlike denominators prepares students for

Lesson 99 Solving Rational Equations


Lesson 95 637

Problem 25

Use the Pythagorean Theorem.

Problem 28

Error Alert Students may set y equal to 0 to fi nd the highest point. However, this is when the hat hits the ground. Remind students that the hat’s maximum height is represented by the y-value at the vertex.

Problem 30

Remind students that the probability of an event is the number of desired outcomes over the total number of outcomes.


Saxon Algebra 1638

Warm Up

96LESSON

1. Vocabulary A(n) __________ is a line that divides a figure or graph into two mirror-image halves. axis of symmetry

Evaluate for the given value.

2. y = 4 x 2 - 6x - 4 for x = 1 _ 2 -6 3. y = -x2 + 5x - 6 for x = -2 -20

Find the axis of symmetry using the formula.

4. y = -2x2 + 4x - 5 x = 1 5. y = x2 - 3x - 4 x = 3_2

A quadratic function can be graphed using the axis of symmetry, the vertex, the y-intercept, and pairs of points that are symmetric about the axis of symmetry. The quadratic function in the standard form f(x) = ax2 + bx + c can be used to find these parts of the graph of the parabola.

The equation of the axis of symmetry and the x-coordinate of the vertex of a quadratic function is x = - b

_ 2a

. To find the y-coordinate of the vertex, substitute the x-coordinate of the vertex into the function. The y-intercept of a function is the point on the graph where x = 0. For quadratic functions in standard form, the y-intercept is c.

Example 1 Graphing Quadratics of the Form y = x2 + bx + c

Graph the function.

y = x 2 + 4x + 5

SOLUTION

Step 1: Find the axis of symmetry.

x = - b _

2a Use the formula.

= - 4 _

2 · 1 = -

4 _ 2

= -2 Substitute values for b and a.

The axis of symmetry is x = -2.

Step 2: Find the vertex.

y = x 2 + 4x + 5

= (-2 ) 2 + 4(-2) + 5 = 1 Substitute -2 for x.

The vertex is (-2, 1).

Step 3: Find the y-intercept.

The y-intercept is c, or 5.

(89)(89)

(9)(9) (9)(9)

(89)(89) (89)(89)


Graphing Quadratic Functions

Online Connection


Math Reasoning

Analyze Why can a not equal 0 in the standard form of a quadratic equation?

Sample: If a = 0, then there is no x2, or quadratic term. The function would no longer be a quadratic function.





Master C96

Graphing quadratic functions is a key element of algebra. The graph can help students analyze and solve problems that can be modeled by a quadratic function.

Begin by fi nding the equation of the axis of symmetry, which passes through the vertex. Then plot the y-intercept and one other point found by substituting a value of x into the function. Finally, refl ect these points across the axis of symmetry. Connect all the points with a smooth curve.

Symmetry is an important characteristic of quadratic functions. By fi nding one point on the curve, a second point can be quickly identifi ed by refl ection across the axis of symmetry. Connecting the points with a smooth curve is critical. The graph of a quadratic function is different from a linear function because it is not straight. The vertex is a critical point because the y-value at the vertex represents the minimum or maximum value of the function.

Warm Up1

638 Saxon Algebra 1

96LESSON

Problem 3

Remind students that after substituting, (-2) should be squared and then the product should be multiplied by -1.

2 New Concepts

In this lesson, students graph quadratic functions.

Example 1

Students graph quadratics of the form y = x2 + bx + c.


Graph the function y = x2 + 5x + 6.

x

y

O

6

2

2-2-4-6


Lesson 96 639

Step 4: Find one point not on the axis of symmetry.

y = x 2 + 4x + 5

y = (1 ) 2 + 4(1) + 5 = 10 Substitute 1 for x.

A point on the curve is (1, 10).

Step 5: Graph.

Graph the axis of symmetry x = -2, the vertex (-2, 1), the y-intercept (0, 5). Reflect the point (1, 10) over the axis of symmetry and graph the point (-5, 10). Connect the points with a smooth curve.

Example 2 Graphing Quadratics of the Form y = a x 2 + bx + c

Graph the function.

y = 3 x 2 + 18x + 13

SOLUTION


x = - b_2a

Use the formula.

= - 18_

2(3)= -3 Substitute values for b and a.

The axis of symmetry is x = -3.


y = 3 x 2 + 18x + 13

= 3(-3 )2 + 18(-3) + 13 = -14 Substitute -3 for x.

The vertex is (-3, -14).




y = 3 x 2 + 18x + 13

= 3(-1 ) 2 + 18(-1) + 13 = -2 Substitute -1 for x.

A point on the curve is (-1, -2).

Step 5: Graph.

Graph the axis of symmetry x = -3, the vertex (-3, -14), the y-intercept (0, 13). Reflect the point (-1, -2) across the axis of symmetry to get the point (-5, -2). Connect the points with a smooth curve.

x

y

16

24

8

4-4-8

x

y

16

24

8

4-4-8

x

20

-10

-2-4-6

x

20

-10

-2-4-6

Hint

Count how far the plotted point (1, 10) is from the axis of symmetry. Then check that the reflected point is the same distance from the axis of symmetry, but in the opposite direction.

Hint

Identify the values of a, b, and c first.


Lesson 96 639

Example 2

Students learn to graph quadratic functions of the formy = ax2 + bx + c.


Graph the function.

y = 5x2 - 20x - 3

x

y

-10

-20

2-2O


Saxon Algebra 1640

Example 3 Graphing Quadratics of the Form y = a x 2 + c

Graph the function.

y = 5 x 2 + 4

SOLUTION


x = - b_2a

= - 0_

2(5)= 0

The axis of symmetry is x = 0.


y = 5 x 2 + 4

= 5(0 )2 + 4 = 4 Substitute 0 for x.

The vertex is (0, 4).




y = 5x2 + 4.

= 5(-1 )2 + 4 = 9 Substitute -1 for x.

A point on the curve is (-1, 9)

Step 5: Graph.

Graph the axis of symmetry x = 0, the vertex (0, 4), the y-intercept (0, 4). Reflect the point (-1, 9)

across the axis of symmetry to get the point (1, 9). Connect the points with a smooth curve.

A zero of a function is an x-value for a function where f(x) = 0. It is the point where the graph of the function meets or intersects the x-axis. The standard form of a quadratic equation ax2 + bx + c = 0, where a ≠ 0, is the related equation to the quadratic function. The quadratic equation is used to find the zeros of a quadratic function algebraically. Alternatively, a graphing calculator can help find zeros of a quadratic function.

Example 4 Finding the Zeros of a Quadratic Function

Find the zeros of the function.

a. y = x2 - 6x + 9

SOLUTION

Use a graphing calculator to graph y = x2 - 6x + 9.

The zero of the function is 3.

x

y

O

8

16

24

2 4-2-4

x

y

O

8

16

24

2 4-2-4

Math Language

A zero of a function is another name for an x-intercept of the graph.

Graphing

Calculator

For help with finding zeros, see graphing calculator keystrokes in Lab 8 on p. 583.


A table can be used to graph a quadratic function. Choose values for the x-coordinate and substitute them into the function to fi nd the y-coordinates. Then plot the points and connect them with a smooth curve.

ALTERNATE METHOD

640 Saxon Algebra 1

Example 3

Students learn to graph quadratic functions of the form y = ax2 + c.


Graph the function y = 8x2 + 2.

x

y

O

8

2 4

-4

-2-4

-8


Lesson 96 641

b. y = x2 - 3x - 10

SOLUTION

Use a graphing calculator to graph y = x2 - 3x - 10.

There are two zeros for this function, 5 and -2.

c. y = -2x2 - 3

SOLUTION

Use a graphing calculator to graph y = -2x2 - 3.

There are no real zeros for this function.

When an object is thrown or kicked into the air, it follows a parabolic path.

S

You can calculate its height in feet after t seconds using the formula h = -16t2 + vt + s. The initial vertical velocity in feet per second is v, and s is the starting height of the object in feet.

Example 5 Application: Baseball

A baseball is thrown straight up with an initial velocity of 50 feet per second. The ball leaves the player’s hand when it is 4 feet above the ground. At what time does the ball reach its maximum height?

SOLUTION Substitute the values given for the initial velocity and starting height into the formula h = -16t2 + vt + s. Then find the x-coordinate of the vertex. Use the formula x = - b

_ 2a

.

h = -16t2 + 50t + 4

x = - b _ 2a

= - 50 _

2(-16) = 1.5625 Substitute values for b and a.

The ball reaches its maximum height 1.5625 seconds after it has been thrown.

Math Language

If a quadratic function has no real zeros,

then there are no real numbers that when substituted for x result in y = 0. The graph of such a function does not cross the x-axis.


Use the following strategy with students who have diffi culty with graphing. To help them connect the algebraic representation with the graphical representation, create a color-coded chart. Color code each part of the graph and, in the same color, show how to fi nd that part algebraically.

For example, show x = -b ___ 2a

in red and then show the axis of symmetry in red on the graph. Do this for the vertex, the y-intercept, and a pair of symmetric points.

INCLUSION

Lesson 96 641

Example 4

Students learn to fi nd the zeros of a quadratic function.


Find the zeros of the function.

a. y = x2 - 16x + 64 8

b. y = x2 + 3x - 28 4 and -7

c. y = 5x2 + 9 No real zeros

Error Alert Students may confuse the zeros of a function and the y-intercept. Explain that the zeros are the x-intercepts, and the y-intercept is where x equals zero.

Example 5

Students use quadratic functions to solve vertical motion problems.

Extend the Example

When does the baseball land on the ground? after about 3.2 seconds


An egg is thrown from the top of a wall that is 20 feet tall. Its initial velocity is 32 feet per second. How long will it take for the egg to reach its maximum height? 1 second



Saxon Algebra 1642

Lesson Practice

Graph each function.

a. y = x2 - 4x + 7 b. y = 2x2 - 16x + 24

c. y = 2x2 - 9

Find the zeros of each function.

d. y = x2 + 10x + 25. -5

e. y = 3x2 - 21x + 30. 2, 5

f. y = - 1 _ 2 x2 - 1 no real zeros

g. Scoccer The height of a soccer ball that is kicked can be modeled by the function f(x) = -8x2 + 24x, where x is the time in seconds after it is kicked. Find the time it takes the ball to reach its maximum height. 1.5 seconds

(Ex 1)(Ex 1)

a.

x

y

O

4

2

4-2-4

6

a.

x

y

O

4

2

4-2-4

6

(Ex 2)(Ex 2)b.

x

y

O

8

8

16

24

4

-8

b.

x

y

O

8

8

16

24

4

-8

(Ex 3)(Ex 3)

c.

x

y

8

4

4

-4

-4

c.

x

y

8

4

4

-4

-4

(Ex 4)(Ex 4)

(Ex 4)(Ex 4)

(Ex 4)(Ex 4)

(Ex 5)(Ex 5)

1. Find the zeros of the function shown. 0 and 4

2. Add 25 _

16x2y +

xy _

32y5 .

50y3 + x3_32x2y4

*3. Solve the equation 10|x|

_ 3 + 18 = 4 and graph the solution.

Ø; no graph 4. Solve -0.3 + 0.14n = 2.78. n = 22 5. Solve

6 _ x - 3

= 3 _ 10

. x = 23

6. Find the LCD of 6 _

x + 6 - 12 _

x2 + 8x + 12 . LCD = (x + 6)(x + 2)

7. The table lists the ordered pairs from a relation. Determine whether they form a function. function

Domain (x) Range (y)

10 15

11 17

8 11

9 13

5 5

8. Simplify

-x5

_ 21x + 3

_

5x9

_ 28x + 4

. -4_

15x4

*9. Graph the function y = x2 - 2x - 8.

*10. Write Explain how to reflect a point across the axis of symmetry to get a second point on the parabola.

(89)(89)x

y

O

2

2 4 6

-4

-6

-2

y = 1_2

x2+ 2x

x

y

O

2

2 4 6

-4

-6

-2

y = 1_2

x2+ 2x

(90)(90)

(94)(94)

(24)(24) (31)(31)

(95)(95)

(25)(25)

(92)(92)

(96)(96)

9. xyO2

-2

-4

-4

-8

9. xyO2

-2

-4

-4

-8

(96)(96)Sample: The second point will have the same y-value and will be the same horizontal distance from the axis of symmetry, but on the other side.Sample: The second point will have the same y-value and will be the same horizontal distance from the axis of symmetry, but on the other side.


642 Saxon Algebra 1

Lesson Practice

Problem a

Error Alert Students may refl ect points across the y-axis rather than the axis of symmetry. Emphasize that the graph is symmetric about the axis of symmetry.

Problem b

Scaff olding To graph the function, fi rst fi nd the vertex, the y-intercept, and the axis of symmetry. Then fi nd a pair of points that are symmetric about the axis of symmetry.



“How do you fi nd the axis of symmetry?” Sample: Use the formula x = -b ___

2a .

“What is the signifi cance of the vertex?” Sample: The y-value at the vertex is the maximum or minimum value of the function. The axis of symmetry passes through the vertex.

Practice3


Problem 1

Extend the Problem

What is another name for the zeros of the function? x-intercepts

Problem 5

Error AlertStudents may forget to distribute when cross-multiplying with a binomial. Encourage them to put parentheses around the binomial before cross-multiplying.

Problem 7

For each value of the domain of a function, there can be exactly one value in the range.


Lesson 96 643

*11. Justify Show that the vertex of y = 4x2 - 24x + 9 is (3, -27).

*12. Multiple Choice Which function has the vertex (6, -160)? AA y = 6x2 - 72x + 56 B y = 2x2 - 8x + 48

C y = 3x2 + 42x - 12 D y = 5x2 - 5x + 43

*13. Diving A diver moves upward with an initial velocity of 10 feet per second. How high will he be 0.5 seconds after diving from a 6-foot platform? Use h = -16t2 + vt + s. 7 feet

14. Multiple Choice Find the LCD of 3 _

2x - 10 and 5x

_ 2x2 - 4x - 30

. A

A 2(x - 5)(x + 3) B (x - 5)(x + 3)

C (x + 5)(x - 3) D 2 __ (x - 5)(x + 3)

*15. Geometry One side of a triangle is 2 _ x + 2

yards and two sides are each -5 _

3x + 6 yards.

