Class Notes for Math 251 - Saddleback College · Class Notes for Math 251 Beginning Algebra ... The textbook for Math 251 is Beginning Algebra, 11th Edition, by Lial, ... Section

Class Notes for

Math 251 Beginning Algebra

Prepared by

Stephanie Sorenson Saddleback College

The textbook for Math 251 is Beginning Algebra, 11th Edition, by Lial, Hornsby & McGinnis (Addison-Wesley, 2012). This set of class notes is designed to supplement the text.

Table of Contents 1.2 Exponents, Order of Operations, and Inequality ............................................................................................. 1

1.3 Variables, Expressions, and Equations ............................................................................................................. 4

1.4 Real Numbers and the Number Line ................................................................................................................ 8

1.5 Adding and Subtracting Real Numbers .......................................................................................................... 11

1.6 Multiplying & Dividing Real Numbers ............................................................................................................ 13

1.7 Properties of Real Numbers ........................................................................................................................... 18

1.8 Simplifying Expressions .................................................................................................................................. 21

2.1 The Addition Property of Equality .................................................................................................................. 23

2.2 The Multiplication Property of Equality ......................................................................................................... 25

2.3 More on Solving Linear Equations .................................................................................................................. 27

2.4 An Introduction to Applications of Linear Equations ..................................................................................... 32

2.5 Formulas and Additional Applications from Geometry ................................................................................. 36

2.6 Ratio, Proportion, and Percent ...................................................................................................................... 40

2.7 Further Applications of Linear Equations ....................................................................................................... 42

2.8 Solving Linear Inequalities .............................................................................................................................. 46

3.1 Linear Equations in Two Variables; The Rectangular Coordinate System ...................................................... 51

3.2 Graphing Linear Equations in Two Variables .................................................................................................. 54

3.3 The Slope of a Line ......................................................................................................................................... 57

3.4 Writing and Graphing Equations of Lines ....................................................................................................... 62

3.5 Graphing Linear Inequalities in Two Variables ............................................................................................... 68

3.6 Introduction to Functions ............................................................................................................................... 70

4.1 Solving Systems of Linear Equations by Graphing ......................................................................................... 76

4.2 Solving Systems of Linear Equations by Substitution ..................................................................................... 78

4.3 Solving Systems of Linear Equations by Elimination ...................................................................................... 80

4.4 Applications of Linear Systems ....................................................................................................................... 84

4.5 Solving Systems of Linear Inequalities ........................................................................................................... 89

5.1 The Product Rule and Power Rules for Exponents ......................................................................................... 91

5.2 Integer Exponents and the Quotient Rule ..................................................................................................... 97

5.3 Scientific Notation ........................................................................................................................................ 104

5.4 Adding and Subtracting Polynomials ........................................................................................................... 107

5.5 Multiplying Polynomials ............................................................................................................................... 111

5.6 Special Products ........................................................................................................................................... 114

5.7 Dividing Polynomials .................................................................................................................................... 117

6.1 The Greatest Common Factor; Factoring by Grouping ............................................................................... 121

6.2 Factoring Trinomials ..................................................................................................................................... 126

6.3 More on Factoring Trinomials ...................................................................................................................... 129

6.4 Special Factoring Techniques ....................................................................................................................... 133

6.5 Solving Quadratic Equations by Factoring .................................................................................................... 136

6.6 Applications of Quadratic Equations ............................................................................................................ 140

7.1 The Fundamental Property of Rational Expressions .................................................................................... 143

7.2 Multiplying and Dividing Rational Expressions ............................................................................................ 147

7.3 Least Common Denominators ...................................................................................................................... 149

7.4 Adding and Subtracting Rational Expressions .............................................................................................. 152

7.5 Complex Fractions ........................................................................................................................................ 156

7.6 Solving Equations with Rational Expressions ............................................................................................... 160

7.7 Applications of Rational Expressions ............................................................................................................ 164

7.8 Variation ....................................................................................................................................................... 171

8.1 Evaluating Roots ........................................................................................................................................... 174

8.2 Multiplying, Dividing, and Simplifying Radicals ............................................................................................ 179

8.3 Adding and Subtracting Radicals .................................................................................................................. 185

8.4 Rationalizing the Denominator .................................................................................................................... 187

8.5 More Simplifying and Operations with Radicals .......................................................................................... 190

8.6 Solving Equations with Radicals ................................................................................................................... 193

8.7 Using Rational Numbers as Exponents ......................................................................................................... 196

9.1 Solving Quadratic Equations by the Square Root Property ......................................................................... 199

9.2 Solving Quadratic Equations by Completing the Square.............................................................................. 202

9.3 Solving Quadratic Equations by the Quadratic Formula .............................................................................. 205

Section 1.2 1

1.2 Exponents, Order of Operations, and Inequality

Exponent Notation

The base is _______ ; The exponent is _______.

Example 1 Find the value of the exponential expression.

(a) (b) (c) (d)

Order of Operations

P

E

M

D

A

S

Example 2 Find the value of each expression.

(a) (b)

2 Section 1.2

Parentheses Exponents Multiplication Division Addition Subtraction

(c) (d)

Inequalities

The alligator always chomps on the bigger number!

is less than is greater than is not equal to Example 3 Determine whether each statement is true or false. If the statement is false, change the inequality sign to make a correct statement.

(a) (b)

(c)

Section 1.2 3

More Inequalities is less than OR equal to is greater than OR equal to Example 4 Determine whether each statement is true or false. If the statement is false, change the inequality sign to make a correct statement.

(a) (b)

(c) Example 5 Write each word statement in symbols.

(a) Nine is equal to eleven minus two.

(b) Fourteen is greater than twelve.

(c) Two is greater than or equal to two.

(d) Twelve is not equal to five.

4 Section 1.3

1.3 Variables, Expressions, and Equations

Consider the following question:

There are 40 enrolled students. How many petitioners must there be if there are 45 students in class on the first day? We can represent this problem with an equation as follows:

In algebra, instead of drawing a box with a question mark, we use a letter of the alphabet to represent the unknown number:

A __________________ is a symbol, usually a letter such as or , used to represent any unknown number. An __________________ _________________ is a sequence of numbers, variables, operation symbols ( ), and/or grouping symbols that DOES NOT INCLUDE an equality (=) or inequality ( ) sign.

The following is an algebraic expression:

Example 1 Find the value of the algebraic expression when . Interpret your result in terms of what represents.

Enrolled students

Petitioners Total students in class

Section 1.3 5

Use caution when writing expressions with exponents or finding their value.

is different than Example 2 Find the value of each algebraic expression if .

(a)

(b)

(c) Example 3 Find the value of each algebraic expression if and .

(a)

(b)

Exponent ONLY applies to

Exponent applies to the entire quantity in the parentheses

6 Section 1.3

Example 4 Write each word phrase as an algebraic expression, using as the variable.

(a) The sum of 3 and a number.

(b) A number minus 8.

(c) A number subtracted from 4.

(d) Eight subtracted from a number.

(e) The difference between a number and 2.

(f) The difference between 5 and a number.

(g) 14 times a number.

(h) Twelve divided by a number.

(i) The quotient of a number and 8.

(j) Nine multiplied by the sum of a number and 5.

(k) The product of 3 and five less than a number.

****************************************************************************** Let’s take another look at the original question posed at the beginning of this section.

“There are 40 enrolled students. How many petitioners must there be if there are 45 students in class on the first day?”

We already found that we could represent the problem with the following equation:

where represents the unknown number of petitioners.

An __________________ is a statement that two algebraic expressions are equal. An equation ALWAYS INCLUDES an equality (=) sign.

To _______________ an equation means to find the values of the variable that make the equation

true. Such values of the variable are called the __________________ of the equation.

Section 1.3 7

Example 5 Decide whether the given number is a solution of the equation.

(a)

(b)

Example 6 Write the statement as an equation. Use as the variable. Then find all solutions from the set

(a) The product of a number and 2 is 6.

(b) One less than twice a number is 15.

Example 7 Identify each as an expression or an equation.

(a)

(b)

(c)

8 Section 1.4

1.4 Real Numbers and the Number Line

Natural Numbers: Whole Numbers: Integers: Number Line: Rational Numbers: Irrational Numbers: Real Numbers:

0

Real Numbers Rational Numbers

Integers

Natural Numbers

Section 1.4 9

Ordering of Real Numbers For any two real numbers and ,

is less than if is to the ____________ of . Example 1 Determine whether the statement is true or false.

(a)

(b)

(c)

Additive Inverse The _________________________________ of a number is the number that is the same distance from 0 on the number line as , but on the opposite side of 0.

10 Section 1.4

Double Negative Rule For any real number , Absolute Value The absolute value, of a real number is: ______________________________________________________________________________ *Distance is NEVER negative. Therefore, the absolute value of a number is never negative. Example 2 Simplify.

(a) (b) (c)

Example 3 Decide whether each statement is true or false.

(a) (b)

(c) (d)

Section 1.5 11

1.5 Adding and Subtracting Real Numbers

Example 1 Perform each indicated operation. 1. 2. 3.

4.

5. 6.