Find the perimeter of the triangle. -4_3(x + 2)

yards

*16. Measurement Carrie measured a distance of 3x2

_ 9x - 18

yards and Jessie measured a distance of 4x - 5

_ x2 - 4

yards. How much longer is Carrie’s measurement than Jessie’s?

x3 + 2x2 - 12x + 15__3(x - 2)(x + 2)

yards

17. Banking The dollar amount in a student’s banking account is represented by the absolute-value inequality |x - 200| ≤ 110. Solve the inequality and graph the solution.

18. Generalize Why is a place holder needed for missing variables in a polynomial dividend? Sample: It helps line up like terms for the dividend and quotient.

19. Multiple Choice Simplify (-5x + 2x2 - 3) ÷ (x - 3). B

A 2x - 1 B 2x + 1 C x - 2 _

2x D

x2 - 3 _ 5x

*20. Physics A family is going to see friends that live in two different towns. They will have to travel 100 miles plus or minus 10 miles to see either of them. They want to spend 2 hours in the car. What are the minimum and maximum rates that they need to go? 45 miles per hour; 55 miles per hour

*21. Error Analysis Two students graph the solution to the equation |2x + 10| = 8. Which student is correct? Explain the error.

Student A Student B

{-9, -1} {-9, -1}

-2-4-6-8-10 0 -2-4-6-8-10 0

22. Art Jeremy’s picture frame has an area of x2 - 18x + 80 square inches. He has two square pictures in it, one measuring 1 _

4 x inch on each side and the other

measuring 1 _ 2 x inch on each side. Write a simplified expression for the total fraction

of the frame covered by pictures. 0.3125x2__(x - 8)(x - 10)

(96)(96)

11. Sample: x = -b_

2a= 24_

8= 3. Then,

substitute 3 into the equation to get 4(3)2 - 24(3) + 9 = -27.

11. Sample: x = -b_

2a= 24_

8= 3. Then,

substitute 3 into the equation to get 4(3)2 - 24(3) + 9 = -27.

(96)(96)

(96)(96)

(95)(95)

(95)(95)

3 43 4

(95)(95)

(91)(91)

90 ≤ x ≤ 310; 255 31020090 145

90 ≤ x ≤ 310; 255 31020090 145

(93)(93)

(93)(93)

(94)(94)

(94)(94)21. Student A; Sample: Student B graphed all values between -9 and -1 in addition to the solution set.

21. Student A; Sample: Student B graphed all values between -9 and -1 in addition to the solution set.

(90)(90)


In problem 13, explain the meaning of the word diver. Say:

“A diver is a person who jumps into the water head fi rst with his body straight and his arms out in front.”

Ask volunteers to demonstrate how to position their arms to dive into the water.

ENGLISH LEARNER

Lesson 96 643

Problem 14

Factor fi rst. Then use the factors to fi nd the LCD.

Problem 19

Order the powers in the trinomial in descending order before dividing.


Saxon Algebra 1644

23. Verify Divide the rational expression 3x2 + 2x _

9y ÷

y + 2 _ y . How can you check your

answer?

24. Multi-Step The volume of a prism is 6x3 + 14x2 + 4x cm3. Find the dimensions of the prism. a. Factor out common terms. 2x(3x2 + 7x + 2)

b. Factor completely to find the dimensions. 2x(3x + 1)(x + 2)

25. Baseball A baseball diamond is a square that is 90 feet long on each side. To use a coordinate grid to model positions of players on the field, place home plate at (0, 0), first base at (90, 0), second base at (90, 90), and third base at (0, 90). An outfielder located at (150, 80) throws to the first-baseman. How long is the throw? Round your answer to the nearest foot. 100 feet

26. Multi-Step A square has a side length of a centimeters. A smaller square has a side length of b centimeters. a. If the difference in the areas of the squares is a2 - 16, what is the value of b? 4 cm

b. If the area of the larger square is 36b2 + 60b + 25, what is the side length in terms of b? 6b + 5

c. Using your answers to parts a and b, find the side length and area of the larger square. 29 cm; 841 c m 2

27. Determine if the inequality 6x > 7x is never, sometimes, or always true. If it is sometimes true, identify the solution set. sometimes true; It is true for all negative values of x.

28. Use the graph to find the theoretical probability of choosing each color.

Favorite Color10

8

6

4

2

0

Red

Green

Yellow

Blue

Orang

e

Purple

Color

29. Suppose d varies inversely with b and jointly with a and c. Find the constant of variation when a = 4, b = 5, c = 2, and d = 8. Express the relationship between these quantities. What is d when a = 9, b = 15, and c = 6? k = 5; d = 5ac_

b; d = 18

30. Is (x + 10)(x - 2) the correct factorization for x2 - 8x - 20? Explain.

(88)(88) x(3x + 2) _

9(y + 2) ; Sample: Substitute real numbers for the variables x and y before and after dividing.

x(3x + 2) _

9(y + 2) ; Sample: Substitute real numbers for the variables x and y before and after dividing.

(87)(87)

(86)(86)

(83)(83)

(81)(81)

(80)(80)

P(red) = 6_24

= 1_4, P(green) = 2_

24= 1_

12,

P(yellow) = 8_24

= 1_3, P(blue) = 5_

24,

P(orange) = 2_24

= 1_12

, P(purple) = 1_24

P(red) = 6_24

= 1_4, P(green) = 2_

24= 1_

12,

P(yellow) = 8_24

= 1_3, P(blue) = 5_

24,

P(orange) = 2_24

= 1_12

, P(purple) = 1_24

(Inv 8)(Inv 8)

(Inv 9)(Inv 9) no; If (x + 10)(x - 2) is multiplied, the result is x2 + 8x - 20. Changing the signs to (x - 10)(x + 2) would produce the correct factorization.no; If (x + 10)(x - 2) is multiplied, the result is x2 + 8x - 20. Changing the signs to (x - 10)(x + 2) would produce the correct factorization.

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Graphing quadratic functions prepares students for

• Lesson 98 Solving Quadratic Equations by Factoring

• Lesson 100 Solving Quadratic Equations by Graphing

• Lesson 102 Solving Quadratic Equations Using Square Roots

• Lesson 110 Using the Quadratic Formula

The height in feet of a dolphin as it jumps out of the water at an aquarium show can be modeled by the function f (x) = -16x2 + 32x, where x is the time in seconds after it exits the water. Find how long the dolphin is in the air. 2 seconds

LOOKING FORWARDCHALLENGE

644 Saxon Algebra 1

Problem 23

Write the division problem as the dividend times the reciprocal of the divisor. Factor each term and simplify.

Problem 25

Use the distance formula.

Problem 27

Test different types of numbers: positive, negative, zero, and fractions.

Problem 30

Remind students to be mindful of the signs when multiplying binomials.


Lab 9 645

L AB

9Linear inequalities in two variables can be graphed by hand or with a graphing calculator. Begin by following the process used to graph a linear equation. Then change the settings to shade the region of the coordinate system that makes the inequality true.

Graph the solution set of the inequality y > 2x - 7.

1. Enter the equation y = 2x - 7 into the Y = editor.

2. Graph the equation by pressing and selecting 6:ZStandard.

The line represents the boundary of the solution set of the inequality y > 2x - 7. The solution set of the inequality is either the region above or the region below the line y = 2x - 7.

Use a test point to determine which region makes the inequality true.

Choose a test point that is not on the graph of the line y = 2x - 7.

The point (0, 0) is a good test point because it does not fall on the boundary line.

Substitute 0 for both x and y in the inequality. This substitution gives 0 > 2 · 0 - 7, or 0 > -7, which is true.

Since the point (0, 0) satisfies the inequality, the solution set is the region that contains the point (0, 0).

3. Shade the region above the line y = 2x - 7.

To graph this region, press . Then press

the key twice. The cursor moves to the

left of Y1 over an icon that looks like a line segment, \.

Press twice to choose the icon, which resembles a shaded

region above a line.

Graphing Calculator Lab (Use with Lesson 97)

Graphing Linear Inequalities

Online Connection


Graphing

Calculator Tip

For help with entering an equation into the Y=

editor, see the graphing calculator keystrokes in Lab 3 on page 305.

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Lab 9 645

9LAB

Materials

• graphing calculator

Discuss

The solution set and graph for a linear inequality is a region of the rectangular coordinate system. Remind students that the graph of a linear equation is a straight line. The inequality sign extends the solution set to a region on a side of the line on the graph.

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Saxon Algebra 1646

(Pressing a third time allows you to choose the icon, which

resembles a shaded region below a line.)

4. Press to view the graph of the solution set.

All points in the shaded region are in the solution set of y > 2x - 7.

Note that the solution set does not include points on the line y = 2x - 7. The graph of the solution set of the inequality y ≥ 2x - 7 does include points on the boundary line y = 2x - 7.

Lab Practice

Graph the solution set of each inequality. a. y < 3x + 5; Is the point (1, 1) a solution of the inequality? yes

b. y ≥ 2x - 5; Is the point (7, 2) a solution of the inequality? no

c. y < -2x + 3; Is the point (0, 0) a solution of the inequality? yes

a.a.

b.b.

c.c.

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646 Saxon Algebra 1

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Lesson 97 647

Warm Up

97LESSON

1. Vocabulary The of the equation of a line is y = mx + b, where m is the slope of the line and b is the y-intercept. slope-intercept form

Determine the slope and the y-intercept of each equation.

2. y = - 1 _ 3 x - 5 3. 2x + 2y = 6

slope is -1; y-intercept is 3

Graph each of the following inequalities on a number line.

4. y < 3 0 2 4-2

5. x ≥ -2 0 2-2

A linear inequality is similar to a linear equation, except that a linear inequality has an inequality symbol instead of an equal sign. A solution of a linear inequality is any ordered pair that makes the inequality true.

You can evaluate an inequality with an ordered pair to find out if the ordered pair makes the inequality true and is a solution.

Example 1 Determining Solutions of Inequalities

Determine if each ordered pair is a solution of the given inequality.

a. (0, 4); y > 5x - 1

SOLUTION

y > 5x -1

4 > 5(0) -1 Evaluate the inequality for the point (0, 4).

4 > -1 Simplify.

The inequality is true. The ordered pair (0, 4) is a solution.

b. (3, -3); y < -3x + 6

SOLUTION

y < -3x + 6

-3 < -3(3) + 6 Evaluate the inequality for the point (3, -3).

-3 < -3 Simplify.

The inequality is not true because -3 is not less than -3. The ordered pair (3, -3) is not a solution.

c. (-4, 8); y ≤ 9

SOLUTION

y ≤ 9

8 ≤ 9 Evaluate the inequality for the point (-4, 8).

The inequality is true. The ordered pair (-4, 8) is a solution.

(49)(49)

slope is - 1_3; y-intercept

is -5slope is - 1_

3; y-intercept

is -5 (49)(49) (49)(49)

(50)(50) (50)(50)


Graphing Linear Inequalities

Online Connection


Hint

The inequalities y ≤ 9 and y ≤ 0x + 9 are equivalent. If an ordered pair is a solution of the inequality, then the y-coordinate is less than or equal to 9, and the x-coordinate can be any real number.


LESSON RESOURCES



Master C97

The word solution means answer. For solutions of linear inequalities, there will be more than one ordered pair that will give a correct answer, or solution.

The graph of a simple inequality on a number line can be compared to the solution of a linear inequality. Note the differences:

the linear inequality contains two variables while a simple inequality has only one variable. The solution of an inequality with only one variable has a linear solution. The solution of a linear inequality has an area as a solution.

MATH BACKGROUND

Warm Up1

97LESSON

Lesson 97 647

Problem 4

Remind students to represent the inequality using an open or closed circle.

2 New Concepts

In this lesson, students are introduced to graphing linear inequalities on the coordinate plane.

Example 1

This example shows whether a given ordered pair is or is not a solution of a linear inequality.


Determine if the ordered pair is a solution of the given inequality.

a. (3, -2); y ≤ 2x - 8 (3, -2) is a solution.

b. (2, -2); y > x - 3 (2, -2) is not a solution.

c. (-2, 8); x > -4 (-2, 8) is a solution


Saxon Algebra 1648

Exploration Exploration Graphing Inequalities

a. Graph the equation y = x + 2 on a coordinate plane.

b. Test three points that lie above the graph of y = x + 2. Substitute the coordinates of each point for the x- and y-values in the inequalityy < x + 2. If the statement is true, mark the point of the graph.

c. Test three points that lie below the graph of y = x + 2. Substitute the coordinates of each point for the x- and y-values in the inequalityy < x + 2. If the statement is true, mark the point of the graph.

d. To graph the inequality y < x + 2, would you choose points above or below y = x + 2? below

e. Would you graph points above or below the graph of y = x - 3 to graph the inequality y > x - 3? above

A linear inequality describes a region of a coordinate plane called a half-plane. The boundary line for the region is the related equation.

To graph an inequality, begin by graphing the boundary line. Test points are helpful in deciding which half-plane makes the inequality true.

The boundary line is a dashed line when the inequality contains the symbol < or >. The boundary line is a solid line when the inequality contains the symbol ≤ or ≥.

Example 2 Graphing Linear Inequalities without Technology

Graph each inequality.

a. y ≥ - 3 _ 4

x - 3

SOLUTION

Graph the boundary line y = - 3 _ 4 x - 3 using a solid line because the

inequality contains the symbol ≥.

Next use an ordered pair as a test point to find which half-plane should be shaded on the coordinate plane.

Test (0, 0).

y ≥ - 3_4

x - 3

0 ≥ - 3 _ 4 (0) - 3 Evaluate for (0, 0).

0 ≥ -3 Simplify.

The point (0, 0) satisfies the inequality, so it is a solution. The half-plane that contains the point should be shaded.

a–c. Sample:

x

y

O

4

2 4

-2

-4

-4

a–c. Sample:

x

y

O

4

2 4

-2

-4

-4

x

y

O

8

8

-4

4

-4 4-8

-8

x

y

O

8

8

-4

4

-4 4-8

-8

Hint

The point (0, 0) is a good test point if it is not on the boundary line.

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Explain the meaning of a shaded area.Say:

“An area that has been shaded is an area that has been darkened by coloring. To shade, artists may color more heavily in places.”

Have students discuss other situations where artists use shading in their drawings.

ENGLISH LEARNERS

648 Saxon Algebra 1

ExplorationExploration

This exploration will connect the shaded area of the graph with the solution of the inequality.

Extend the Exploration

Graph the equation x = 3. Where would you graph the inequality x ≤ 3? Graph the points of the inequality to the left of the line for x = 3.

2

x

y

O

4

2 4

-2

-2-4

-4

Example 2

This example introduces students to graphing a linear inequality on the coordinate plane.



a. y ≤ 1 __ 2 x + 2

x

y

O

4

2 4

-2

-2-4

-4

b. y > 3

2

x

y

O

4

2 4

-2

-2-4

-4

Error Alert Watch for students who identify the area to shade without checking. Have students verify their solutions using substitution.


Lesson 97 649

b. x < 4

SOLUTION

Graph the boundary line x = 4 using a dashed line because the inequality contains the symbol <.

Test (0, 0).

x < 4

0 < 4 Evaluate for (0, 0).

The point (0, 0) satisfies the inequality, so it is a solution. The half-plane that contains the point should be shaded.

To graph an inequality using a graphing calculator, the inequality must first be solved for y.

Example 3 Graphing Linear Inequalities with Technology

Graph each inequality using a graphing calculator.

a. 16x + 4y ≤ 8

SOLUTION

Solve for y.

16x + 4y ≤ 8

4y ≤ -16x + 8 Subtract 16x from both sides.

4y

_ 4 ≤

-16x _ 4 +

8 _ 4 Divide all three terms by 4.

y ≤ -4x + 2 Simplify.

Enter the inequality into your graphing calculator to view the graph.

b. 3x - y < -4

SOLUTION

Solve for y.

3x - y < -4

-y < -3x - 4 Subtract 3x from both sides.

-y

_ -1

> -3x _ -1

- 4 _

-1 Divide all three terms by -1.

y > 3x + 4 Simplify.

Enter the inequality into your graphing calculator to view the graph.

x

y

O

8

4

8

-4

-4-8

-8

x

y

O

8

4

8

-4

-4-8

-8

Caution

Remember to reverse the inequality when multiplying or dividing both sides by a negative number.

Graphing

Calculator Tip

For help with graphing inequalities, see graphing calculator Lab 9 on p. 645.


Students may have diffi culty deciding which area to shade to represent the inequality. Once the equation of the related line is graphed, have them draw a dotted or solid line. For Example 3, copy and use the charts.

y ≤ -4x + 2

output ≤ line

Say, “Since the output is less than or equal to the y-values on the line, shade below the line to represent the output.”

y > 3x + 4

output > line

Say, “Since the output is greater than the y-values on the line, shade above the line to represent the output.”

INCLUSION

Lesson 97 649

Example 3

This example demonstrates graphing a linear inequality using a graphing calculator. Simplifying the inequality fi rst is necessary.



a. 2y - 8x > -6

b. 5 - 2y ≤ 3x


Saxon Algebra 1650

Example 4 Writing a Linear Inequality Given the Graph

Write an inequality for the region graphed on each coordinate plane.

a.

x

y

O

4

2

2 4

-2

-2-4

-4

SOLUTION

Determine the equation of the boundary line. It is a horizontal line that cuts through the y-axis at -1.

The equation of the boundary line is y = -1.

Then decide which inequality symbol to use for this graph.

The graph is shaded below the solid boundary line, so the inequality contains the ≤ symbol.

The inequality shown on the graph is y ≤ -1.

b.

x

y

O

8

4

4 8-4-8

-8

SOLUTION

Determine the equation of the boundary line. It has a y-intercept at -4 and a slope of 1.

The equation of the boundary line is y = x - 4.

Then decide which inequality symbol to use for this graph.

The graph is shaded above the dashed boundary line, so the inequality contains the > symbol.

The inequality shown on the graph is y > x - 4.

Example 5 Application: Carnival

Kia will attend a school carnival and she plans to spend no more than $12. Each game costs $2 and each item of food costs $3. Write and graph an inequality to describe the total cost of the carnival.

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650 Saxon Algebra 1

Example 4

Students learn to formulate an inequality from a graph.



a.

x

y

O

4

2

2 4

-2

-2-4

-4

x > 3

b.

2

-4

x

y

O

4

2 4

-2

-2

-4

y ≤ - 3 __ 2

Example 5

This example helps students formulate inequalities that describe maximum cost.

Extend the Example

Kia’s father gave her an extra $5 for the carnival and her mother gave her an extra $8. Kia also learned that the cost of food items is $3.50 this year. Write a new inequality and graph the solution. 2x + 3.5y ≤ 25;

20

x

y

0

30

10 30


Lesson 97 651

SOLUTION

Write the inequality that models the situation.

cost of games plus cost of food is no more than 12

2x + 3y ≤ 12

Solve the inequality for y.

2x + 3y ≤ 12

3y ≤ -2x + 12 Subtract 2x from both sides.

y ≤ - 2 _ 3 x + 4 Divide all three terms by 3 and simplify.

Graph the solutions.

Since Kia cannot buy a negative amount of games and food, use only Quadrant I. Graph the boundary line y = - 2 _

3 x + 4. Use a solid

line for ≤.

Shade below the line. Kia must buy whole numbers of games and food items. All the points on or below the line with whole-number coordinates are the different combinations of games and food Kia can buy.

Lesson Practice

Determine if each ordered pair is a solution of the given inequality.

a. (2, 6); y > 3x - 2 yes

b. (4, 1); y < -4x + 1 no

c. (-6, 2); y ≤ 5 yes


d. 4x + 5y ≥ -10

e. x < 6


f. 4x + 2y ≤ 6

g. y > 2x + 6


h.

x

y

O

8

4 8

-4

-4-8

-8

i.

x

y

O

8

8

-4

4

-4-8

-8

y ≤ 4 y > x - 5

x

y

2 4 6 8

2

4

6

8

0x

y

2 4 6 8

2

4

6

8

0

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

d.

x

y

O

4

2

2 4-4

-4

d.

x

y

O

4

2

2 4-4

-4

e.

x

y

O

8

4

4 8

-4

-4-8

-8

e.

x

y

O

8

4

4 8

-4

-4-8

-8

(Ex 3)(Ex 3)f.f.

g.g.

(Ex 4)(Ex 4)

Caution

The graph of an inequality will represent all of the solutions of the inequality, but only selected points on the graph may represent solutions of the word problem.


Lesson 97 651


Student tickets for a play cost $5 and adult tickets cost $10. Write an inequality to describe the minimum number of tickets that need to be sold to earn $525 in sales. 5x + 10y ≥ 525

x

y

60

80

40

20

30 60 90O

Lesson Practice

Problem c

Error Alert Make sure that students select the correct coordinate of the ordered pair to use in the inequality.

Problem f

Scaff olding Have students solve for y. Then enter the inequality into the graphing calculator.



“Explain how to graph an inequality on the coordinate plane.” Graph the boundary line. Then shade the area of the graph that contains the solution set of the inequality.

“How can you determine which area of the graph to shade to show the solution set?” Substitute an ordered pair into the inequality. If the point is a solution, then the points on that side of the line make up the solution set. If the point is not a solution, then the points on the other side make up the solution set.



Saxon Algebra 1652

j. Nila has plans to attend the school bookfair and she wants to spend no more than $25. Each book series costs $15 and each book costs $5. Write an inequality to describe the total cost of the books Nila can buy and graph the inequality. 15x + 5y ≤ 25; See Additional Answers.

(Ex 5)(Ex 5)

Simplify.

1. 30x-2y12

_ 6y-5

5y17_x2

2. √ �� 0.09q2r + q √ �� 0.04r 0.5q√ � r

3. 16g4

_ 2g + 3

- 81 _

2g + 3 (4g2 + 9)(2g - 3)

4. Find the range of the data set that includes the ages of 9 members of a chess club:23, 7, 44, 31, 18, 27, 35, 39, 66. 59

5. Find the product (4x2 + 8)(2x - 7) using the FOIL method.

*6. Add 9 _ 9x - 36

+ -24 _

3x2 - 48 . 1_

x + 4

*7. Jim ran a total of x _

x2 + 2x + 1 miles in the gym and x + 2

_ x + 1

miles outside. How

many more miles did he run inside? -x2 - 2x - 2__(x + 1)(x + 1)

miles

8. Find the quotient of (x2 - 14x + 49) ÷ (x - 7). (x - 7)

Solve and graph the inequality.

9. 13 ≤ 2x + 7 < 15 3 ≤ x < 4;

10. ⎢x�

_ 6 > 8 x < -48 or x > 48;

11. Determine if the inequality 3x - 4x ≥ 6 - x + 8 is never, sometimes, or always true. If it is sometimes true, identify the solution set. never true

*12. Determine if the ordered pair (2, 6) is a solution of the inequality y > 3x - 2.Yes, it satisfies the inequality.

*13. Graph the function. y = x2 + 2x - 24

*14. Write What points on a graph of an inequality satisfy the inequality? Explain.

*15. Generalize How do you know which half-plane to shade for the graph of a linear inequality?

(32)(32)

(69)(69)

(90)(90)

(48)(48)

(58)(58)8x3 - 28x2 + 16x - 56.8x3 - 28x2 + 16x - 56.

(95)(95)

(95)(95)

(93)(93)

(82)(82) 4 5 62 31 4 5 62 31

(91)(91) 400 20-20-40 400 20-20-40

(81)(81)

(97)(97)

(96)(96)

13.

x

y

-4-8

-30

-10

10

-20

13.

x

y

-4-8

-30

-10

10

-20

(97)(97)

14. Sample: All the points that are on a solid boundary line and all the points that fall in the shaded half-plane satisfy the inequality.

14. Sample: All the points that are on a solid boundary line and all the points that fall in the shaded half-plane satisfy the inequality.

(97)(97)

15. Sample: Choose a test point and evaluate the inequality for that point. If the point satisfi es the inequality, shade the half-plane that contains that point. If it does not satisfy the inequality, shade the remaining half-plane.

15. Sample: Choose a test point and evaluate the inequality for that point. If the point satisfi es the inequality, shade the half-plane that contains that point. If it does not satisfy the inequality, shade the remaining half-plane.


652 Saxon Algebra 1

Practice3


Problem 4

Extend the Problem

Find the median and mean of the data set. median: 31; mean: 32.2

Problem 5

Error AlertStudents who do not read the problem carefully may expect a product with three terms. Caution students to be sure that they have copied the factors carefully before multiplying.

Problem 7


“What is the fi rst step needed to solve this problem?” Factor the denominator of the fi rst expression.

“What are the factors of the denominator in the fi rst expression?” The factors are (x + 1) and (x + 1).

“What is the LCD of the two expressions?” (x + 1)(x + 1)


Lesson 97 653

*16. Multiple Choice Which inequality represents the graph on the coordinate plane? B

x

y

O

4

2

-2

-4

-4 -2

A y ≥ - 2 _ 3 x + 2 B y ≤ -

2 _ 3 x + 2

C y ≤ 2 _ 3

x + 2 D y < - 2 _ 3 x + 2

17. Football Tickets for the school football game cost five dollars for adults and three dollars for students. In order to buy new helmets, at least $9000 worth of tickets must be sold. Write an inequality that describes the total number of tickets that must be sold in order to buy new helmets. 5x + 3y ≥ 9000

*18. Error Analysis Two students find the vertex for y = x2 - 6x + 19. Which student is correct? Explain the error. Student B; Sample: Student A did not substitute the x-value into the original equation to find the y-value.

Student A

-b _ 2a

= 6 _ 2

= 3

The vertex is (3, 0).

Student B

-b _ 2a

= 6 _ 2 = 3

32 - 6(3) + 19 = 10 The vertex is (3, 10).

*19. Geometry The area of a rectangle is 48 square inches. The length is three times the width. Find the width of the rectangle by finding the positive zero of the function y = 3w2 - 48. 4 inches

*20. Multi-Step The height y of a golf ball in feet is given by the function y = -16x2 + 49x. a. What is the y-intercept? 0

b. What does this y-intercept represent? Sample: The ball starts on the ground.

c. What answer does the equation give for the height of the ball after 5 seconds?

d. What does that height mean? Sample: After 5 seconds, the ball has already landed. It cannot have a negative height.

*21. Commuting Jeff traveled 1 _ 2x2 - 4x

miles for his job on Monday and 1 _ x3 - 2x2 miles for

his job on Tuesday. How many more miles did he travel on Tuesday? - 1_

2x2 miles

22. Multiple Choice Add 3y + 2

_ y + z + 4 _

2y + 2z . A

A 3y + 4

_ y + z B y + 4

_ y + z

C 6y

_ 4mn

D 3y - 4

_ y - z

(97)(97)

(97)(97)

(96)(96)

(96)(96)

(96)(96)

-155 feet-155 feet

(95)(95)

(95)(95)


Lesson 97 653

Problem 16


“Which choices can you immediately discard? Why?” Choice D because the line of the graph is solid; choice A because the shading of the graph is below the line.

“What is the y-intercept of the graph?” 2

“What is the slope of the graph?” - 2 __

3

“What choice can you eliminate now? Why?” choice C because the slope of the graph is negative

Problem 17

Extend the Problem

Adult tickets to the game cost $15. Student tickets cost $6. If 432 adult tickets are sold, how many student tickets must be sold to reach the goal of $9000? 420


Saxon Algebra 1654

23. Verify Show that -11 is a solution to the inequality ⎢3x� - 2 = 31. Sample: ⎢3 (-11)� - 2 = ⎢-33� - 2 = 33 - 2 = 31

24. Multiple Choice Solve 4⎢x - 8� = 12. CA {-5, 5} B {11, -11}

C {5, 11} D {-20, 20}

25. Bike Riding Ron rode his bike for 10 _

45x2 + 4x - 1 minutes to get to his grandmother’s

house that was 1 _ 45x - 5

+ 2x _

25x + 5 miles away. Find his rate in miles per minute.

26. Carpentry A carpenter uses a measuring tape with an accuracy of ± 1 _ 32

inches. He measures the height of a bookshelf to be 95 5

_ 8 inches. Solve the inequality

⎢ ⎢ x - 95 5 _ 8 � � ≤ 1

_ 32 to find the range of the height of the bookshelf.