7. 8. 9. 10.

Answers: 1) 2) 3) 4)

5)

6) 7) 8) 9) 10)

To add two numbers with the same sign: Mentally ignore the signs and add. The answer has the same sign as the given numbers.

eg. ; To add two numbers with different signs:

Mentally ignore the signs and subtract the smaller number from the larger number. The answer has the same sign as the bigger number in the problem. eg. ;

Subtraction is addition of the additive inverse: eg. eg. eg.

12 Section 1.5

Example 2 Write a numerical expression for each phrase and simplify.

a) The sum of and 5 and

b) added to the sum of and 11

c) The sum of and , increased by 14

d)

more than the sum of

and

e) The difference between and

f) less than

g) The sum of 12 and , decreased by 14

h) 19 less than the difference between 9 and

Section 1.6 13

1.6 Multiplying & Dividing Real Numbers

Multiplication For all real numbers ,

Rules of Signs: Example 1 Find all integer factors of 50. Division

Observe,

and

Any two numbers whose product is 1 are called reciprocals, or __________________________ _________________________.

The reciprocal of

is _______________. The reciprocal of

is ___________________.

What is the reciprocal of 0?? _________________________________________________. ****************************************************************************** Definition of Division: For any two real numbers and , ( )

In Words: divided by equals times the reciprocal of . ******************************************************************************

Note:

Rules of Signs:

14 Section 1.6

Example 2 Perform the indicated operation(s).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Example 3 Evaluate. Let (b) (c) (e) (f)

Section 1.6 15

Example 4 Evaluate. Let (b) (c) (e) (f) Example 5 Evaluate each expression for , , and .

16 Section 1.6

Words or phrases that mean MULTIPLY: Product, times, double, twice, of, as much as

Words or phrases that mean DIVIDE: Quotient, divided by, ratio of

********************************************************************

Example 6 Write a numerical expression for each phrase and simplify.

(l) The product of 4 and , added to

(m) Twice the product of and , subtracted from

(n) Three subtracted from the product of 4 and

(o) The product of and the difference between and

(p) The quotient of and the sum of and

(q) The sum of and , divided by the product of and

(r) Three-fourths of the sum of and 12

(s) of the product of and 5

Section 1.6 17

Example 7 Write each sentence as an equation, using as the variable. Then find the solution from the set of integers between and , inclusive.

(a) The quotient of a number and 4 is .

(b) 7 less than a number is 2.

18 Section 1.7

1.7 Properties of Real Numbers

Commutative Properties For all real numbers , (addition) (multiplication)

Example 1 Use the commutative properties to complete the equality.

(a) (b)

Associative Properties For all real numbers , (addition) (multiplication)

Example 2 Use the associative properties to complete the equality.

(a) (b)

Identity Properties

For all real numbers and (addition) and (multiplication)

Example 3 Use the identity properties to complete the equality.

(a) (b)

Section 1.7 19

Inverse Properties

For all real numbers and (addition)

and

(multiplication)

Example 4 Use the inverse properties to complete the equality.

(a) (b)

Distributive Property

For all real numbers and (addition) and (subtraction)

Example 5 Use the distributive property to complete the equality.

(a) (b) ****************************************************************************** Example 6 Decide whether each statement is an example of the commutative, associative, identity, inverse or distributive property.

(a)

(b) (c)

(d)

(e)

20 Section 1.7

Example 7 Find each sum or product without using a calculator.

(a)

(b)

(c)

Example 8 Simplify each expression.

(a)

(b)

(c)

Section 1.8 21

1.8 Simplifying Expressions

Example 1 Simplify each expression. (Use the properties of Section 1.7)

(a) (b) A term is a number, a variable, or a product or quotient of numbers and variables raised to powers. The numerical coefficient of a term is the number in front of the variables. Identify the numerical coefficient of each term:

Term Coefficient

Terms with exactly the same variables that have the same exponents (but possibly different coefficients) are called like terms. Identify each group of terms as like or unlike. and and and and

22 Section 1.8

Using the distributive property (“in the reverse direction”) is called combining like terms. For example,

It is important to be able to distinguish between terms (separated by a or ) and factors (multiplied):

however,

Example 2 Simplify each expression.

(a)

(b)

(c)

(d)

(e) Example 3 Translate the phrase into a mathematical expression. Use as the variable. Combine like terms when possible.

“A number multiplied by 5, subtracted from the sum of 14 and eight times the number”

Section 2.1 23

2.1 The Addition Property of Equality

Definition A linear equation in one variable can be written in the form

where and are real numbers, and . Recall, a solution of an equation is a number that makes the equation true when it replaces the variable. An equation is solved by finding its solution set, the set of all solutions. Equations with exactly the same solution sets are equivalent equations.

Example of equivalent equations: and Addition Property of Equality If and are real numbers, then the equations

and are equivalent equations. *Since subtraction is addition of the additive inverse, we can also subtract the same number from each side of the equation without changing the solution. Example 1 Solve each equation. a) b)

c)

d)

A B

24 Section 2.1

24

Example 2 Solve each equation. a) b) c) d)

Example 3 Write an equation using the information given in the problem. Use as the variable. Then solve the equation. “One added to three times a number is three less than four times the number. Find the number.”

Section 2.2 25

2.2 The Multiplication Property of Equality

Multiplication Property of Equality If and ( ) are real numbers, then the equations

and are equivalent equations. *Since division is multiplication by the reciprocal, we can also divide each side of an equation by the same nonzero number without changing the solution. Example 1 Solve each equation.

a) b)

c)

d)

e)

f)

A B

26 Section 2.2

Example 2 Solve each equation.

a) b)

c) d)

Example 3 Write an equation using the information given. Use as the variable. Then solve the equation.

“When a number is multiplied by , the result is 10. Find the number.”

Section 2.3 27

2.3 More on Solving Linear Equations

Steps for solving a linear equation:

Step 1: Simplify each side separately. Step 2: Collect all the variable term on one side. Step 3: Isolate the variable.

You can check your answer by plugging your solution into the original equation. Example 1 Solve. (a) (b) (c) (d)

****************************************************************************** Where did the word “Algebra” come from?

The word “algebra” comes from the work: Hisab al-jabr m’al muquabalah written by Muhammed ibn Musa Al-Khowarizmi in the 800’s A.D. Translated, the title means: The Science of Transposition and Cancellation

28 Section 2.3

If fractions appear in an equation, CLEAR THE FRACTIONS by multiplying both sides of the equation (or every term!) by the least common denominator. If decimals appear in an equation, CLEAR THE DECIMALS by multiplying both sides of the equation (or every term!) by either 10, 100, 1000, etc… Example 2 Solve.

Section 2.3 29

There are 3 types of linear equations: 1) An equation with exactly one solution is called a ____________________________________ .

Solution Set: where is the solution 2) An equation for which every real number is a solution is called an ______________________.

Solution Set: 3) An equation that has no solution is called a _________________________________________.

Solution Set:

Example 3 Solve each equation. Then state whether the equation is a conditional equation, an identity, or a contradiction.

(a)

30 Section 2.3

(b)

(c)

Example 4 Perform each translation.

a) Two numbers have a sum of 34. One of the numbers is . What expression represents the other number?

b) The product of two numbers is . One of the numbers is . What expression represents the other number?

Section 2.3 31

c) A football player gained yards on a punt return. On the next return, he gained 6 yd. What expression represents the number of yards he gained altogether?

d) A hockey player scored 42 goals in one season. He scored n goals in one game. What expression represents the number of goals he scored in the rest of the games?

e) Chandler is m years old. What expression represents his age 4 yr ago? 11 yr from now?

f) Claire has y dimes. Express the value of the dimes in cents.

g) A clerk has v dollars, all in $20 bills. What expression represents the number of $20 bills the clerk has?

h) A concert ticket costs p dollars for an adult and q dollars for a child. Find an expression that represents the total cost for 4 adults and 6 children.

32 Section 2.4

2.4 An Introduction to Applications of Linear Equations

Example 1 If 5 is added to the product of 9 and a number, the result is 19 less than the number. Find the number. (Assign the variable . Write an equation and solve.) Example 2 In the 2006 Winter Olympics in Torino, Italy, Canada won 5 more medals than Norway. The two countries won a total of 43 medals. How many medals did each country win? (Assign the variable . Write an equation and solve.)

Solving an Applied Problem

Step 1 Read the problem carefully until you understand what is given and what is to be found. Step 2 Assign a variable to represent the unknown value, using diagrams or tables as needed.

Write down what the variable represents. If necessary, express any other unknown values in terms of the variable.

Step 3 Write an equation using the variable expression(s). Step 4 Solve the equation. Step 5 State the answer. Does it seem reasonable? Step 6 Check the answer in the words of the original problem.

Section 2.4 33

Example 3 The owner of Terry’s Coffeehouse found that the number of orders for croissants was 1/6 the number of orders for muffins. If the total number for the two breakfast rolls was 56, how many orders were placed for croissants? (Assign the variable . Write an equation and solve.) Example 4 At a meeting of the local computer user group, each member brought two nonmembers. If a total of 27 people attended, how many were members and how many were nonmembers? (Assign the variable . Write an equation and solve.)