27. Generalize How are the number of zeros of a function related to the location of the vertex of the function’s parabola? See Additional Answers.

28. Probability The probability of winning a certain game is 2x4y2

_ 15xy3 . The probability

of winning a different game is 5x2y

_ 8x3y2

. What is the probability of winning both games? x2_

12y2

29. Rate An orange juice machine squeezes juice out of x2 + 30x oranges every hour. How much time in days will it take to squeeze 3000 oranges? 125_

x(x + 30) days

30. Multi-Step A circle has a radius of x. Another circle has a radius of 3x. a. Write equations for the areas of both circles. A = πx2; A = 9πx2

b. Graph both functions in the same coordinate plane.

c. Compare the graphs. Sample: The graph of the area of the larger circle is much narrower.

(94)(94)

(94)(94)

(92)(92) 18x2 + 3x + 1__50

miles

per minute

18x2 + 3x + 1__50

miles

per minute

(91)(91)

95 19_32

≤ x ≤ 95 21_32

95 19_32

≤ x ≤ 95 21_32

(89)(89)

(88)(88)

(88)(88)

(84)(84)

30b.

x

y

6

8

4

2

1 2 3 4O

30b.

x

y

6

8

4

2

1 2 3 4O


Find the solution set to the system of inequalities by graphing. The solution set is the region where the areas overlap.

y < 3x - 5

2x - 3y ≤ 4

2

x

y

O2

-2

-2

Graphing linear inequalities prepares students for

• Lesson 109 Graphing Systems of Linear Inequalities


654 Saxon Algebra 1

Problem 26

Extend the Problem

Using the same tape measure, the carpenter measures the door opening at 31 1 __ 4 inches. Find the range of possible measures by solving the inequality ⎢x - 31 1 __ 4 � ≤ 1 ___

32 . 31 7 ___

32 ≤ x ≤ 31 9 ___

32

Problem 27


“Think of the graph of a quadratic function. Under what circumstances would you say there is a zero of the function?” If the parabola touches or crosses the x-axis, there would be a zero in the function.

“Under what circumstances would there be one zero?” There would be one zero if the vertex was on the x-axis.

“When would there be two zeros?” There would be two zeros if the vertex was below the x-axis opening upward, or if the vertex was above the x-axis opening downward.

“When would there be no zeros?” There would be no zeros when the vertex is above the x-axis opening upward, or when the vertex is below the x-axis opening downward.

Problem 28

Error AlertWatch for students who add instead of multiplying the given probabilities. Review how to fi nd the probability of two independent outcomes.


Lesson 98 655

Warm Up

98LESSON

1. Vocabulary A is an x-value for the function where f (x) = 0.zero of a function

Factor.

2. x2 + 3x - 88 (x + 11)(x - 8) 3. 6x2 - 7x - 5 (3x - 5)(2x + 1)

4. 4x2 + 28x + 49 (2x + 7)(2x + 7) 5. 12x2 - 27 3(2x + 3)(2x - 3)

A root of an equation is the solution to an equation. A quadratic equation can have zero, one, or two roots. The roots of a quadratic equation are the x-intercepts, or zeros, of the related quadratic function.

To find the roots of a quadratic equation, set the equation equal to 0. If the quadratic expression can be factored, the equation can be solved using the Zero Product Property.

Zero Product Property

If the product of two quantities equals zero, at least one of the quantities equals zero.

Example 1 Using the Zero Product Property

Solve.

(x - 4)(x + 5) = 0

SOLUTION

By the Zero Product Property, one or both of these factors must be equal to 0. To find the solutions, set each factor equal to zero and solve.

x - 4 = 0 x + 5 = 0 Set each factor equal to zero.

x = 4 x = -5 Solve each equation for x.

Check Substitute each solution into the original equation to show it is true.

(x - 4)(x + 5) = 0 (x - 4)(x + 5) = 0

(4 - 4)(4 + 5) � 0 (-5 - 4)(-5 + 5) � 0

0 · 9 � 0 -9 · 0 � 0

0 = 0 ✓ 0 = 0 ✓


(89)(89)

(72)(72) (75)(75)

(83)(83) (83)(83)


Solving Quadratic Equations by Factoring

Online Connection


Math Language

The roots of a quadratic equation are the values of x that makeax2 + bx + c = 0.

Math Reasoning

Analyze What is the difference between a quadratic function and a quadratic equation?

Sample: A quadratic equation has one variable, and a quadratic function has two variables.


LESSON RESOURCES



Master C98

The graph of a quadratic function is a parabola. The function is equal to 0 at the x-intercept(s).

Use this principle to set a quadratic equation equal to 0 and then to solve it. Factoring the equation produces two factors. By the Zero Product Property (or Principle of Zero Products), one of the factors must be equal to zero.

By setting each factor equal to zero and solving, the x-intercepts, or roots of the equation are determined.

Solutions can be checked by substituting one solution at a time into the original equation.

As a further check, have the students confi rm their solutions using a graphing calculator. Have them graph the original function on the calculator and then use the calculator to fi nd the x-intercepts. The x-intercepts and their solutions should be the same. Checking by graphing will also emphasize that the roots, solutions, and x-intercepts have the same meaning.

MATH BACKGROUND

Warm Up1

98LESSON

Lesson 98 655

Problems 2–5

These problems review factoring quadratic expressions in preparation for solving quadratic equations by factoring.

2 New Concepts

This lesson introduces the Zero Product Property. Students are informally aware of this principle, but have not applied it.

This property shows that each factor is equal to zero.

Discuss the term root of an equation. Have students note that on the graph of the equation, any x-intercept, where y is equal to zero, is a solution of the equation.

Example 1

This example uses the Zero Product Property to solve a quadratic equation.


Solve. (x - 3)(x - 8) = 0 {3, 8}


Saxon Algebra 1656

Example 2 Solving Quadratic Equations by Factoring

Find the roots.

a. x2 + 2x = 8

SOLUTION

x2 + 2x = 8

x2 + 2x - 8 = 0 Set the equation equal to 0.

(x + 4)(x - 2) = 0 Factor.

Use the Zero Product Property to solve the equation.

x + 4 = 0 x - 2 = 0 Set each factor equal to zero.

x = -4 x = 2 Solve each equation for x.

Check

x2 + 2x = 8

(-4)2 + 2(-4) � 8

16 - 8 � 8

8 = 8 ✓

x2 + 2x = 8

(2)2 + 2(2) � 8

4 + 4 � 8

8 = 8 ✓

The roots are -4 and 2.

b. 7x2 - 6 = 19x

SOLUTION

7x2 - 6 = 19x

7x2 - 19x - 6 = 0 Set the equation equal to 0.

(7x + 2)(x - 3) = 0 Factor.

7x + 2 = 0 x - 3 = 0 Set each factor equal to zero.

7x = -2 x = 3 Solve each equation for x.

x = - 2 _ 7

Check

7x2 - 6 = 19x

7 (- 2 _ 7 )

2

- 6 � 19 (- 2 _ 7 )

( 4 _ 7 ) - 6 � - 38 _ 7

- 38 _ 7 = -

38 _ 7 ✓

7x2 - 6 = 19x

7(3)2 - 6 � 19(3)

63 - 6 � 57

57 = 57 ✓

The roots are - 2 _ 7 and 3.

Hint

Standard form for the equation isax2 + bx + c = 0.


For this lesson, say and explain the meaning of root. Say:

“The word root means source or foundation. For example, the root of a tooth is under the gums.”

Also relate the word root to the root of a tree, explaining that the roots are the foundation of a tree.

ENGLISH LEARNERS

656 Saxon Algebra 1

TEACHER TIPEncourage students to try to factor the equations mentally rather than listing all the possible combinations fi rst. They may need to adjust their fi rst guesses as they go along, but this will give them practice in mental math as well as help them gain familiarity with the factoring process.

Example 2

These examples demonstrate how to solve quadratic equations by factoring.


Find the roots.

a. x2 + 6 = -5xThe roots are -3 and -2.

b. 3x2 = 4x + 7The roots are 7 __

3 and -1.


Lesson 98 657

Example 3 Finding the Roots by Factoring Out the GCF

Find the roots.

20 - 2x2 = 70 - 20x

SOLUTION

2x2 - 20x + 50 = 0 Set the equation equal to zero.

2(x2 - 10x + 25) = 0 Factor out the GCF.

2(x - 5)(x - 5) = 0 Factor the trinomial expression.

Disregard the factor of 2, since it can never equal 0.

The factor (x - 5) appears twice, but it only needs to be set to equal zero once.

x - 5 = 0

x = 5

Check

20 - 2x2 = 70 - 20x

20 - 2(5)2 � 70 - 20(5)

20 - 2(25) � 70 - 100

20 - 50 � -30

-30 = -30 ✓

The root is 5.

Finding the values of x that satisfy the quadratic equation is another way of finding the roots of a quadratic equation.

Example 4 Application: Gardening

The area of a rectangular garden is 51 square yards. The length is 14 yards more than the width. What are the length and width of the garden?

SOLUTION

Let w be the width and w + 14 be the length.

A = lw Area formula

51 = (w + 14)w Substitute known values into the equation.

51 = w2 + 14w Distribute.

0 = w2 + 14w - 51 Write the equation into standard form.

0 = (w + 17)(w - 3) Factor.

w + 17 = 0 w - 3 = 0 Use the Zero Product Property.

w = -17 w = 3 Solve.

Because the width must be a positive number, the only possible solution is 3 yards. Since the width is 3 yards, the length is w + 14. So the length is 3 + 14, which is 17 yards.

Caution

When checking your answers, use the original equation, not the one that has been rearranged. Also, use the order of operations when simplifying each side of the equation.

Math Reasoning

Justify How could you check your answers?

Sample: Multiply the length and width to see if the product is 51.


Have students practice factoring quadratic expressions, including expressions for which the GCF can be factored out:

x2 - x - 12 (x - 4)(x + 3)

6x2 - 24x + 24 6(x - 2)(x - 2)

6x3 - 2x2 - 8x 2x(3x - 4)(x + 1)

Encourage students to identify which factors of their solutions would be used to fi nd solutions for quadratic equations.

Help them understand that each factor containing a variable could be used to determine a root of an equation. For example, have students set each factor of 6x2 - 24x + 24 equal to zero. Point out the 6 ≠ 0. Then have them set each factor of 6x3 - 2x2 - 8x equal to zero. Point out that x = 0 is a solution of 2x = 0.

INCLUSION

Lesson 98 657

Example 3

This example requires that the GCF be fi rst factored out of the quadratic equation. Then the equation is factored to solve.


Find the roots. 6x2 - 12 = -21x.The roots are 1 __

2 and -4.

Example 4

This example shows how to solve an area problem using the Zero Product Property.

Extend the Example

The area of the garden is increased by 21 square yards. What are the dimensions of the expanded garden? The width is 4 yards and the length is 18 yards.


The area of a rectangular pasture is 58,500 square meters. The width is 320 meters less than the length. What are the length and width of the pasture? length = 450 meters; width = 130 meters


Saxon Algebra 1658

Example 5 Solving Quadratic Equations with Missing Terms

Solve.

a. 18x2 = 8x

SOLUTION

18x2 - 8x = 0 Set the equation equal to zero.

2x(9x - 4) = 0 Factor out the GCF.

2x = 0

x = 0

9x - 4 = 0 Set each factor equal to zero.

x = 4 _ 9 Solve each equation for x.

Check

18x2 = 8x

18 · (0)2 � 8(0)

0 = 0 ✓

18x2 = 8x

18 ( 4 _ 9 )

2

� 8 ( 4 _ 9 )

18 ( 16

_ 81

) � 8 ( 4 _ 9 )

32 _ 9 =

32 _ 9 ✓

The solution set is ⎧ ⎨

⎩ 0, 4 _

9 ⎫ ⎬

⎭ .

b. 4x2 - 25 = 0

SOLUTION

4x2 - 25 = 0 Set the equation equal to 0.

(2x - 5)(2x + 5) = 0 Factor.

2x - 5 = 0 2x + 5 = 0 Set each factor equal to zero.

2x = 5 2x = -5 Solve each equation for x.

x = 2.5 x = -2.5

Check

4x2 - 25 = 0

4(2.5)2 - 25 � 0

4(6.25) - 25 � 0

25 - 25 � 0

0 = 0 ✓

4x2 - 25 = 0

4 · (-2.5)2 - 25 � 0

4(6.25) - 25 � 0

25 - 25 � 0

0 = 0 ✓

The solution set is {-2.5, 2.5}.


658 Saxon Algebra 1

Example 5

These examples demonstrate a method of solving that requires students to factor quadratic equations with missing terms.

Error Alert Some students may use incorrect signs in the solution. Encourage them to set each factor equal to zero and to solve the resulting equation. When the variable is isolated, the solution has the opposite sign of the term in the factor.


a. Solve 18x2 + 9x = 0. The solution set is {0, - 1 __

2 }.

b. Solve 9x2 - 4 = 0.The solution set is {- 2 __

3 , 2 __

3 }.



Lesson 98 659

Solve and graph the inequality.

1. 11 < 2(x + 5) < 20 2. ⎢x� + 1.5 ≤ 7.6

3. Determine whether the polynomial -121 + 9x2 is a perfect-square trinomial or a difference of two squares. Then factor the polynomial. difference of two squares; (3x + 11)(3x - 11)

4. Graph the function y = 2x2 + 8x + 6.

5. The number of Apples A and Oranges O grown in a certain fruit orchard can be modeled by the given expressions where x is the number of years since the trees were planted. Find a model that represents the total number of apples and oranges grown in this orchard. S = 35x3 + 28x - 24

A = 15x3 + 17x - 20

O = 20x3 + 11x - 4

6. Write an equation for a line that passes through (1, 2) and is perpendicular to y = - 3 _

4 x + 2 3

_ 4 . Sample: y = 4_3x + 2_

3

7. Solve the equation 4⎢x�

_ 9

+ 3 = 11 and graph the solution.

Simplify.

8. 3x + 6 _ 7x - 7

_

5x + 10

_ 14x - 14

6_5 9. (5xyz)2(3x-1y)2 225y4z2

*10. Determine if the ordered pair (5, 5) is a solution of the inequality y < -5x + 4. no; It does not satisfy the inequality.

*11. Write Explain the Zero Product Property in your own words. Sample: If two numbers multiplied together equal 0, then at least one of the numbers has to be 0.