34 Section 2.4

Example 5 A piece of pipe is 50 in. long. It is cut into three pieces. The longest piece is 10 in. more than the middle-sized piece, and the shortest piece measures 5 in. less than the middle-sized piece. Find the lengths of the three pieces. (Assign the variable . Write an equation and solve.)

Example 6 Two back-to-back page numbers in this book have a sum of 569. What are the page numbers? (Assign the variable . Write an equation and solve.)

Definition: Two integers that differ by 1 are called consecutive integers. Examples: 5 and 6; 102 and 103; -3 and -2 If x represents an integer, then _________ represents the next larger consecutive integer.

Section 2.4 35

Example 7 Find two consecutive even integers such that six times the lesser added to the greater gives a sum of 86. (Assign the variable . Write an equation and solve.)

Definition: Consecutive even integers and consecutive odd integers differ by 2. Examples: 4 and 6 (consecutive even integers) 7 and 9 (consecutive odd integers) If x represents an even integer, then _________ represents the next larger even consecutive integer. If x represents an odd integer, then _________ represents the next larger odd consecutive integer.

36 Section 2.5

2.5 Formulas and Additional Applications from Geometry

Important Formulas from Geometry

Figure Formulas Illustration

*Square Perimeter: Area:

*Rectangle Perimeter: Area:

*Triangle Perimeter:

Area:

*Parallelogram Perimeter: Area:

*You must know the formulas for these figures

Example 1 Find the value of the remaining variable in the formula.

Section 2.5 37

Example 2 A farmer has 800 m of fencing material to enclose a rectangular field. The width of the field is 50 m less than the length. Find the dimensions of the field. Example 3 The longest side of a triangle is 1 in. longer than the medium side. The medium side is 5 in. longer than the shortest side. If the perimeter is 32 in., what are the lengths of the three sides?

38 Section 2.5

Example 4 The area of a triangle is 120 m2. The height is 24 m. Find the length of the base of the triangle.

Geometry Review

Type of Angles Measurement Figure

Vertical Equal Measure

Complementary Sum to

Supplementary Sum to

Example 5 Find the measure of each marked angle in the figure.

Section 2.5 39

Example 6 Solve each formula for the specified variable. for for for

for

40 Section 2.6

2.6 Ratio, Proportion, and Percent A ratio is a comparison of two quantities (with the same units) using a quotient:

to

Ratios are often used when comparison shopping at the grocery store! We set up the ratio of the price of the item to the number of units on the label to obtain the price per unit. The lowest price per unit is the better value. Example 1 The local supermarket charges the following prices for a jug of pancake syrup. Which size is the best buy?

Size Price

12 oz $1.89

24 oz $2.79

36 oz $3.89

A proportion says that two ratios are equal

is to as is to

Beginning with the proportion,

and multiplying each side by the common denominator yields:

We see that the products and can also be found by multiplying diagonally. We call this “cross multiplying”.

Section 2.6 41

Example 2 Decide whether each proportion is true or false.

Example 3 Solve the equation

Example 4 If 12 gallons of gasoline costs $37.68, how much would 16.5 gallons of the same fuel cost?

42 Section 2.7

2.7 Further Applications of Linear Equations Percents should be represented as decimals. For example, or Interest Problems:

(Principal) x (% Interest Rate) = Annual Interest Example 1 Find the annual interest if $5000 is invested at 4%. Example 2 With income earned by selling a patent, an engineer invests some money at 5% and $3000 more than twice as much at 8%. The total annual income from the investments is $1710. Find the amount invested at 5%. Assign a variable:

Write an equation & solve:

Principal % Interest Annual Interest

Section 2.7 43

Mixture Problems:

(Amount of Solution) x (% concentration of substance X) = Amount of substance X Example 3 What is the amount of pure acid in 40 L of a 16% acid solution? Example 4 A certain metal is 40% copper. How many kilograms of this metal must be mixed with 80 kg of a metal that is 70% copper to get a metal that is 50% copper? Assign a variable:


Amt of Metal (kg) % Copper Amt of Copper (kg)

Substance X Solution

44 Section 2.7

Money Denominations Problems: (Number) x (Value of one item) = Total Value Example 5 A man has $2.55 in quarters and nickels. He has 9 more nickels than quarters. How many nickels and how many quarters does he have? Assign a variable:

Number of coins Coin Value Total Value

Write an equation & solve: Distance-Rate-Time Problems: (Rate) x (Time) = Distance Example 6 A new world record in the men’s 100-m dash was set in 2005 by Asafa Powell of Jamaica, who ran it in 9.77 sec. What was his speed in meters per second?

=

= 65 mph

Section 2.7 45

Example 7 Two airplanes leave Boston at 12:00 noon and fly in opposite directions. If one flies at 410 mph and the other 120 mph faster, how long will it take them to be 3290 mi apart? Assign a variable:

Rate Time Distance

Write an equation & solve: Example 8 Two buses left the downtown terminal, traveling in opposite directions. One had an average speed of 10 mph more than the other. Twelve minutes (1/5 hour) later, they were 12 mi apart. What were their speeds? Assign a variable:

Rate Time Distance


46 Section 2.8

2.8 Solving Linear Inequalities

Interval Notation

Inequality Interval

Notation Graph

All real numbers

Example 1 Write an inequality involving the variable that describes the set of numbers graphed.

Example 2 Write each inequality in interval notation, and graph the interval.

a) b) c)

a b

a b

a b

[

[ ]

] (

)

) a b (

a [

a (

b ]

) b

) 4 2 8

] (

Section 2.8 47

A linear inequality in one variable can be written in the form

where , and are real numbers, with . Consider the (true) inequality: Now add 4 to both sides: Is the inequality still true?? Again, consider the (true) inequality: Now subtract 4 from both sides: Is the inequality still true??

Addition Property of Inequality For any real numbers , , and , the inequalities

and are equivalent. Also,

and are equivalent. Example 1 Solve the inequality. Write the solution set in interval notation and graph it.

48 Section 2.8

Consider the (true) inequality: Now multiply both sides by 4: Is the inequality still true?? Again, consider the (true) inequality: Now multiply both sides by : Is the inequality still true??

Multiplication Property of Inequality For any real numbers , , and , with ,

1. If is positive, and are equivalent.

2. If is negative, and are equivalent.

Since division is multiplication by the reciprocal, the property applies to division:

1. If is positive, and

are equivalent.

2. If is negative, and

are equivalent.

In words: When multiplying or dividing by a negative #, remember to ____________________________.

Example 2 Solve the inequality. Write the solution set in interval notation and graph it.

(a) (b)

Section 2.8 49

Example 3 Solve the inequality. Write the solution set in interval notation and graph it. (a) (b)

When solving three-part inequalities, the goal is to isolate in the middle:

Keep in mind that whatever you do to the middle expression, you must do to each of the three parts of the inequality!

Also, sometimes through solving a three-part inequality, the signs reverse and you get the following: Do not leave your answer in this form!! You need to reverse the order so that all the inequalities point in the “less than” direction: Example 4 Solve the inequality. Write the solution set in interval notation and graph it.

(a) (b)

50 Section 2.8

Phrase Inequality

A number is more than 4.

A number is less than .

A number is at least 6.

A number is at most 8.

Example 5 You score 98, 86 and 88 on the first three exams. If you want to average an A on all the exams (a score of at least 90), what score must you get on the 4th exam?

Section 3.1 51

3.1 Linear Equations in Two Variables; The Rectangular Coordinate System

A linear equation in two variables is an equation that can be written in the form:

where and are real numbers and and are not both 0. Example 1 Write the following equations in the form . Then identify and .

(a)

(b)

(c) A solution of a linear equation is a pair of numbers, and , that make the equation true. We write a solution of a linear equation as an ordered pair . A linear equation in two variables has infinitely many solutions. Example 2 List a few solutions of the linear equation .

(Note that there are infinitely many solutions!)

52 Section 3.1

Example 3 Decide whether the ordered pair is a solution of the equation . Example 4 Complete each ordered pair for the equation .

(a) (b) In order to graph the solutions to linear equations in two variables, we need a number line for each value and . The horizontal number line used to represent the value of a solution is called the - axis. The vertical number line used to represent the value of a solution is called the - axis. When both axes are represented together, we call this the rectangular (or Cartesian) coordinate system. The numbers in the ordered pairs are called the coordinates of the point.

Section 3.1 53

Example 5 Complete the table of values for the equation. Then write the results as ordered pairs and plot the solutions.

Example 6 Complete the table of values for the equation. Then write the results as ordered pairs and plot the solutions.

0

0

3

54 Section 3.2

3.2 Graphing Linear Equations in Two Variables

Example 1 Graph .

Example 2

Graph

.

Graph of a Linear Equation The graph of any linear equation in two variables, , is a straight line.

The x-intercept is where a graph crosses the -axis.

To find -intercepts, _________________________________________________________.

The y-intercept is where a graph crosses the -axis.

To find -intercepts, _________________________________________________________.

Section 3.2 55

Example 3 Find the intercepts for . Then sketch the graph.