(82)(82)

1_2

< x < 5; 40 2-2 1_2

< x < 5; 40 2-2

(91)(91)

(83)(83)

(96)(96)

(53)(53)

(65)(65)

(94)(94)

(92)(92) (40)(40)

(97)(97)

(98)(98)

Lesson Practice

a. Solve (x - 3)(x + 7) = 0. 3, -7

Find the roots. b. x2 + 3x - 18 = 0 3, -6 c. 2x2 + 13x + 15 = 0 - 3_

2, -5

Solve. d. 5x2 - 20x = 10x - 45 3 e. 45x2 = 27x ⎧

⎨

⎩ 0, 3_

5⎫ ⎬

⎭

f. 25x2 - 16 = 0 ⎧ ⎨

⎩ - 4_

5, 4_

5⎫ ⎬

⎭

g. Architecture A rectangular pool has an area of 360 square feet. The length is 6 feet more than three times the width. Find the dimensions of the pool. The width is 10 feet, and the length is 36 feet.

(Ex 1)(Ex 1)

(Ex 2)(Ex 2) (Ex 2)(Ex 2)

(Ex 3)(Ex 3) (Ex 5)(Ex 5)

(Ex 5)(Ex 5)

(Ex 4)(Ex 4)

-6.1 ≤ x ≤ 6.1;

6.1-6.1 0

4.

x

y

6

2 4

-2

-4

{18, -18};180 9-18 -9


Lesson 98 659

Lesson Practice

Problem d

Error Alert Watch that students correctly combine the terms of the equation as they move all the terms to one side of the equation.

Problem e

Scaff olding Have students write the equation with all of the terms on one side of the equal sign. Have students explain how they decide on which side to isolate the zero.



“What are some steps used to solve a quadratic equation by factoring?” Sample: Make sure that the equation is set equal to zero. Some terms may need to be moved from one side of the equation to the other so that the like terms can be combined.

“How is the Zero Product Property used when solving quadratic equations by factoring?” Sample: Since either factor may cause the equation to equal zero, each factor can be set equal to zero to fi nd the solution(s).

Practice3


Problem 1


“Is this compound inequality combined with AND or OR?” AND

“How is the solution set found?” Sample: Include all values common to both inequalities.


Saxon Algebra 1660

*12. Justify What property allows you to use the following step when solving an equation? Sample: The Zero Product Property

(x + 4)(x + 5) = 0

x + 4 = 0 x + 5 = 0

*13. Multiple Choice What is the solution set of 0 = (3x - 5)(x + 2)? A

A ⎧

⎨

⎩ 5 _ 3 , -2

⎫ ⎬

⎭ B

⎧ ⎨

⎩ 5 _ 3

, 2 ⎫ ⎬

⎭ C

⎧ ⎨

⎩ -

5 _ 3 , -2

⎫ ⎬

⎭ D

⎧ ⎨

⎩ -

5 _ 3

, 2 ⎫ ⎬

⎭

*14. Ages A girl is 27 years younger than her mother. Her mother is m years old. The product of their ages is 324. How old is each person? The mother is 36 years old and the girl is 9 years old.

*15. Multi-Step Seve plans to go shopping for new jeans and shorts. She plans to spend no more than $70. Each pair of jeans costs $20 and each pair of shorts costs $10. a. Write an inequality that describes this situation. 20x + 10y ≤ 70

b. Graph the inequality. See Additional Answers.

c. If Seve wants to spend exactly $70, what is a possible number of each she can spend her money on? Sample: 3 pairs of jeans and 1 pair of shorts

*16. Geometry The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. The sides of a triangle are labeled 4x inches, 2y inches, and 8 inches. James wrote an inequality that satisfies the Triangle Inequality Theorem. He wrote the inequality 4x + 2y > 8. Use a graphing calculator to graph the inequality. See Additional Answers.

*17. Solve (x + 4)(x - 9) = 0. -4, 9

18. Error Analysis Students were asked to write an inequality that results in a dashed horizontal boundary line. Which student is correct? Explain the error. Student A; Sample: Student B wrote an inequality with a solid horizontal line.

Student A Student B

y > -3 y ≥ 4

19. Horseback Riding It took Joe 2x - 10 _

2x5 minutes to ride his horse to Darrell’s house that

was 3x2 - 15x _

3x miles away. Find his rate in miles per minute. x5 miles per minute

20. Measurement The area of a triangle can be expressed as 4x2 - 2x - 6 square meters. The height of the triangle is x + 1 meters. Find the length of the base of the triangle. 8x - 12 meters

21. Art Michael bought a rectangular painting from a local artist. The area of the painting was (20x + 5 + x3) square inches. The width was (x - 5) inches. What was the length? (x2 + 5x + 45 + 230_

x - 5) inches

22. Analyze Should you find the LCD when multiplying or dividing rational fractions? Sample: No, the LCD is found in addition and subtraction problems so that parts of equal size can be added or subtracted.

(98)(98)

(98)(98)

(98)(98)

(97)(97)

(97)(97)

(98)(98)

(97)(97)

(92)(92)

3 43 4

(92)(92)

(93)(93)

(95)(95)


CHALLENGE

Factor and solve each quadratic equation. What do they have in common?

x2 + 12x + 36 = 0

16 = -24x - 9x2

16x2 + 48x = -48x - 144

81x2 - 162x + 324 = 162x

-54x = -9x2 - 81

All these equations are products of squares. They have only one solution.

660 Saxon Algebra 1

Problem 13

Error AlertEncourage students who choose an incorrect solution to set each factor equal to zero and to carefully perform the operations necessary to isolate the variable. Then have them substitute their solutions into the respective factors and see if the result is zero.

Problem 14

Extend the Example

Have students solve the problem letting g represent the girl’s age. Have them give the age of the mother in terms of g and explain how the solution compares to the solution of the original problem. Sample: The mother’s age is g + 27. The solution is the same except that the girl’s age is found fi rst and is then used to fi nd the age of her mother.

Problem 21

Remind students that the long division will be easier if a placeholder is used for the missing quadratic term.


Lesson 98 661

23. Multi-Step Erika and Casey started a new walking program. They walked

4x _

3x + 9 miles Thursday and 16

_ x2 + 12x + 27

miles Friday.

a. What is the total distance that they walked? 4x2 + 36x + 48__3(x + 9)(x + 3)

miles

b. If their rate was 4x

_ x + 3 miles per hour, how much time did it take them to walk on Thursday and Friday?

x2 + 9x + 12__3x(x + 9)

hours

*24. Physics A ball is dropped from 100 feet in the air. What is its height after 2 seconds? Use h = -16t2 + vt + s. (Hint: Its initial velocity is 0 feet per second.) 36 feet

*25. Error Analysis Two students find the zeros of the function y = x2 - 8x - 33. Which student is correct? Explain the error. Student B; Sample: Student A found the y-intercept.

Student A Student B

y = x2 - 8x - 33y = (0)2 - 8(0) - 33y = -33(0, -33)

0 = x2 - 8x - 330 = (x - 11)(x + 3)x - 11 = 0 or x + 3 = 0 x = 11 or x = -311 and -3

26. Multi-Step Hideyo has a picture frame that measures 20 centimeters by 26 centimeters. The frame is 1.5 centimeters wide. a. Find the length and width of the picture area. 23 cm; 17 cm

b. Find the length of the diagonal of the picture area to the nearest tenth of a centimeter. 28.6 cm

27. Profit A business sold x2 + 6x + 5 items. The profit for each item sold

is x2

_ 100x

dollars. a. What is the profit in terms of x?

x(x + 5)(x + 1)__100 dollars

b. What is the profit (in dollars) if x = 50? $1402.50

28. Multi-Step Javed has a garden with an area of 36 square feet. The width of his garden is 9 feet less than the length. What are the dimensions of his garden? a. Write a formula to find the dimensions of the garden and describe how you will

solve it. x(x - 9) = 36; Sample: The formula needs to be set in the formax2 + bx + c = 0 in order to solve for x.

b. What are the dimensions of the garden? 12 feet long and 3 feet wide.

29. Generalize Describe the steps for subtracting x - 6 _

x + 5 from x

2 - x - 30 _

x + 5 .

30. The width of a rectangle is represented by the expression (4x - 6) and the length (8x + 7). Would the area of the rectangle be correctly expressed as32x2 + 20x - 42? If not, what is the correct area? No. A sign error has occurred; The correct expression would be 32x2 - 20x - 42.

(95)(95)

(96)(96)

(96)(96)

(85)(85)

(88)(88)

(89)(89)

(90)(90)

29. Sample:Distribute the negative sign so that you are subtracting x and adding 6 tox2 - x - 30. Then combine like terms to get x2 - 2x - 24. Finally, try to factor out x + 5 from the numerator. Since you can’t, your answer in simplest

form is x2 - 2x - 24__

x + 5.

29. Sample:Distribute the negative sign so that you are subtracting x and adding 6 tox2 - x - 30. Then combine like terms to get x2 - 2x - 24. Finally, try to factor out x + 5 from the numerator. Since you can’t, your answer in simplest

form is x2 - 2x - 24__

x + 5.

(Inv 9)(Inv 9)


Solving quadratic equations by factoring prepares students for

• Lesson 100 Solving Quadratic Equations by Graphing

• Lesson 102 Solving Quadratic Equations by Using Square Roots

• Lesson 104 Solving Quadratic Equations by Completing the Square


• Lesson 113 Interpreting the Discriminant

LOOKING FORWARD

Lesson 98 661

Problem 26


“How many centimeters does the frame add to the width of the picture?” Sample: 3 cm

“What formula is used to fi nd the length of the diagonal?” the Pythagorean Theorem

Problem 27

Have students check their work by substituting 50 for x in the original expressions for the number of items sold and the profi t for each item. Then have them multiply the results to fi nd the total profi t.


Saxon Algebra 1662

Warm Up

99LESSON

1. Vocabulary The denominator of a contains a variable. The value of the variable cannot make the denominator equal to zero.

Find the LCM.

2. 7x2y and 3xy3 21x2y3 3. (3x - 6) and (9x2 - 18x) 9x(x - 2)

4. (x + 3) and (2x - 1) 5. (14x - 7y) and (10x - 5y)

A rational equation is an equation containing at least one rational expression. There are two ways to solve a rational equation: using cross products or using the least common denominator.

Either way may lead to an extraneous solution; that is, a solution that does not satisfy the original equation. The solution may satisfy a transformed equation, but make a denominator in the original equation equal 0. If an answer is extraneous, eliminate it from the solution set.

If a rational equation is a proportion, it can be solved using cross products.

Example 1 Solving a Rational Proportion


a. 3 _ x

= 5 _

x - 6

SOLUTION

3 _ x

= 5 _

x - 6

3(x - 6) = 5x Use cross products.

3x - 18 = 5x Distribute 3 over (x - 6).

-18 = 2x Subtract 3x from both sides.

-9 = x Divide both sides by 2.

Check Verify that the solution is not extraneous.

3 _ x

= 5 _

x - 6

3 _

-9 �

5 _ -9 - 6

Substitute -9 for x in the original equation.

3 _

-9 �

5 _ -15

Simplify the denominator.

- 1 _ 3

= - 1 _ 3 Simplify each fraction.

The solution is x = -9. ✓

(39)(39)rationalexpressionrationalexpression

(57)(57) (57)(57)

(57)(57)(x + 3)(2x - 1)(x + 3)(2x - 1)

(57)(57)35(2x - y)35(2x - y)


Solving Rational Equations

Online Connection


Math Language

A rational expression is a fraction with a variable in the denominator.

Math Reasoning

Analyze Why is it necessary to keep the terms in the denominator grouped?

Sample: They are already grouped by the fraction bar. A set of parentheses is a reminder to distribute when cross multiplying.





Master C99

For this lesson, consider any expression with a variable in the denominator to be a rational expression.

Rational equations can be solved using cross products or by fi nding the least common denominator to simplify all terms on both sides of the equation.

Using cross products to solve proportions can be applied to solving rational equations that have one term on each side.

Rational equations that contain more than two terms must be solved using the LCD.

Regardless of the method used to solve rational equations, the equations are transformed into the form of an equation, linear or quadratic, that the students already know how to solve.

Warm Up1

662 Saxon Algebra 1

99LESSON

Problems 2–5

Review fi nding the LCM in preparation for fi nding the LCD to solve rational proportions.

2 New Concepts

Remind students of the defi nition of a rational number: a number that can be expressed as the quotient of two numbers. Introduce the term rational equation and have students look over the examples. Point out that all the terms have a variable in the denominator.

Explain that in this lesson, students will solve equations that contain rational terms.

Example 1

These examples introduce students to solving rational equations that take the form of a proportion.



a. x __ 5 =

2x + 4 _______ 2

x = -2.5


Lesson 99 663

b. x _ 4 =

3 _ x - 1

SOLUTION

x _ 4 =

3 _ x - 1

x(x - 1) = 12 Use cross products.

x2 - x = 12 Distribute x over (x - 1).

x2 - x - 12 = 0 Subtract 12 from both sides.

(x - 4)(x + 3) = 0 Factor.

x = 4 or x = -3 Use the Zero Product Property to solve.


x _ 4 =

3 _ x - 1

; x = 4

4 _ 4 �

3 _ 4 - 1

1 = 1 ✓

or

Substitute.

Simplify each fraction.

x _ 4 =

3 _ x - 1

; x = -3

-3 _ 4 �

3 _ -3 - 1

- 3 _ 4 = -

3 _ 4 ✓


If a rational equation includes a sum or difference, find the LCD of all the terms to solve.

Example 2 Using the LCD to Solve Addition Equations

Solve 3 _ x

+ 16 _ 2x

= 11.

SOLUTION

3 _ x

+ 16 _ 2x

= 11

The LCD of the denominators is 2x.

(2x) 3 _ x

+ (2x) 16 _ 2x

= (2x)11 Multiply each term by the LCD.

6 + 16 = 22x Simplify each term.

22 = 22x Add.

1 = x Divide both sides by 22.


3 _ x +

16 _ 2x

= 11

3 _ 1 +

16 _ 2(1)

� 11 Substitute 1 for x in the original equation.

11 = 11 ✓ Simplify.

The solution is x = 1.