Example 4 Find the intercepts for . (What do you notice?) Then sketch the graph.

The graph of a linear equation of the form

passes through the origin , and so only has one intercept. Thus, to sketch the graph:

56 Section 3.2

Example 5 Graph the line:

x y

Example 6 Graph the line:

x y

The graph of , where is a real number, is what kind of line?

________________________________

----------------------------------------------------------------------------------------------------------------------

The graph of , where is a real number, is what kind of line?

________________________________

Section 3.3 57

3.3 The Slope of a Line

Example 1 Find the slopes of the following lines.

Slope of a Line (Geometric Interpretation)

= Run

Rise

in change horizontal

in change vertical

x

y

P 1)(0,

Q (3,4)

P 1,2)(

Q 5)(4,

Up units

Right units

Down units

Right units

5)3,(

1)(2,

2)4,(

6)(3,

58 Section 3.3

Let’s derive a formula for slope! ****************************************************************************** Slope Formula The slope of the line through the points and is

Example 2 Find the slope of the line through the following two points:

(a) and

(b) and

(c) and

(d) and

Section 3.3 59

Example 3 Use the slope formula to find the slope of the line . (Hint: You will need two points on this line!)

Example 4 Find the slopes of the lines:

(a)

(b)

Positive Slope Line slants UPWARD

from left to right

Negative Slope Line slants DOWNWARD

from left to right

Slope = 0 Line is HORIZONTAL

Slope is undefined Line is VERTICAL

How to Find the Slope of a Line from its Equation

Method #1 Method #2 (FASTEST method!) Step 1 Find any two points on the line. Step 1 Solve the equation for . Step 2 Use the slope formula. Step 2 The slope is the coefficient of .

60 Section 3.3

Parallel lines have the same slope, but do not intersect (ie. have different -intercepts.) Example 5 Is the line that contains the points and parallel to the line that contains the points and ? Perpendicular lines intersect at a right angle ( ). Their slopes are negative reciprocals. Example 6 Is the line that contains the points and perpendicular to the line that contains the points and ?

Slope Slope

Slope Slope

or

Section 3.3 61

Example 7 Decide whether the pair of lines is parallel, perpendicular, or neither.

62 Section 3.4

3.4 Writing and Graphing Equations of Lines

Recall from the previous section, if given the equation of a line,

we may find the slope of the line as follows:

1)

2) Example 1 Given the following equation of a line,

(a) Solve for .

(b) What is the slope of the line?

(c) Find the -intercept of the line.

Slope-Intercept Form of the Equation of a Line

Section 3.4 63

Example 2 Identify the slope and -intercept of each line.

(a) Slope: _________ -int: _________

(b)

Slope: _________ -int: _________

(c) Slope: _________ -int: _________

(d) Slope: _________ -int: _________ Example 3 Find an equation of the line with slope and -intercept .

64 Section 3.4

Example 4 Example 5 Graph the line with -intercept Graph the line passing through the

and slope

. point with slope .

Example 6 Example 7 Graph the line passing through Graph the line passing through the the point with . point with undefined slope. Example 8 Graph the equation of the line by using the slope and -intercept.

Section 3.4 65

Example 9 Write an equation in slope-intercept form of the line passing through the point with slope . In this last example, we were given a point on the line, and the slope of the line. Since we didn’t know the -intercept, we could not immediately write the equation of the line in slope-intercept form. We had to do a little bit of work to first find the -intercept.

Wouldn’t it be nice if there was a form of the equation of a line that didn’t require that we know the -intercept?

Let’s derive another form of the equation of a line! Suppose you are given that the point is on a line with slope .

66 Section 3.4

Point-Slope Form of the Equation of a Line

Example 10

Write an equation for the line passing through the point with slope

.

Give the final answer in slope-intercept form.

Example 11 Write an equation for the line passing through the point with undefined slope.

Section 3.4 67

Example 12 Write an equation for the line passing through the points and .

Give the final answer in slope-intercept form. Example 13 Write an equation for the line passing through the points and . Example 14 Write an equation of the line passing through the point and perpendicular to .

68 Section 3.5

3.5 Graphing Linear Inequalities in Two Variables

How To Graph Linear Inequalities: , ,

1) Graph the boundary line

For and , use a dashed line to exclude the points on the boundary.

For and , use a solid line to include the points on the boundary.

2) Choose a test point not on the line.

If the point satisfies the original inequality, it is a solution! So shade the side of the boundary line containing the test point.

However, if the point causes the inequality to be false, it is not a solution. So shade the side of the boundary line that doesn’t contain the test point.

Example 1 Graph the solution set Example 2 Graph the solution set

Section 3.5 69

Example 3 Graph the solution set Example 4 Graph the solution set Example 5 Graph the solution set

70 Section 3.6

3.6 Introduction to Functions

When the elements in one set are linked to elements in a second set, we call this a relation.

If is an element in the domain and is an element in the range, and if a relation exists between and , then we say that depends on , and we write yx . We can also represent this relation as

a set of ordered pairs ),( yx , where represents the input and represents the output:

{(Dog,11), (Cat,11), (Duck,10), (Lion,10), (Rabbit,7)}

Any set of ordered pairs is a relation!

Example 1

(a) Represent this relation as a set of ordered pairs ),( yx , where represents the input and

represents the output.

(b) Identify the domain and range of this relation.

11

10

7

Dog Cat

Duck Lion

Rabbit

Animal

Life Expectancy (years)

Set of Inputs=Domain Set of Outputs=Range

3

4

6

0

Set A Set B

1 5 8

10 12

Section 3.6 71

Example 2 Identify the domain and range of the relation: {(2,4), (2, -3), (1, 5)} Domain: Range:

Example 3 Determine whether each relation is a function.

(a)

(b)

(c) {(-2,8), (-1,1), (0,0), (1,1), (2,8)}

(d) {(5,2), (5,1), (3,4)}

A function is a special relation. It is a set of ordered pairs in which each input corresponds to exactly one output.

11

10

7

Dog Cat

Duck Lion

Rabbit

Animal

Life Expectancy (years)

Domain Range

3

4

6

0

1 5 8

10 12

72 Section 3.6

Most useful functions have an infinite number of ordered pairs and are usually defined with equations that tell how to get the outputs given the inputs.

Everyday Examples of Functions

1. The cost in dollars charged by an express mail company is a function of the weight in pounds determined by the equation:

2. In Cedar Rapids, Iowa, the sales tax is 7% of the price of an item. The tax on a particular item is a function of the price : .

3. The distance a car moving at 45 mph travels is a function of the time

One way to determine if a relation is a function is to look at the graph of the equation!

x

y

122 yx

x

y

2 xy

x

y

2xy

Section 3.6 73

Example 4 Determine whether each relation is a function.

(a) (b) (b)

(c) (d)

Vertical Line Test

If a vertical line intersects a graph in more than one point, then the graph is not the graph of a function.

Intersects in one point Intersects in more

than one point

Passes the test Function

Fails the test Not a Function

74 Section 3.6

Function Notation

The letters _____, _____, and _____ are commonly used to name functions. For example, since the equation

34 xy

describes a function, we may use function notation:

34)( xxf

If , then ______________________________________________. The statement says that the value of is _______ when is _______.

The statement also indicates that the point lies on the graph of

For functions, the notations and can be used interchangeably.

Say: “ of ”

INPUT

OUTPUT

Section 3.6 75

Example 5 For each function, find the following. (a) (b)

76 Section 4.1

4.1 Solving Systems of Linear Equations by Graphing

Recall, an equation in two variables is linear provided it can be written in the form: CByAx

Example 1 Decide whether the ordered pair )1,4( is a solution of each system.

(a) 352

1465

yx

yx

(b) 3

3

yx

yx

A system of two equations containing two variables represents a pair of lines. The points of intersection are the solutions of the system.

Hence, we can look at their graphs to solve the system! Their graphs can appear in one of 3 ways:

Intersect at exactly one point

One solution:

Parallel

No solution:

(Inconsistent System)

They are the same line

Infinite number of solutions:

(dependent equations)

A system of linear equations is a grouping of two or more linear equations. A solution of a system of linear equations is an ordered pair that makes both equations true at the same time. A solution of an equation is said to satisfy the equation.

Section 4.1 77

Example 2 Solve the system by graphing.

24

42

yx

yx


1242

42

yx

yx


824

42

yx

yx

78 Section 4.2

4.2 Solving Systems of Linear Equations by Substitution

Solving a system by graphing is very difficult, especially without graph paper! Thus, we prefer algebraic methods for solving systems of linear equations. There are two such algebraic methods:

1. Substitution 2. Elimination

We look at the method of substitution in this section.

Example 1 Solve the system by the substitution method.

yx

yx

2

1272


1152

41

yx

yx

Substitution Method

Step 1 Solve one equation for either variable. Step 2 Substitute for that variable in the other equation. Step 3 Solve the equation from Step 2. Step 4 Substitute the result from Step 3 into the equation from Step 1 to find the value of the other variable. Step 5 Check the solution in both of the original equations.

Section 4.2 79


8216

48

yx

xy


28124

73

yx

yx

Example 5 Solve the system by the substitution method. (Hint: Clear fractions)

222

1

3

1

3

1

2

1

yx

yx

80 Section 4.3

4.3 Solving Systems of Linear Equations by Elimination

Solving a system by graphing is very difficult, especially without graph paper! Thus, we prefer algebraic methods for solving systems of linear equations. There are two such algebraic methods:

3. Substitution 4. Elimination

We look at the method of elimination in this section.

Example 1 Solve the system by the elimination method. 3 7

2 3

x y

x y

Solving a Linear System by Elimination

Step 1 Write both equations in standard form, Ax By C

Step 2 Transform the equations as needed so that the coefficients of one pair of variable terms are opposites. Multiply one or both equations by appropriate numbers so that the sum of the coefficients of either the x- or y- terms is 0.

Step 3 Add the new equations to eliminate a variable. Step 4 Solve the equation from Step 3 for the remaining variable.

Step 5 Substitute the result from Step 4 into either of the original equations, and solve for the other variable.

Step 6 Check the solution in both of the original equations. Then write the solution set.

Section 4.3 81

Example 2 Solve the system by the elimination method.

2

2 10

x y

x y


4 5 18

3 2 2

x y

x y

82 Section 4.3


3 8 4

6 9 2

y x

x y

Section 4.3 83


3 7

6 2 5

x y

x y


2 5 1

4 10 2

x y

x y

84 Section 4.4

4.4 Applications of Linear Systems

Example 1 In 2008, spending on sporting equipment and recreational transport totaled $51,879 million. Spending on recreational transport exceeded spending on sporting equipment by $2113 million. How much was spent on each? Assign variables: Write a system of two equations & solve:

Solving an Applied Problem with Two Variables

Step 1 Read the problem carefully until you understand what is given and what is to be found. Step 2 Assign variables to represent the unknown values, using diagrams or tables as needed.

Write down what each variable represents. Step 3 Write two equations using both variables. Step 4 Solve the system of two equations. Step 5 State the answer. Does it seem reasonable? Step 6 Check the answer in the words of the original problem.

Section 4.4 85

Example 2 For a production of Wicked at the Pantages Theatre in Los Angeles, main floor tickets cost $96 and mid-priced mezzanine tickets cost $58. If a group of 18 people attended the show and spent a total of $1234 for their tickets, how many of each kind of ticket did they buy? Assign variables:

# of Tickets

Price per ticket (in dollars)

Total value

Write a system of two equations & solve:

86 Section 4.4

Example 3 How many liters of a 25% alcohol solution must be mixed with a 12% solution to get 13 L of a 15% solution? Assign variables:

Liters of Solution % Concentration

of alcohol Liters of pure alcohol


Section 4.4 87

Example 4 Two cars that were 450 mi apart traveled toward each other. They met after 5 hr. If one car traveled twice as fast as the other, what were their rates? Draw diagram/Assign variables:

Rate Time Distance


88 Section 4.4

Water/Air Current Problems:

Example 5 In two hours, Abby can row 4 mi against the current or 20 mi with the current. Find the speed of the current and Abby’s speed in still water. Assign variables:

Rate Time Distance


Downstream (with current)

Rate

Upstream (against current)

Rate

Section 4.5 89

4.5 Solving Systems of Linear Inequalities

To Solve a System of Linear Inequalities:

Step 1: Graph the first inequality. a) Graph the boundary line (Dashed or Solid?) b) Test a point not on the line to shade the correct region.

Step 2: Graph the second inequality.

a) Graph the boundary line (Dashed or Solid?) b) Test a point not on the line to shade the correct region.

Step 3: The solution is the INTERSECTION of the two regions.

Example 1 Graph the solution set of the system: 3 6

2 8

x y

x y

Step 1: Graph Step 2: Graph Step 3: The solution is the intersection of the two regions. Make this region stand out by shading it even darker!

90 Section 4.5

Example 2 Graph the solution set of the system:

2 0

3 4 12

x y

x y

Example 3 Graph the solution set of the system:

2

1 0

x

x

Section 5.1 91

5.1 The Product Rule and Power Rules for Exponents

Recall from Section 1.2: In the exponential expression , _________ is the base and _________ is the exponent or power. Let’s evaluate it! Example 1 Name the base and the exponent. Then evaluate.

Exponential Expression

Base Exponent Evaluate

****************************************************************************** Now, evaluate the product: This suggests the following rule…

Product Rule for Exponents For any positive integers and , In words:

92 Section 5.1

Product Rule for Exponents: Example 2 Use the product rule, if possible, to simplify each expression. Write each answer in exponential form.

(a) (b)

(c) (d)

(e) (f)

Note the difference between adding and multiplying: *********************************************************************************** Now, evaluate: This suggests the following rule…

Power Rule (a) for Exponents For any positive integers and , In words:

Section 5.1 93

Power Rule (a) for Exponents: Example 3 Use power rule (a) for exponents to simplify each expression. Write each answer in exponential form.

(a) (b)

Note the difference between the product rule and power rule (a): *********************************************************************************** Now, evaluate: This suggests the following rule…

Power Rule (b) for Exponents For any positive integer , In words:

94 Section 5.1

Power Rule (b) for Exponents: Example 4 Use power rule (b) for exponents to simplify each expression. Write each answer in exponential form.

(a) (b)

(c) (d)

Note that power rule (b) does not apply to a sum. *********************************************************************************** Now, evaluate:

This suggests the following rule…

Power Rule (c) for Exponents For any positive integer ,

In words:

Section 5.1 95

Power Rule (c) for Exponents:

Example 5 Use power rule (c) for exponents to simplify each expression. Write each answer in exponential form. Assume any variables in the denominator are non-zero.

Summary of Rules for Exponents For positive integers and , 1) 3) 2) 4) Example 6 Simplify each expression by using a combination of the rules of exponents.

96 Section 5.1

Example 7 Find an expression that represents the area of the figure. Example 8 Find an expression that represents the volume of the figure.

Section 5.2 97

5.2 Integer Exponents and the Quotient Rule

Observe the following: Do you see a pattern? Using this pattern we should be able to continue the list to include zero and negative exponents… We just need to make sure the Rules for Exponents from Section 5.1 will apply to this definition of zero and negative exponents: If we define _____ then: Using the Power Rule for Exponents: Do these agree?? If we define ____________ then: Using the Power Rule for Exponents: Do these agree??

98 Section 5.2

Thus we define zero and negative exponents as follows: Zero Exponent For any nonzero real number ,

Negative Exponents For any nonzero real number and any integer , In Words: raised to is ____________________________________________ raised to . Example 1 Evaluate. Assume that all variables represent nonzero real numbers. (a) (b) (c)

(d) (e)

(f)

(g)

(h)

Section 5.2 99

Consider the following:

Changing from Negative to Positive Exponents For any nonzero numbers and and any integers and ,

and as we already observed from the definition:

Example 2 Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers.

(a)

(b)

(c)

100 Section 5.2

Be careful if the exponents occur in a sum or difference of terms.

However,

****************************************************************************** Now, consider the quotient of two exponential expressions:

Also,

This suggests the following rule…

Quotient Rule for Exponents For any nonzero real number and any integers and ,

In words:

Section 5.2 101

Quotient Rule for Exponents:

Example 3 Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers.

(a)

(b)

(c)

(d)

(e)

(f)

102 Section 5.2

Summary of Definitions and Rules for Exponents For any integers and ,

Zero Exponent

Power Rule (b)

Negative Exponent

Power Rule (c)

Product Rule

Negative-to-Positive Rules

Quotient Rule

Power Rule (a)

Example 4 Simplify. Assume that variables represent nonzero real numbers.

(a)

(b)

Section 5.2 103

(c)

(d)

104 Section 5.3

5.3 Scientific Notation

Use a scientific calculator to compute the following: 5,000,000 6,000,000 = ___________________ Very large and very small numbers often occur in the sciences. For such numbers, we use scientific notation simply because it’s easier to work with AND most calculators cannot display enough digits to give the answer in decimal form!

Example 1 Circle the numbers written in scientific notation.

21.3 10 32 10

20.4 10 321 10

653.2 10 465.99 10

Observe the following:

Do you notice a pattern??

How to write a number in decimal notation (without exponents):

For numbers greater than 10, the exponent n is positive and equal to the number of places the decimal point in the number a moves to the right.

For numbers less than 1, the exponent n is negative and equal to the number of places the decimal point in the number a moves to the left.

The form for scientific notation is: where 1 10a and is an integer.

scientific notation decimal notation (without exponents)

scientific notation decimal notation (without exponents)

Section 5.3 105

Example 2 Write in decimal notation.

87.4 10 ___________________________ 22 10

___________________________

63.54 10 ___________________________ 41.333 10 ________________________

22.5 10 ___________________________

18 10 ___________________________

How to write a number in scientific notation:

First write (where by placing the decimal after the first nonzero digit. For numbers greater than 10, the exponent n on the base 10 should be positive and equal to the number of places the decimal point in would need to move to the right to yield the number without exponents.