Hint

To find the LCD of the terms, consider the denominator of the whole number to be 1.


INCLUSION

Use the following strategy with students who have diffi culty with written symbols. Help students identify the terms needed to fi nd the LCD of a rational equation. Since sometimes one denominator will be a factor of another, it is not necessary to multiply by that factor a second time.

Present students with these groups of rational terms. Have them identify the denominators used to fi nd the LCD of the terms.

1. 3 ___ ab ; 6 ___ bc ; 12 ___ b ab, bc

2. 9 ______ x + 2 ; 13 ___ 2x ; 4 _____ x - 3 x + 2, 2x, x - 3

3. 18 ____ 5m ; 6 _______ 2m + 1 ; 3 ___ m 5m, 2m + 1

4. 7 ____ x2y3

; 9 ___ xy ; 2 ____ xy4

x2y3, xy4

5. 5 ______ d - 3 ; 1 ______ d - 1 ; 4 ___ d 2 d - 3, d - 1, d2

Lesson 99 663

b. x __ 3 = 5 ______

x - 2

x = -3 and x = 5

Example 2

This example demonstrates how to solve a rational equation involving addition by multiplying by the LCD.


Solve 9 ___ 3x

+ 15 ___ x = 6. x = 3


Saxon Algebra 1664

Example 3 Using the LCD to Solve Subtraction Equations

Solve 3 _ x - 1

- 2 _ x

= 5 _

2x .

SOLUTION

3 _ x - 1

- 2 _ x

= 5 _

2x

The LCD is 2x(x - 1).

2x(x - 1) 3 _

x - 1 -2x(x - 1)

2 _ x = 2x(x - 1) 5 _

2x Multiply each term by

the LCD.

6x - 4(x - 1) = 5(x - 1) Simplify each term.

6x - 4x + 4 = 5x - 5 Use the Distributive Property.

2x + 4 = 5x - 5 Collect like terms.

4 = 3x - 5 Subtract 2x from both sides.

9 = 3x Add 5 to both sides.

3 = x Divide both sides by 3.


3 _

x - 1 -

2 _ x

= 5 _

2x

3 _

3 - 1 -

2 _ 3

� 5 _

2(3) Substitute 3 for x in the original equation.

5 _ 6 =

5 _ 6 ✓ Simplify.


Example 4 Checking for Extraneous Solutions

Solve the equation.

x - 1 _ x - 2

= x + 9 _ 2x - 4

SOLUTION

x - 1 _ x - 2

= x + 9 _ 2x - 4

(x - 1)(2x - 4) = (x - 2)(x + 9) Use cross products.

2x2 - 6x + 4 = x2 + 7x - 18 Multiply.

x2 - 13x + 22 = 0 Subtract x2, 7x, and -18 from both sides.

(x - 11)(x - 2) = 0 Factor.

x = 11 or x = 2 Use the Zero Product Property to solve.

Caution

Remember to distribute the negative over the new numerator in a fraction following a subtraction sign.


ENGLISH LEARNERS

Explain the meaning of the word extraneous. Write the word on the board and underline “extra”. Say:

“Extraneous means extra. Something that is extra is not needed. For example, the light from the window is extraneous lighting.”

Have students give other examples using the word extraneous. Sample: He made an extraneous remark about the situation.

664 Saxon Algebra 1

Example 3

This example demonstrates how to solve a rational equation involving subtraction by multiplying by the LCD. Point out to students the caution note regarding distributing the negative sign over the new numerator in a subtraction equation.


Solve 12 ___ x + 15 ______ x + 1

= 72 ___ 3x

. x = 4

Example 4

This example shows students how an extraneous solution may occur and how such cases should be handled.


Solve the equation.

3x - 1 ______ x + 3

= 2x - 4 ______ x - 2

x = 7 and x = 2;

2 is an extraneous solution, so x = 7.

Error Alert Check that students fully understand that it is possible to get a solution that makes the equation false. Some students will assume they have made an error and continue trying to fi nd a second valid solution.


Lesson 99 665

Check Verify that the solutions are not extraneous.

x - 1 _ x- 2

= x + 9 _ 2x - 4

; x = 11

11 - 1 _ 11 - 2

� 11 + 9 _

2(11) - 4

10 _ 9

= 10 _ 9 ✓

or

Substitute.

Simplify.

x - 1 _ x - 2

= x + 9 _ 2x - 4

; x = 2

2 - 1 _ 2 - 2

� 2 + 9 _

2(2) - 4

1 _ 0 =

11 _ 0 ✗

2 is an extraneous solution.


Example 5 Application: Painting

It takes Samuel 7 hours to paint a house. It takes Jake 5 hours to paint the same house. How long will it take them if they work together?

SOLUTION

Understand The answer will be the number of hours h it takes for Samuel and Jake to paint the house.

Samuel can paint the house in 7 hours, so he can paint 1 _ 7 of the house

per hour.

Jake can paint the house in 5 hours, so he can paint 1 _ 5 of the house per hour.

Plan The part of the house Samuel paints plus the part of the house Jake paints equals the complete job. Samuel’s rate times the number of hours worked plus Jake’s rate times the number of hours worked will give the complete time it will take them to paint the house. Let h represent the number of hours worked.

(Samuel’s rate)h + (Jake’s rate)h = complete job

1 _ 7

h + 1 _ 5

h = 1

Solve

1 _ 7 h +

1 _ 5 h = 1

(35) 1 _ 7

h + (35) 1 _ 5 h = (35)1 Multiply each term by the LCD, 35.

5h + 7h = 35 Simplify each term.

h = 35 _ 12

Combine like terms; and divide both sides by 12.

Together, they can paint the house in 2 11

_ 12 hours.

Math Reasoning

Justify Why do you have to check both solutions to the equation?

Sample: Either or both solutions could be extraneous.

Math Reasoning

Verify Show that the solution 35_

12 hours satisfies the original equation.

Sample:1_7h + 1_

5h =1

1_7 ( 35_

12 ) + 1_5 ( 35_

12 ) = 1

5_12

+ 7_12

= 1

1 = 1


Lesson 99 665

Example 5

This example solves a rate problem using rational expressions.

Extend the Example

Suppose that Samuel takes 1 hour less and Jake 1 hour more to paint a house. Will they still fi nish in the same amount of time as when they worked at their original rates? Explain. no; Sample: Both now have a rate of 6 hours to paint a house. Working together, it will take them 3 hours to paint the house.


It takes Bill 8 hours to mow the pasture. It takes Frank 10 hours to mow the same pasture. How long will it take them if they work together? Together, they can mow the pasture in 4 4 __

9 hours.



Saxon Algebra 1666

Lesson Practice


a. 6 _ x

= 7 _

x - 1 x = -6

b. 2 _

x + 4 =

x _ 6 x = 2, -6

c. 12 _ 2x

+ 16 _ 4x

= 5 x = 2

d. 4 _ x - 2

- 2 _ x

= 1 _

3x x = - 14_

5

e. x + 5 _ x + 4

= x - 2 _ 2x + 8

x = -12

f. Lawn Care It takes John 2 hours to mow the yard. Sarah can do it in 3 hours. How long will it take them if they mow the yard together? 1 1_

5 hours

(Ex 1)(Ex 1)

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

(Ex 4)(Ex 4)

(Ex 5)(Ex 5)

Solve.

1. 4 _ x

= 8 _

x + 4 x = 4 2. (x - 13)(x + 22) = 0 {13, -22}

3. Physics A student is biking to a friend’s house. He bikes at 10 miles per hour. The friend lives 20 miles away, give or take 2 miles. What are the minimum and maximum times it will take him to get there? 1.8 hours, 2.2 hours

*4. Verify Without graphing, show that the point (2, -6) lies on the graph of y = x2 + x - 12. Sample: y = (2)2 + (2) - 12 = -6

5. Simplify: 4x

_ 12x - 60

+ 1 _ 4x - 16

__

-2

_ 9x - 20 - x2 . 6. Factor x + 4x2 - 5. (4x + 5)(x - 1)

7. Larry weighed 180 pounds. He has lost 2 pounds a month for x months. Write a linear equation to model his weight after 8 months. y = 180 - 2x; 164 pounds

*8. Write What is an extraneous solution? Sample: It is an answer that solves the transformed equation, but not the original one.

9. Find the LCD of 2x _ 2x2 - 72

- 12 __

x2 + 13x + 42 . LCD = 2(x - 6)(x + 6)(x + 7)

10. Population The function y = -0.0003x2 + 0.03x + 1.3 shows the population of Philadelphia County between the years 1900 and 1990, where x is the number of years after 1900 and y is the population for that year in millions of people. Find the vertex of the parabola that represents the function. What does it represent in terms of the scenario? (50, 2.05); The population reached its maximum of about 2,050,000 people in 1950.

(99)(99) (98)(98)

(94)(94)

(96)(96)

(92)(92)

5. 4x2 - 13x - 15__24

5. 4x2 - 13x - 15__24

(75)(75)

(52)(52)

(99)(99)

(95)(95)

(89)(89)


666 Saxon Algebra 1

Lesson Practice

Problem d

Error Alert Some students may use x and 3x in the LCD. Point out that since the LCD uses each factor to its highest degree, the LCD is 3x(x - 2) rather than 3x(x)(x - 2).

Problem f

Scaff olding Have students fi rst fi nd John’s rate of mowing, and then have them fi nd Sarah’s rate of mowing. Ask them what the variable represents in this problem. Sample: the number of hours worked together



“What are two methods used to solve rational equations?” Sample: Use cross products or use the LCD.

“Why are the solutions of a rational equation checked?” Sample: because one of the solutions might be extraneous, and that would make the equation false

Practice3


Problem 9

Error Alert Make sure that students don’t try to use the denominators as written for the LCD. Remind them to factor polynomials when possible and to delete common factors when fi nding the LCD.

Problem 10

Extend the Problem

Predict the population of Philadelphia County in the year 2000, assuming that the trend shown by the function continues. 1.3 million


Lesson 99 667

11. Multi-Step The coordinates of three friends’ houses on a city map are P(3, 3), Q(5, 9), and R(11, 3). The friends plan to meet at the point that is half-way between Q and the midpoint of

−− PR .

a. Find the midpoint of −−

PR . (7, 3)

b. Find the coordinates of the point where the friends plan to meet. (6, 6)

12. Find the quotient of (18x2 - 120 + 6x5) ÷ (x - 2).

13. What is the ratio of the volume of a cube with side lengths of 5 to the volume of a cube with side lengths of 3? 125_

27

*14. Formulate How can you quickly tell if a possible solution is extraneous using the denominators in the original equation? Sample: If it would cause one of the denominators to equal 0, the solution is extraneous.

*15. Multiple Choice Which of the following is an extraneous solution to the equation x2

_ x - 1

+ 4x2 - 20x _

(x - 1)(x - 5) = 10

_ 2x - 2 ? C

A x = -1 B x = 0

C x = 1 D x = 5

*16. Housekeeping It takes a man 8 hours to clean the house. His friend can clean it in 6 hours. How long will it take them if they clean together? 24_

7 hours

*17. Error Analysis Two students solve the equation 0 = (x - 5)(x + 11). Which student is correct? Explain the error.

Student A Student B

x - 5 = 0 x + 11 = 0 x = 5 x = -11

x - 5 = 0 x + 11 = 0 x = -5 x = 11

18. Geometry The area of a triangle is 24 square centimeters. The height is four more than two times the base. Find the base and height of the triangle. The base is 4 centimeters and the height is 12 centimeters.

*19. Multi-Step The length of a lawn is twice the width. The lawnmower cuts 2-foot strips. One strip along the length and width has already been cut.

2x feet

x feet

2 feet2 feet

a. Write expressions for the length and width of the area left to be cut.

b. The area left to be cut is 144 square feet. Find the width of the yard. 10 feet

c. What is the length of the yard? 20 feet

20. Generalize How do you know when there is no solution to an absolute-value inequality? Sample: There is no solution when the absolute value is less than 0 because it would be a negative number.

21. Architecture Mia is designing a rectangular city hall. She accidentally spills water on her newly revised sketches. She is only able to determine the area, which is (x2 - 15x + 56) square feet, and the length, which is (x - 7) feet. What is the width? (x - 8) feet

(86)(86)

(93)(93)6x4 + 12x3 + 24x2 + 66x + 132 + 144_

x - 2 6x4 + 12x3 + 24x2 + 66x + 132 + 144_

x - 2

(36)(36)

(99)(99)

(99)(99)

(99)(99)

(98)(98)17. Student A; Sample: Student B has the incorrect signs on both solutions.

17. Student A; Sample: Student B has the incorrect signs on both solutions.

(98)(98)

(98)(98)

2x - 2 and x - 22x - 2 and x - 2

(91)(91)

(93)(93)


Lesson 99 667

Problem 14


“What kind of denominator would not be allowed in a rational equation?” A denominator equal to 0.

“How could the denominator equal zero?” The value substituted for the variable causes the denominator to equal zero.

“What would this tell about the solution?” It would be an extraneous solution.

Problem 19

Error Alert Make sure that students write the correct expressions for the length and width of the lawn yet to be cut. If students begin with erroneous expressions, they cannot arrive at the correct solution.

Problem 21

Extend the Problem

The height of the rectangular-prism-shaped city hall is (3x + 5). What is the volume of the building? 3x3 - 40x2 + 93x + 280


Saxon Algebra 1668

*22. Multiple Choice What are the zeros for the function y = 4x2 + 28x - 72? CA 0, 4 B 0, -4

C 2, -9 D 2, -76

*23. Band The school band is performing a music concert. Tickets cost $3 for adults and $2 for students. In order to cover expenses, at least $200 worth of tickets must be sold. Write an inequality that describes the graph of this situation. 3x + 2y ≥ 200

x

y

O40-40

-40

2 units

3 units

24. Measurement Jesse bought a new glass window to go in his front room. The area of the window is (2 + 3x3 - 8x) square inches. The width is (x + 4) inches. What is the length? (3x2 - 12x + 40 - 158_

x + 4 ) inches

*25. Error Analysis Students were asked to write an inequality that results in a solid, vertical boundary line. Which student is correct? Explain the error.