For numbers less than 1, the exponent n on the base 10 should be negative and equal to the number of places the decimal point in would need to move to the left to yield the number without exponents.

Example 3 Write each number in scientific notation. 0.00035 _____________________ 280,000 _____________________

358 _____________________ 0.125 _____________________ 0.0000056 _____________________ 43,000,000 _____________________

decimal notation (without exponents) scientific notation

decimal notation (without exponents) scientific notation

106 Section 5.3

Example 4 Perform the indicated operations. Write each answer

(a) in scientific notation, and (b) without exponents (decimal notation).

Section 5.4 107

5.4 Adding and Subtracting Polynomials

Name the coefficient of each term in the expression: 3 22 5 4x x x

Simplify by adding like terms: 2 2 2 23 5 10x xy x xy

The degree of a term is the sum of the exponents on the variables:

Examples: 72x degree:_______________ 5 degree:_______________

2 43x y degree:_______________

The degree of a polynomial is the greatest degree of any nonzero term of the polynomial.

Examples: 2 51 4 3x x degree:_______________ 4 8x degree:_______________

3 5 63 4 20x y xy degree:_______________

A polynomial in is a term or the sum of a finite number of terms of the form , for any real number and any whole number .

Circle the polynomials:

108 Section 5.4

A polynomial with only one term is called a ______________________. Example: A polynomial with exactly two terms is called a _______________________. Example: A polynomial with exactly three terms is called a _______________________. Example: Example 1 For each polynomial, first simplify, if possible, and write it in descending powers of the variable. Then give the degree of the resulting polynomial and tell whether it is a monomial, binomial, trinomial, or none of these.

(a) 5 3 5 26 4 8 10p p p p

(b) 6 64 1

5 5r r

Example 2

Evaluate the polynomial 32 8 6y y at 1y .

Section 5.4 109

Example 3 Add. Horizontally: Vertically: Example 4 Subtract. Horizontally: Vertically:

To add two polynomials, add like terms.

110 Section 5.4

Example 5 Find a polynomial that represents the perimeter of the rectangle. Example 6 Graph the equation by completing the table of values.

0

1

2

Section 5.5 111

5.5 Multiplying Polynomials

To find the product of polynomials, we use the distributive property: Example 1 Find the product. (a) (b) Longer Method

(c) Short-cut Method

SHORT-CUT: To multiply two polynomials, multiply each term of the second polynomial by each term of the first polynomial and add the products.

112 Section 5.5

Example 2

Multiply Vertically: 23 4 5

4

x x

x

Example 3 Use the rectangle method to find the product .

Example 4 Choose any method to multiply:

Section 5.5 113

2( )( )ax b cx d acx adx bcx bd

Example 5 Use the FOIL method to multiply. (a) (4 3)( 2)x x

(b) 2( 6)m

(c) ( 4 )(2 3 )y x y x

(d) 33 ( 2)(2 1)x x x

Inner

First

Outer

Last

F. O. I. L.

114 Section 5.6

5.6 Special Products

Multiply:

Square of a Binomial*

*Note: It is your choice whether you want to be able to recognize and use these formulas, or just FOIL as usual.

Example 1 Multiply.

(a) (b)

(c)

(d)

Section 5.6 115

Multiply:

Product of the Sum and Difference of Two Terms*

*Note: It is your choice whether you want to be able to recognize and use this formula, or just FOIL as usual.

Example 2 Multiply.

(a) (b)

(c)

(d)

116 Section 5.6

Example 3 Multiply.

(a)

(b)

Section 5.7 117

5.7 Dividing Polynomials

1. Division by a Monomial

Example 1 Divide by . Example 2 Divide

Example 3 Divide

118 Section 5.7

2. Division by a Non-Monomial

Question: How should we divide

Answer: Recall the long division process:

Example 4 Divide

Section 5.7 119

Example 5 Divide

Example 6 Divide

120 Section 5.7

Example 7 Divide

Example 8 Divide by .

Section 6.1 121

6.1 The Greatest Common Factor; Factoring by Grouping To factor means “to write a quantity as a product.” Factoring is the opposite of multiplying. List all the positive factors of 12: List all the positive factor of 18: From the lists above, identify the greatest common factor (GCF) of 12 and 18: __________ Example 1 Find the greatest common factor (GCF) for each list of numbers.

(a) 20, 64

(b) 12, 18, 26, 32

(c) 12, 13, 14

122 Section 6.1

Fact: The greatest common factor (GCF) will be the product of every common prime factor raised to the smallest exponent.

Example 2 Find the greatest common factor of 72 and 240 by first factoring the numbers into prime factors. (A factor tree is very helpful!) Example 3 Find the greatest common factor for each list of terms.

(a) , ,

(b)

(c)

Section 6.1 123

Example 4 Find the greatest common factor for each list of terms.

(a)

(b) The process of applying the distribute property (reverse direction) to write a sum as a product with the greatest common factor (GCF) as one of the factors is called factoring out the greatest common factor (GCF). Eg. Example 5 Factor out the greatest common factor (GCF).

(a)

(b)

(c)

(d)

(e)

124 Section 6.1

Example 6 Factor out the greatest common factor.

(a)

(b) When a polynomial has four terms, we can often factor by grouping. Example 7 Factor by grouping.

(a)

(b)

Section 6.1 125

(c)

(d)

(e)

126 Section 6.2

6.2 Factoring Trinomials Observe: Here, we have factored the polynomial as a product of two binomials. Note the relationship between the coefficients of the original polynomial and those of the two binomials. Trinomials with a Leading Coefficient of 1

where the product and the sum If it is not possible to find such and then the polynomial cannot be factored, and we say the

polynomial is __________________.

Example 1 Factor .

Section 6.2 127

Example 2 Factor . Example 3 Factor . Example 4 Factor .

128 Section 6.2

Example 5 Factor . Example 6 (Factor a trinomial with two variables) Factor

The 1st step in EVERY factoring problem is to _________________________________!!!

Example 7 Factor

Section 6.3 129

6.3 More on Factoring Trinomials Main Objective:

Factor the general polynomial when the leading coefficient is not 1.

Techniques: #1) By Grouping (a.k.a. AC Method) #2) By Using FOIL (a.k.a. Trial & Error Method)

Technique #1: By Grouping Example 1 Factor by grouping method. Example 2 Factor by grouping method.

130 Section 6.3

Example 3 Factor by grouping method. Example 4 Factor by grouping method. Technique #2: By Using FOIL Example 5 Factor by using FOIL (trial & error).

Section 6.3 131

Example 6 Factor by using FOIL (trial & error). Example 7 Factor by using FOIL (trial & error). Example 8 Factor by using FOIL (trial & error).

132 Section 6.3

Example 9 Factor by using FOIL (trial & error). Example 10 Factor by using FOIL (trial & error). Example 11 Factor by using FOIL (trial & error).

Section 6.4 133

6.4 Special Factoring Techniques

Difference of Squares

Example 1 Factor completely.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

134 Section 6.4

Difference of Cubes

Sum of Cubes


(a)

(b)

(c)

(d)

(e)

(f)

Section 6.4 135

Perfect Square Trinomials


(a)

(b)

(c)

(d)

(e)

136 Section 6.5

6.5 Solving Quadratic Equations by Factoring

A quadratic equation is an equation that can be written in the form:

where and are real numbers and . How to recognize a quadratic equation: The highest power of is ________. Circle which of the following equations are quadratic? Zero-Factor Property

If , then _________________________________. Example 1 Solve the quadratic equation.

(a)

(b)

(c)

Standard Form

Section 6.5 137

How to Solve a Quadratic Equation by ____________________.

1)

2)

3)

Example 2 Solve the quadratic equation.

(a)

(b)

(c)

138 Section 6.5

Example 3 Solve the quadratic equation.

(a)

(b)

(c)

Section 6.5 139

The zero-factor property may be used to solve equations that are not quadratic, but that are still factorable, as we will see in the next example… Example 4 Solve.

(a)

(b)

(c)

140 Section 6.6

6.6 Applications of Quadratic Equations

Example 1 Area of a parallelogram: The area of this parallelogram is sq. units. Find its base and height. Example 2 Area of a rectangle: A hardcover book is 3 longer than it is wide. The area of the cover is 54 Find the length and width of the book.

Section 6.6 141

Example 3 Volume of a rectangular box: The volume of this rectangular box is 192 cu. units. Find its length and width. Example 4

Area of a triangle:

The area of a triangle is 24 . The base of the triangle measures more than the height. Find the measures of the base and the height.

4

142 Section 6.6

Example 5 The product of the first and third of three consecutive odd integers is 16 more than the middle integer. Find the integers.

Two consecutive integers:

Two consecutive even or odd integers:

Section 7.1 143

7.1 The Fundamental Property of Rational Expressions

Consider the following expressions:

Each of these is a ratio of two __________________________.

A rational expression is an expression of the form: where and are __________________________ , with _______. Example 1 Find the value of the rational expression

when

(a)

(b)

(c)

144 Section 7.1

Note: A rational expression is undefined for any values of the variable that cause the denominator to be zero. To find these values, we set the denominator equal to zero and solve. Example 2 Find any values of the variable for which each rational expression is undefined. Write answers with .

Section 7.1 145

A rational expression

( ) is in lowest terms if the numerator and denominator share no

common factors (except 1). The Fundamental Property of Rational Expressions

If

( ) is a rational expression and if represents any polynomial ( , then

To write a rational expression in lowest terms:

1)

2) Example 3 Write in lowest terms.

146 Section 7.1

Example 4 Write in lowest terms.

Example 5 Write four equivalent forms for the rational expression.

Section 7.2 147

7.2 Multiplying and Dividing Rational Expressions

Multiplying rational expressions

Example 1 Multiply. Write your answer in lowest terms.

148 Section 7.2

Dividing rational expressions

Example 2 Divide. Write your answer in lowest terms.

Section 7.3 149

7.3 Least Common Denominators

Recall how to add or subtract two rational numbers with a common denominator:

Recall how to add or subtract two rational numbers with different denominators:

Before two rational expressions with different denominators can be added or subtracted, both rational expressions must be expressed in terms of a common denominator. This common denominator is called the LCD (Least Common Denominator). How to find the LCD of two (or more) rational expressions:

1) Factor each denominator into prime factors. 2) The LCD is the product of the LCM of the coefficients and each variable factor raised to the

greatest power that occurs in any one factorization. Example 1 Find the LCD:

150 Section 7.3

Example 2 Find the LCD:

The next step after finding a common denominator is to rewrite each rational expression with the new denominator. Example 3 Rewrite each rational expression with the indicated denominator.

Section 7.3 151

Example 4 Rewrite each rational expression with the LCD as the denominator.

152 Section 7.4

7.4 Adding and Subtracting Rational Expressions To add/subtract rational expressions with the same denominator, add/subtract the numerators and keep the same denominator. Recall how to write a rational expression in lowest terms:

1) Factor the numerator and denominator 2) Cancel common factors

Example 1 Add/subtract. Write each answer in lowest terms.

Section 7.4 153

To add/subtract rational expressions with different denominators:

1) Rewrite the problem with the denominators written in factored form. 2) Identify the LCD, and rewrite each rational expression with the LCD as the denominator. 3) Add/subtract the numerators and write this result over the LCD. 4) Write the answer in lowest terms (Factor and cancel common factors).

Example 2 Add/subtract. Write each answer in lowest terms.

154 Section 7.4

Section 7.4 155

156 Section 7.5

7.5 Complex Fractions

Example 1 Simplify the complex fraction (Method #1):


Method #1 to simplify a complex fraction: 1. Write the numerator as a single fraction. 2. Write the denominator as a single fraction. 3. Divide using multiplication by the reciprocal. Write in lowest terms.

Section 7.5 157


158 Section 7.5



Method #2 to simplify a complex fraction: 1. Find the LCD of all fractions within the complex fraction. 2. Multiply both the numerator and the denominator of the complex fraction by this LCD

using the distributive property as necessary. Write in lowest terms.

Section 7.5 159


160 Section 7.6

7.6 Solving Equations with Rational Expressions

1. When simplifying expressions, keep the LCD (denominator) throughout the simplification. 2. When solving an equation, multiply each side by the LCD to “clear the fractions”.

Example 1 Identify each of the following as an expression or an equation. Then simplify the expression or solve the equation.

(a) 4 3

7 5x x (b)

4 31

7 5x x

Example 2

Solve. 2 3 6

5 3 5

m m

Section 7.6 161

When solving an equation with variables in the denominator, you must check your answer for

extraneous solutions which make the denominator equal 0.

Example 3

Solve, and check your solutions. 2 2

11 1

x

x x

Example 4

Solve, and check your solutions. 2 2

2 3

2p p p p

Solving an Equation with Rational Expressions

1. Factor the denominators to find the LCD. Multiply each side of the equation by the LCD to clear fractions.

2. Solve the equation 3. Check your solutions and reject any that cause a denominator to equal 0.

162 Section 7.6

Example 5

Solve, and check your solutions. 2

8 3 3

4 1 2 1 2 1

r

r r r

Example 6


1 1 2

2 5 5( 4)x x

Section 7.6 163

Example 7


6 1 4

5 10 5 3 10a a a a

Example 8 Solve each formula for the specified variable.

(a) x

zx y

for y (b) 2 1 1

x y z for z

164 Section 7.7

7.7 Applications of Rational Expressions

Example 1

A certain number is added to the numerator and subtracted from the denominator of

. The new

number equals the reciprocal of

. Find the number.

Recall from Chapter 2 the distance formula: Dividing both sides by yields the equivalent equation:

Recall the following relationship between the speed of a boat/airplane and current:

Water/Air Current Problems:

Downstream (with current)

Rate

Upstream (against current)

Rate

Section 7.7 165

Example 2 A boat can go 10 mi against a current in the same time it can go 30 mi with the current. The current flows at 4 mph. Find the speed of the boat with no current. Assign a variable:

Distance Rate Time

Write an equation & solve: (Hint: How do the times relate?) Example 3 Howie Sorkin can travel 8 mi upstream in the same time it takes him to go 12 mi downstream. His boat goes 15 mph in still water. What is the rate of the current? Assign a variable:

Distance Rate Time

Write an equation & solve: (Hint: How do the times relate?)

166 Section 7.7

Example 4 Jenna walks from her house to school, a distance of 6 miles. Then she walks from school to the train station, a distance of 10 miles. It takes Jenna about 2 hours longer to walk from school to the train station than it does to walk from her house to school. What is Jenna’s walking pace? Diagram: Assign a variable:

Distance Rate Time

Write an equation & solve: (Hint: How do the times relate?)

Section 7.7 167

Suppose that you can finish your math homework in 4 hours. Then your rate of work is:

In words: After 1 hour, you will have completed

of your math homework.

Now, after 3 hours, the fractional part of the job done is:

Since you can complete the job in 4 hours, after 4 hours, the fractional part of the job done is:

When the job is completed, the fractional part of the job done is ________! In general:

Rate of Work

If a job can be completed in t units of time, then the rate of work is

job per unit of time

Rate of Work

Time Worked

Fractional part of the job done

Work Problems Formula

Rate of work Time worked = Fractional part of job done

Rate of Work

Time Worked

Fractional part of the job done

168 Section 7.7

Example 5 Al and Mario operate a small roofing company. Mario can roof an average house alone in 9 hr. Al can roof a house alone in 8 hr. How long will it take them to do the job if they work together? Assign a variable:

Rate of Work

Time Working Together

Fractional Part of the Job Done when Working Together

Write an equation & solve: (Hint: How do the fractional parts of the job done relate?)

Section 7.7 169

Example 6 One pipe can fill a swimming pool in 2 hours, and another pipe can do it in 5 hours. How long will it

take the two pipes working together to fill the pool

full?

Assign a variable:

Rate of Work




170 Section 7.7

Example 7 Cheryl can paint a room three times faster than Betty. Working together, they can paint a room in 4 hours. How long would it take Cheryl to paint the room working alone? Assign a variable:

Rate of Work




Section 7.8 171

7.8 Variation

For example: If the cost of gas is $4 per gallon, then the total cost of gas varies directly as the number of gallons of gas purchased, with constant of variation .

# gallons of gas x

Total Cost ($)

1 2 3

Example 1 If varies directly as , and when , find when .

Direct Variation

varies directly as if _____________________ for some constant

The constant is called the constant of variation.

With direct variation, where ,

As one variable increases, the other variable ________________.

As one variable decreases, the other variable ________________.

172 Section 7.8

For example: According to the distance formula,

. Thus, the time it takes to travel 100 miles

varies inversely as the speed of the car, with constant of variation .

Speed of car

Time to travel 100 miles

10 mph

20 mph

50 mph

Example 2 If varies inversely as , and when , find when .

Inverse Variation

varies inversely as if _____________________ for some constant

With inverse variation, where ,

As one variable increases, the other variable ________________.

As one variable decreases, the other variable ________________.

Section 7.8 173

Example 3 The pressure exerted by water at a given point varies directly with the depth of the point beneath the surface of the water. Water exerts 4.34 lb per in.2 for every 10 ft traveled below the water’s surface. What is the pressure exerted on a scuba diver at 20 ft? Example 4 For a constant area, the length of a rectangle varies inversely as the width. The length of a rectangle is 27 ft when the width is 10 ft. Find the width of a rectangle with the same area if the length is 18 ft.

174 Section 8.1

8.1 Evaluating Roots

“Square” of a number If , then _______.

If , then _______. In this chapter, we will consider the opposite process… “Square root” of a number If , then __________________.

If

, then __________________.

If , then __________________.

The “radical sign” symbol represents the positive (or principal) square root.

Also, represents the negative square root.

So _______ , but _______ Example 1 Find each square root.

(a) (b)

(c)

(d) (e) (f)

Section 8.1 175

Example 2 Find the square of each radical expression.

(a) (b) (c)

A perfect square is any number whose positive square root is a rational number. Circle the perfect squares:

Example 3 Determine whether each number is rational, irrational, or not a real number. If a number is rational, give its exact value. If a number is irrational, give a decimal approximation to the nearest thousandth using a calculator.