Student A Student B

x ≤ 6 x < -2

26. Graph the inequality 4x + 5y ≥ -7.

27. Multi-Step Pedro biked 6 miles on dirt trails and 12 miles on the street. His biking rate on the dirt trails was 25% of what it was on the street. a. Write a simplified expression for Pedro’s total biking time. 9_

0.25x = 36_x

b. Analyze Explain how finding the simplified expression would change if Pedro’s biking rate on the trails was 50% of what it was on the streets.

28. Graph the function y - 3 = -x2 + 3.

29. A square has a side length s. The square grows larger and has a new area of s2 + 16s + 64. What is the new side length? s + 8

30. Write an equation where j is inversely proportional to m and n and directly proportional to p and q. j =

kpq_mn

(96)(96)

(97)(97)

3 43 4

(93)(93)

(97)(97)25. Student A; Sample: Student B wrote an inequality with a dashed vertical boundary line.

25. Student A; Sample: Student B wrote an inequality with a dashed vertical boundary line.

(97)(97)26.

2

x

y

O

4

2 4

-2

-2-4

-4

26.

2

x

y

O

4

2 4

-2

-2-4

-4

(90)(90)

27b. Sample: The expression for the street time, 12_

x , would be

multiplied by .5_.5

insteadof .25_

.25, making the

simplified expression

12_.5x

= 24_x

.

27b. Sample: The expression for the street time, 12_

x , would be

multiplied by .5_.5

insteadof .25_

.25, making the

simplified expression

12_.5x

= 24_x

.

(84)(84)

28.

x

y

O

4

2

2 4

-2

-4

28.

x

y

O

4

2

2 4

-2

-4

(83)(83)

(Inv 8)(Inv 8)


LOOKING FORWARD

Solving rational equations will be further developed in other Saxon High School Math courses.

CHALLENGE

Have students apply the skills learned in this lesson to simplify each expression.

1. 15 ___ m2 - 2 __ m - 1

___________ 4 ___ m2 - 5 __ m + 4

-(m - 3)(m + 5)

______________

4m2 - 5m + 4

2. 1 - 12 ________

3b + 10 ______________

b - 8 ________ 3b + 10

1

_____ b + 4

668 Saxon Algebra 1

Problem 24


“What is the fi rst step to perform in solving this problem?” Sample: Rewrite the terms in descending order of the exponents.

“What else must be done before dividing?” Sample: Write a 0x2 term as a placeholder so you can divide.

Problem 29

Extend the Problem

The square grows larger still and has a new area of s4 + 18s2 + 81. What is the new side length?(s2 + 9)


Lesson 100 669

Solving Quadratic Equations by Graphing

Warm Up

100LESSON

1. Vocabulary The U-shaped curve that results from graphing a quadratic function is called a(n) . parabola

Evaluate each expression for the given values.

2. 3(x - y)2 - 4y2 for x = -5 and y = -2 11

3. -x2 - 3xy + y for x = 3 and y = -1 -1

Determine the direction that the parabola opens.

4. f (x) = 3x2 + x - 4 upward 5. f (x) = -2x2 + x + 1 downward

The solution(s) of a quadratic equation, 0 = ax2 + bx + c, can be found by graphing the related function, f(x) = ax2 + bx + c. The U-shaped graph of a quadratic function is called a parabola. The solutions of the equation are called roots and can be found by determining the x-intercepts or zeros of the quadratic function. These zeros can be found by graphing the related function to see where the parabola intersects the x-axis.

Graphical Solutions

One Real Solution

The graph intersects the x-axis at the vertex. x

y

O

4

2

2 4 6-2

Two Real Solutions

The graph intersects the x-axis at two distinct points.

x

y

O

4

2

4

-2

-4

No Real Solutions

The graph does not intersect the x-axis.

xy

O4 62

-2

-4

-6

(84)(84)

(9)(9)

(9)(9)

(84)(84) (84)(84)


Online Connection


Math Language

The same function is described by y = 3 x 2 - 5and f(x) = 3 x 2 - 5.

The function notation

for y is f(x). It is read, “f of x.”


LESSON RESOURCES



Master C100Technology Lab Master 100

MATH BACKGROUND

To solve a quadratic equation, it is written in standard form, where the exponents are in descending order and the polynomial is set equal to zero: 0 = a x 2 + bx + c.

Quadratic equations may have one real solution (when the vertex of the parabola is on the x-axis), two real solutions (when the parabola crosses the x-axis), or no real solutions (when the parabola does not interesect the x-axis). The x-intercepts shown by the graph are the solutions.

Take the opportunity with this lesson, to tell or remind students that when any type of equation is solved, the solution is the x-coordinate of the point where the graph of the equation crosses the x-axis.

Students have already solved linear, quadratic, and absolute-value equations in their study of algebra. In Section 10, they will solve rational and radical equations. In all cases, the graphical interpretation of the solution, or roots, will be the x-intercept(s).

Warm Up1

100LESSON

Lesson 100 669

Problems 4 and 5

Briefl y review graphing quadratic functions in preparation for solving quadratic equations by graphing.

2 New Concepts

Students are already familiar with graphing and solving quadratic functions. Now they will solve quadratic equations by graphing.

Discuss the graphs in the table, and note the number of real solutions indicated by each type of graph. Have students identify the vertex and x-intercept(s) of each graph.

TEACHER TIPHelp students understand that the most important points of the quadratic function to graph are the x-intercepts and the vertex. The axis of symmetry is helpful to use in graphing other points on the parabola. The infi nite number of points remaining to be graphed can be approximated with a smooth curve.


Saxon Algebra 1670

Example 1 Solving Quadratic Equations by Graphing

Solve each equation by graphing the related function.

a. x2 - 36 = 0

SOLUTION


x = - b _ 2a

Use the formula.

x = - 0_

2(1) = 0 Substitute values for a and b.

The axis of symmetry is x = 0.


f (x) = x2 - 36

f (0) = (0)2 - 36 Evaluate the function for x = 0 to find the vertex.

The vertex is (0, -36).


The y-intercept is c, or -36.

Step 4: Find two more points that are not on the axis of symmetry.

f (5) = 52 - 36 f (7) = 72 - 36

(5, -11) (7, 13)

Step 5: Graph.

Graph the axis of symmetry x = 0, the vertex and the y-intercept, both at coordinate (0, -36). Reflect the points (5, -11) and (7, 13) over the axis of symmetry and graph the points (-5, -11) and (-7, 13). Connect the points with a smooth curve.

From the graph, the x-intercepts appear to be 6 and -6.

Check Substitute the values for x in the original equation.

x2 - 36 = 0; x = 6 x2 - 36 = 0; x = -6

(6)2 - 36 � 0 (-6)2 - 36 � 0

36 - 36 � 0 36 - 36 � 0

0 = 0 ✓ 0 = 0 ✓

The solutions are 6 and -6.

x

y

O4 8

-20

20

40

-40

-8

x

y

O4 8

-20

20

40

-40

-8

Hint

When the coefficient of the x 2 -term is positive, the parabola will open upward.

When the coefficient of the x 2 -term is negative, the parabola will open downward.

Math Reasoning

Write Why are the x-intercepts substituted into the original equation?

The x-intercepts of the function are roots, or solutions, of the equation.


ENGLISH LEARNERS

For Example 1, explain the meaning of the word related. Say:

“Related means connected with. For example, we are related to the members of our family.”

Have students give other examples using the word related in a sentence. Sample: The notes are related by the key signature of the music.

670 Saxon Algebra 1

Example 1

These examples introduce the concept of solving a simple quadratic equation through graphing the related function.

Error Alert Some students may locate points on only one side of the parabola. Encourage them to use the axis of symmetry to plot points on both sides of the parabola.



a. x 2 - 25 = 0 The solutions are -5 and 5.

10

x

y

O8-8

-10


Lesson 100 671

b. -x2 - 2 = 0

SOLUTION

Graph the related function f(x) = -x2 - 2.

axis of symmetry: x = 0

vertex: (0, -2)

y-intercept: (0, -2)

two additional points: (1, -3) and (3, -11)

Reflect these two points across the axis of symmetry and connect them with a smooth curve.

From the graph, it can be seen that there is no x-intercept because the graph does not intersect the x-axis.

There is no real-number solution.

c. x2 + 16 = 8x

SOLUTION

Write the equation in standard form.

x2 - 8x + 16 = 0

Graph the related function f(x) = x2 - 8x + 16.

axis of symmetry: x = 4

vertex: (4, 0)

y-intercept: (0, 16)

two additional points: (2, 4) and (3, 1)

Reflect these two points across the axis of symmetry and connect them with a smooth curve.

From the graph, the x-intercept appears to be 4.

Check Substitute 4 for x in the original equation.

x2 - 8x + 16 = 0; x = 4

(4)2 - 8(4) + 16 � 0

16 - 32 + 16 � 0

0 = 0 ✓

The solution is 4.

xy

O2 4

-4

-8

-12

-4 -2

xy

O2 4

-4

-8

-12

-4 -2

x

y

O

8

12

16

4

2 6 8

x

y

O

8

12

16

4

2 6 8

Caution

When a parabola does not cross the x-axis, there is no real-number solution to the quadratic equation.


Lesson 100 671

b. 0 = - x 2 - 3 There are no real-number solutions.

x

y

O

2

2 4

-2

-2-4

-6

c. x 2 -3x -18 = 0 The solutions are 6 and -3.

x

y

O

-20

20

10

4 8-4-8


Saxon Algebra 1672

Example 2 Solving Quadratic Equations Using a Graphing

Calculator

Solve each equation by graphing the related function on a graphing calculator.

a. -6x - 9 = x2

SOLUTION


-x2 - 6x - 9 = 0

Graph the related function

f (x) = -x2 - 6x - 9.

The graph appears to have an x-intercept at -3.

Use the Table function to determine the zeros of this function.

The solution is -3.

b. -6x = -x2 - 13

SOLUTION


x2 - 6x + 13 = 0.

Graph the related function f (x) = x2 - 6x + 13.

The graph opens upward and does not intersect the x-axis.

There is no solution.

c. -3x2 + 5x = -7

Round to the nearest tenth.

SOLUTION


-3x2 + 5x + 7 = 0

Graph the related function f (x) = -3x2 + 5x + 7.

The graph appears to have x-intercepts at 3 and -1.

Use the Zero function to determine the zeros of this function. Round to the nearest tenth.

The solutions are x = 2.6 and -0.9.

Hint

For help with graphing quadratic functions, see Graphing Calculator Lab 8: Characteristics of Parabolas on p. 583.


INCLUSION

Use the following strategy with students who have diffi culty with abstract concept processing. Have students practice making tables in preparation for drawing graphs of quadratic equations. Help them decide which numbers to include in their tables. Have them fi nd the vertex fi rst using -b

___ 2a . They should understand that since a parabola is symmetrical, their list of numbers should also be symmetrical about the axis of symmetry.

Students should be able to fi nd the x-intercepts from the table. Any value of x for which the corresponding value of y is 0, is an x-intercept and thus a solution.

Once students have mastered making the table for the graph, help them plot the points and draw a smooth curve.

672 Saxon Algebra 1

Example 2

Once students have graphed quadratic functions by hand, the graphing calculator is a useful tool for quickly graphing and fi nding the solution(s) of quadratic equations.



a. x 2 + 28 = 11xThe solutions are 4 and 7. See Additional Answers.

b. x 2 + 2x + 7 = 0There is no solution. See Additional Answers.

c. 2 x 2 - 7x + 3 = 0The solutions are 3 and 0.5. See Additional Answers.

Example 3

The example provides a framework for a real-world application of solving quadratic equations by graphing.

Extend the Example

Suppose that Gill drops the baseball off of a platform twice the height of the original platform. How much longer does it take the baseball to hit the ground? about 0.8 second


Wendy dropped a golf ball from the top of a platform 102 feet off the ground. The height of the golf ball is described by the quadratic equation h = -16 t 2 + 102, where h is the height in feet and t is the time in seconds. Find the time t when the golf ball hits the ground. t ≈ 2.52 seconds

t

h

O

40

60

80

20

2 4-4



Lesson 100 673

Example 3 Application: Physics

Gill drops a baseball from the top of a platform 64 feet off the ground. The height of the baseball is described by the quadratic equation h = -16t2 + 64, where h is the height in feet and t is the time in seconds. Find the time t when the ball hits the ground.

SOLUTION

Graph the related function h(t) = -16t2 + 64 on a graphing calculator.

Height h is zero when the ball hits the ground. Use the Zero function of the graphing calculator to determine the zeros of this function.

There are two zeros for the given parabola: t = 2 and t = -2. Only values greater than or equal to zero are considered. So, t = 2 is the only solution.

The baseball hits the ground in 2 seconds.

Lesson Practice


a. 3x2 - 147 = 0 x = 7 and x = -7

b. 5x2 + 6 = 0 no solution

c. x2 - 10x + 25 = 0 x = 5


d. x2 + 64 = 16x x = 8

e. x2 + 4 = 2x no solution

f. Round to the nearest tenth: -7x2 + 3x = -7. x = -0.8 and 1.2

g. Marcus shot an arrow while standing on a platform. The path of its movement formed a parabola given by the quadratic equation h = -16t2 + 2t + 17, where h is the height in feet and t is the time in seconds. Find the time t when the arrow hits the ground. Round to the nearest hundredth. t = 1.10 seconds

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

(Ex 2)(Ex 2)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

Solve.

1. x(2x - 11) = 0 0, 11_2

2. 12 _ x - 6

= 4 _ x

x = -3

*3. Generalize Using the path of a ball thrown into the air as an example, describe in mathematical terms each part of the graph the path of the ball creates.

*4. Generalize What does the graph of a quadratic equation look like when there is no solution? one solution? two solutions? See Additional Answers.

(98)(98) (99)(99)

(100)(100)See Additional Answers.See Additional Answers.

(100)(100)

Hint

The time t is plotted on the x-axis. The height h is plotted on the y-axis.


Lesson 100 673

Lesson Practice

Problem a

Scaff olding To locate points on the parabola, fi rst have students fi nd the vertex and axis of symmetry. Then have them fi nd two x-values to the left or right of the vertex and refl ect them over the axis of symmetry.

Problem d

Error Alert Some students may not set the equation equal to zero. Remind them to write the equations in standard form.



“Where on the graph of a quadratic equation are the solutions located?” Sample: where the graph intersects the x-axis

“What is another name for the solutions of a quadratic equation?” roots or x-intercepts

Practice3


Problem 3


“What is the name of the graph of a quadratic equation?” parabola

“What is the highest or lowest point on the graph called?” vertex

“What are the points where the graph crosses the horizontal axis?” x-intercepts or solutions


Saxon Algebra 1674

5. Given that y varies directly with x, identify the constant of variation such that when x = 15, y = 30. k = 2

*6. Basketball Ramero shoots a basketball into the air. The ball’s movement forms a parabola given by the quadratic equation h = -16t2 + 7t + 7, where h is the height in feet and t is the time in seconds. Find the maximum height of the path the basketball makes and the time t when the basketball hits the ground. Round to the nearest hundredth. h = 7.77 feet and t = 0.92 seconds

*7. Multiple Choice What is the equation of the axis of symmetry of the parabola defined by y = 1 _

4 (x - 4)2 + 5? B

A x = 1 B x = 4 C x = 5 D x = -4

*8. Solve -7x2 - 10 = 0 by graphing. no solution

*9. Solve 6 _ x

= 8 _

x + 7 . x = 21

10. A deck of ten cards has 5 red and 5 black cards. Cards are replaced in the deck after each draw. Use an equation to find the probability of drawing a black card twice and rolling a 6 on a number cube. P(black, black, 6) = ( 1_

2 )2

· 1_6

= 1_24

11. Geometry The altitude of the right triangle divides the hypotenuse into segments of lengths x units and 5 units. To find x, solve the equation x + 5

_ 6 = 6 _

x . 4 units

*12. Multi-Step Henry starts working a half-hour before Martha. He can complete the job in 4 hours. Martha can complete the same job in 3 hours. a. Let t represent the total time they work together. In terms of t, how long does

Henry work? t + 0.5

b. Use an equation to find how long they work together to complete the job. 1 hour 30 minutes

c. How long does Henry work? 2 hours

13. Find the quotient of a2 + 10a - 24 __

a - 2 . 14. Simplify √ �� 49y5 . 7y2

√ � y

15. Profit An entrepreneur makes $3 profit on each object sold. She would like to make $270 plus or minus $30 total. What is the minimum and maximum number of objects she needs to sell? 80 objects,100 objects

16. Data Analysis A student knows there will be 4 tests that determine her semester grade. She wants her average to be an 85, plus or minus 5 points. What is the minimum and maximum number of points she needs to earn during the semester? 320 points, 360 points

17. Solve the equation �10x� - 3 = 87. {-9, 9}

18. Exercise Tom ran a total of 7x _ x2 + 3x - 18

miles in August and 2x + 1

_ 7x + 42

miles in

September. How many more miles did he run in August?

19. Graph the function y = 5x2 - 10x + 5.

*20. Verify A boundary line is a vertical line. The inequality contains a < symbol. Which half-plane should be shaded on the graph? Sample: Shade the half-plane to the left of the vertical line.

(Inv 7)(Inv 7)

(100)(100)

(100)(100)

(100)(100)

(99)(99)

(80)(80)

(99)(99) 5

x6

5

x6(99)(99)

(93)(93) (61)(61)(a + 12)(a + 12)

(94)(94)

(94)(94)

(94)(94)

(95)(95)

-2x2 + 54x + 3__7(x - 3)(x + 6)

miles -2x2 + 54x + 3__7(x - 3)(x + 6)

miles

(96)(96)

19.

x

y

O

28

2 4-2-4

19.

x

y

O

28

2 4-2-4

(97)(97)


CHALLENGE

Graph to fi nd the ordered pair that is a solution to all the equations.

2 x 2 + 3x - 5 = 0

3x - 2y = 3

4 x 2 + 3x - 4 = 3(1, 0)

674 Saxon Algebra 1

Problem 12

Extend the Problem

Suppose Martha and Henry work the same job, but that this time, Martha begins one hour before Henry starts. How long do they work together? How long does Martha work? 1 1 __ 7 hours; 2 1 __ 7 hours

Problem 15

Error Alert Make sure that students answer the question asked: the range of the number of objects she needs to sell, not the range of the amount of money she wants to make.


Lesson 100 675

21. Multiple Choice Which point does not satisfy the inequality x + 2y < 5? DA (0, 0) B (2, 1) C (3, -4) D (-1, 3)

*22. Ages A boy is b years old. His father is 23 years older than the boy. The product of their ages is 50. How old is each person? The boy is 2 years old and the father is 25 years old.

*23. Error Analysis Two students find the roots of 3x2 - 6x = 24. Which student is correct? Explain the error.

Student A Student B

3x2 - 6x = 24 3x2 - 6x = 243x(x - 2) = 24 3x2 - 6x - 24 = 03x = 0 x - 2 = 0 3(x2 - 2x - 8) = 0

x = 0 x = 2 3(x - 4)(x + 2) = 0x - 4 = 0 x + 2 = 0

x = 4 x = -2

24. Does the graph of y + 2x2 = 12 + x open upward or downward? downward

25. Do the side lengths 18, 80, and 82 form a Pythagorean triple? yes

26. Multi-Step The volume of a prism is 3x3 + 12x2 + 9x. What are the possible dimensions of the prism? a. Factor out common terms. 3x( x2 + 4x + 3)

b. Factor completely. 3x(x + 1)(x + 3)

c. Find the dimensions. 3x, x + 1, x + 3

27. Travel The Jackson family drove 480 miles on Saturday and 300 miles on Sunday. Their average rate on Sunday was 10 miles per hour less than their rate was on Saturday. Write a simplified expression that represents their total driving time.

28. Multi-Step At the carnival, a man says that he will guess your weight within 5 pounds. a. You weigh 120 pounds. Write an absolute-value inequality to show the range of

acceptable guesses. �x - 120� ≤ 5

b. Solve the inequality to find the actual range of acceptable guesses. 115 ≤ x ≤ 125

29. Verify If the numerator of a rational expression is a polynomial and the denominator of the rational expression is a different polynomial, will factoring the polynomials always provide a way to simplify the expression? Verify your answer by giving an example.

30. If a 9% decrease from the original price resulted in a new price of $227,500, what was the original price? $250,000

(97)(97)

(98)(98)

(98)(98)23. Student B; Sample: Student A did not put the equation in standard form before factoring.

23. Student B; Sample: Student A did not put the equation in standard form before factoring.

(84)(84)

(85)(85)

(87)(87)

(90)(90)

60(13x - 80) _

x(x - 10)

60(13x - 80) _

x(x - 10)

(91)(91)

(92)(92)29. no; Sample: If there are no common factors, the expression is in the simplest form; x2 - 4__

2x2 + 12x + 18=

(x - 2)(x + 2)

__2(x + 3)(x + 3)

29. no; Sample: If there are no common factors, the expression is in the simplest form; x2 - 4__

2x2 + 12x + 18=

(x - 2)(x + 2)

__2(x + 3)(x + 3)

(47)(47)


LOOKING FORWARD

Solving quadratic equations by graphing prepares students for

• Lesson 102 Solving Quadratic Equations Using Square Roots

• Lesson 104 Solving Quadratic Equations by Completing the Square


• Lesson 113 Interpreting the Discriminant

Lesson 100 675

Problem 21


“Which choice can you eliminate immediately? Why?” Sample: choice A; The result of any calculations with two zeros will be less than 5.

“Which choice would you eliminate next? Why?” Sample: choice C; 3 + 2(-4) is a lot less than 5.

“What is the correct choice?” Sample: choice D; It is equal to 5, which means it is not less than 5.

Problem 25

Error Alert Some students may believe that only multiples of the most common Pythagorean triple—3, 4, and 5—qualify as other Pythagorean triples. Remind students that any set of whole numbers that makes the Pythagorean Theorem true is considered a Pythagorean triple.

Problem 27


“What formula do you have to use to solve the problem?” distance equals rate times time: d = rt

“How can you use this formula to fi nd the driving time?” Rewrite the formula so that the time is isolated on one side of the equation: d __ r = t.

“What LCD will you use to simplify the equation?” (x)(x - 10)

Problem 28

Extend the Problem

After looking at you, the man believes you weigh between 123 and 135 pounds. What is the probability of him choosing a weight from this range that is within 5 pounds of your actual weight? 3

___ 13


10INVESTIGATION

Saxon Algebra 1676

A quadratic function is a function that can be written in the form f(x) = ax2 + bx + c, where a is not equal to 0.

In Investigation 6, linear functions were graphed as transformations of the parent function f(x) = x. Similarly, you can graph a quadratic function as a transformation of the quadratic parent function f(x) = x2.

Parameter Changes

Complete the table of values for f(x) = x2 and graph the quadratic parent function.

x f (x)

-3 (-3)2 = 9

-2 (-2)2 = 4

-1 (-1)2 = 1

0 02 = 0

1 12 = 1

2 22 = 4

3 32 = 9

As is the case for the linear parent function, the quadratic parent function f(x) = x2 can be written as f(x) = ax2, where a = 1. The graph changes when other values are substituted for a.

1. Graph y = x2 and y = 2x2 on the same set of axes. Compare the two graphs. See Additional Answers.

2. Graph y = x2 and y = 1 _ 2 x2 on the same set of axes. Compare the

two graphs. See Additional Answers.

3. Graph y = x2 and y = -x2 on the same set of axes. Compare the two graphs. See Additional Answers.

4. Generalize What is the effect of a on the graph of y = ax2?

5. Predict How will the graph of f(x) = 2 _ 3 x2 change in relation to the

quadratic parent function? Sample: It’s wider than the parent function.

6. Predict How will the graph of f(x) = -4x2 change in relation to the quadratic parent function?

The graph of a function of the form f(x) = ax2 always crosses the y-axis at (0, 0). When c ≠ 0, the graph of the function f(x) = ax2 + c does not pass through the point (0, 0).

7. Graph the quadratic parent function and the function f(x) = x2 + 1 on the same set of axes. Compare the two graphs. See Additional Answers.

8. Graph the quadratic parent function and the function f(x) = x2 - 2 on the same set of axes. See Additional Answers.

x

y

4

6

2

2 4-2-4

x

y

4

6

2

2 4-2-4

See Additional Answers.See Additional Answers.

Sample: It’s narrower than the parent function and opens downward.Sample: It’s narrower than the parent function and opens downward.

Transforming Quadratic Functions

Online Connection


Math Reasoning

Analyze If a = 0, would the function still be quadratic? If not, what type of function would f(x) be? no; linear

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INVESTIGATION RESOURCES

Reteaching Master Investigation 10

Technology Lab Master Investigation 10

MATH BACKGROUND

Understanding the connection between equations and graphs is a critical component to understanding functions. Identifying the parent function as a control allows comparisons. By comparing the graphs of quadratic functions with different parameters, the impact of the parameters in the equation on the graphs can be determined.

Specifi cally, in quadratic equations, a determines how wide or narrow the graph is. The constant term, c, determines the vertical shift of the graph.

676 Saxon Algebra 1

10INVESTIGATION

Materials

• graphing calculator• graph paper

Discuss

In this lesson, students learn the effects of parameter changes in quadratic functions. They graph the parent function and other transformations on the same set of axes to see the changes in the graphs.

Defi ne parent function and quadratic function.

Parameter Changes

Extend the Problem

Have students write equations for quadratic functions given a description of the graphs (i.e. opens down, does not cross the x-axis, and is narrower than the parent function).

Error AlertStudents may want to say that when 0 < a < 1, the graph is narrower than the parent function because the coeffi cient is small. Have students clarify that the greater a is, the narrower the graph.

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Investigation 10 677

9. Predict How will the graph of f(x) = x2 + 7 compare to the graph of the quadratic parent function? Sample: The new function moved up 7 units from the parent function.

Combinations of Parameter Changes

Predict How will each graph compare to the graph of the quadratic parent function? Verify your answer with a graphing calculator.

10. f(x) = -x2 + 2 See Additional Answers.

11. f(x) =1_2

x2 - 3 See Additional Answers.

Investigation Practice

Describe how the graph for the given values of a and c changes in relation to the graph of the quadratic parent function. Verify your answer with a graphing calculator.

a. f(x) = ax2 + c for a = 2 and c = 1 The graph is narrower and moved 1 unit up.

b. f(x) = ax2 + c for a = -3 and c = -2 The graph is narrower, opens downward, and moved 2 units down.

c. f(x) = ax2 + c for a =1_2 and c = 2 The graph is wider and moved 2 units up.

d. f(x) = ax2 + c for a = - 1_2

and c = -1 The graph opens downward, is wider, and moved 1 unit down.

Write an equation for the transformation described. Then graph the original function and the graph of the transformation on the same set of axes.

e. Shift f(x) = 2x2 - 4 up 2 units.

f. Shift f(x) = 3x2 + 5 down 4 units and open it downward. f(x) = -3x2 + 1;

x

4

4 8-4-8

3x2+ 5

3x2+ 1

e. f(x) = 2x2 - 2;

x

y8

4

2 4-2-4

-8

2x2 2

2x2 4

e. f(x) = 2x2 - 2;

x

y8

4

2 4-2-4

-8

2x2 2

2x2 4

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Discuss that children inherit some traits from their parents, but not all of them. Similarly, explain that a parent function has characteristics of any function of that type, but that other functions may look a little different. There are mathematical hints in the equation that indicate how different a transformed function will look.

Transforming quadratic functions will prepare students for


• Lesson 114 Graphing Square-Root Functions

• Lesson 119 Graphing and Comparing Linear, Quadratic, and Exponential Functions

Investigation 10 677

Discuss

The effect of each parameter in the equation on the graph of the function has been identifi ed. Students will describe the transformations in the graph when multiple parameters are changed.

Investigation Practice


Problem a

Scaff olding Have students tell whether the function is wider or narrower, opens upward or downward, and moves up or down.

Problem b

Error Alert Students may forget the effect of the negative value of a. Remind students that a parabola opens down when a < 0.

Problems e and f


“What value makes the graph move up or down?” c

“To move it down, will you add or subtract c?” subtract

“How do you make a parabola open downward?” Make a negative.

ENGLISH LEARNERS LOOKING FORWARD

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Documents

Lessons 91–100,Investigation 10 Math...Investigation 10 Block: 90-Minute Class Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Lesson 91 Lesson 92 Lesson 93 Lesson 94 Lesson 95 Cumulative Test