(a) (b)

(c) (d)

Note: For nonnegative ,

If is not a perfect square, then is irrational.

If is negative, then is not a real number.

176 Section 8.1

Recall, the Pythagorean Theorem: 222 cba Example 4 A boat is being pulled toward a dock with a rope attached at water level. When the boat is 24 ft. from the dock, 30 ft. of rope is extended. What is the height of the dock above the water? *********************************************************************************** Finding the distance between two points:

If , then the positive solution of the equation

is

24 ft

30 ft

Distance Formula The distance between the points and is

a

b

c

Section 8.1 177

Example 5 Find the distance between and . *********************************************************************************** The opposite of “cubing” a number is taking the “cube root”.

, so

____________

, so

____________ The opposite of finding the “fourth power” of a number is taking the “fourth root”

, so

____________

Example 6 Find each root.

(a)

(b)

(c)

(d)

(e)

(f)

In general, the th root of is written

.

If the index is even, then must be nonnegative to yield a real number root.

178 Section 8.1

The following lists will come in handy when evaluating roots until you become more familiar with them:

Perfect Squares

Perfect Cubes

Perfect Fourth Powers

Section 8.2 179

8.2 Multiplying, Dividing, and Simplifying Radicals

Observe the following:

and

WARNING: The rule does not apply to sums. In general, Example 1 Find each product. Assume that .

(a) (b) (c) A square root radical is simplified when no perfect square factor remains under the radical sign.

We accomplish this by using the product rule in the form: Example 2 Simplify each radical. (Method #1: Identify the greatest perfect square factor)

(a) (b) (c)

Product Rule For nonnegative real numbers and ,

and

180 Section 8.2

Example 3 Simplify each radical. (Method #2: Use a factor tree to write the prime factorization.)

(a)

(b)

(c) Example 4 Find each product and simplify.

(a) (b)

Section 8.2 181

Example 5 Simplify each radical.

(a) 3

48 (b)

49

4

(c) 36

5

Example 6

Simplify 54

508

Some problems require both the product and quotient rules. Example 7

Quotient Rule For nonnegative real numbers and , ,

and

182 Section 8.2

Simplify 2

7

8

3

Radicals can involve variables. Simplifying such radicals can get a little tricky.

If represents a nonnegative number , then

If represents a negative number , then

To avoid negative radicands, variables under radical signs will be assumed to be nonnegative in this

course. Therefore, absolute value bars are not necessary (in this course). Example 8 Simplify each radical. Assume that all variables represent positive real numbers.

(a) (b)

(c)

(d)

For any real number ,

Section 8.2 183

(e) (f)

Example 9 Simplify each radical.

(a) 3 108

(b) 4 160

(c) 4

625

16

In general, nn ba and n

n

b

a

184 Section 8.2

To simplify cube roots with variables, use the fact that for any real number a,

aa 3 3

This is true whether a is positive or negative. Example 10 Simplify each radical.

(a) 3 9z (b) 3 68x

(c) 3 554t (d) 3

15

64

a

Section 8.3 185

8.3 Adding and Subtracting Radicals

We can add or subtract “like radicals” ONLY. Example 1 Add or subtract, as indicated.

(a) 5258

(b) 1112113

(c) 107

(d) 3 3232

Sometimes, one or more radical expression can be simplified first. Then it is possible to add or subtract like radicals. Example 2 Add or subtract, as indicated.

(a) 1227

(b) 1862005

(c) 33 24542

186 Section 8.3

Example 3 Simplify each radical expression. Assume that all variables represent nonnegative real numbers.

(a) 272217

(b) rr 836

(c) 21872 yy

(d) 3 43 4 24581 xx

Section 8.4 187

8.4 Rationalizing the Denominator

It is sometimes easier to work with radical expressions if the denominators do not contain any radicals.

For example, to eliminate the radical in the denominator of

, we can multiply the numerator and

denominator by :

This process of changing the denominator from a radical to a rational number is called rationalizing the denominator. Example 1 Rationalize each denominator:

(a) 24

18

(b) 8

16

Example 2

Simplify 18

5

A radical is considered to be in simplified form if the following three conditions are met:

1. The radicand contains no factor that is a perfect square (in dealing with square roots), a perfect cube (in dealing with cube roots), and so on..

2. The radicand has no fractions. 3. No denominator contains a radical.

188 Section 8.4

Example 3

Simplify 6

5

2

1

Example 4

Simplify q

p5. Assume that p and q are positive real numbers.

Example 5

Simplify 7

5 22tr. Assume that r and t represent nonnegative real numbers.

Section 8.4 189

To rationalize a denominator with a cube root, we must try to “create” a perfect cube in the denominator. Example 6 Rationalize each denominator.

(a) 3

6

5

(b) 3

3

3

2

(c) 3

3

4

3

x ( 0x )

190 Section 8.5

8.5 More Simplifying and Operations with Radicals

Example 1 Find each product and simplify.

a)

b)

Simplifying Radical Expressions – Guidelines

1. Simplify radicals containing perfect squares, perfect cubes, etc...

Example: 49 7

2. Use the product rule for radicals to get a single radical.

Example: 3 7 21

3. Factor terms under the radical sign to obtain perfect squares, perfect cubes, etc...

Example: 3 33 3 316 8 2 8 2 2 2

4. Combine like radicals.

Example: 3 2 4 2 7 2 5. Rationalize the denominator.

Example: 3 3 3 2 6

2 22 2 2

Section 8.5 191

c)

d)

e)

192 Section 8.5

The conjugate of 2 6 is _______________. The conjugate of 5 x is _______________.

Conjugates can be used to rationalize the denominators in more complicated quotients. Example 2 Simplify by rationalizing each denominator.

a) 3

2 5

b) 5 2

2 3

Definition

The conjugate of a b is a b

Section 8.6 193

8.6 Solving Equations with Radicals

Example 1

Solve 9 4x .

Example 2

Solve 3 9 2x x .

Example 3

Solve 4x .

To solve a radical equation of the form M N , square both sides to obtain 2M N .

However, you MUST CHECK all proposed solutions in the original equation when squaring both sides of an equation since “false” extraneous solutions may show up occasionally.

194 Section 8.6

Example 4

Solve 2 4 16x x x

Example 5

Solve 2 1 10 9x x

Solving a Radical Equation

Step 1 Isolate the radical. Step 2 Square both sides. Step 3 Combine like terms. Step 4 Repeat Steps 1-3 if there is still a term with a radical Step 5 Solve the equation. Step 6 Check for extraneous solutions.

Section 8.6 195

Example 6

Solve 25 6x x

Example 7

Solve 1 4 1x x

196 Section 8.7

8.7 Using Rational Numbers as Exponents

Question: What should 1/35 equal?

By Laws of Exponents: 1/3 1/3 1/35 5 5

Also, observe: 3 3 35 5 5

Answer: 1/35 =

Example 1 Simplify.

a) 1/249

b) 1/532

Recall the power rule: .

Observe:

Example 2 Evaluate.

a) 5/29

b) 5/38

c) 2/327

If a is a nonnegative number and n is a positive integer, then

If a is a nonnegative number and m and n are integers with 0n , then

Section 8.7 197

Example 3 Evaluate.

a) 3/236

b) 3/481 Example 4 Simplify. Write each answer in exponential form with only positive exponents.

a) 1/3 2/37 7

b) 2/3

1/3

9

9

c) 5/3

27

8

d) 1/2 2

5/2

3 3

3

If a is a nonnegative number and m and n are integers with 0n , then

198 Section 8.7

Example 5 Simplify. Write each answer in exponential form with only positive exponents. Assume that all variables represent positive numbers.

a) 2/3 1/3 2 6( )a b c

b) 2/3 1/3

1

r r

r

c)

32/3

1/4

a

b

Example 6

Simplify 24 12 by first writing it in exponential form.

Section 9.1 199

9.1 Solving Quadratic Equations by the Square Root Property

Observe: If , then ___________________.

Example 1 Solve each equation by using the square root property. Simplify all radicals.

a) b) c)

Example 2 Solve each equation by using the square root property. Simplify all radicals. a)

b)

Square Root Property of Equations If is a positive number and if

then

How to Solve an Equation by the Square Root Property

1. __________________ the squared term.

2. Take ____________________________ of both sides, remembering the _________.

3. Solve for the variable.

200 Section 9.1

To solve an equation of the form:

apply the Square Root Property, using as the base: Example 3 Solve each equation by using the square root property. Simplify all radicals.

a) b)

c) d)

Section 9.1 201

Now let’s try a few application problems… Example 4 The formula

is used to approximate the weight of a bass, in pounds, given its length and its girth , both measured in inches. Approximate the length of a bass weighing 2.80 lb and having girth 11 in. Example 5 The area of a circle with radius is given by the formula

If a circle has area in.2, what is its radius?

202 Section 9.2

9.2 Solving Quadratic Equations by Completing the Square

We have already seen numbers that are perfect squares:

is a perfect square since

But trinomials may also be perfect squares:

is a perfect square since Example 1 Complete each trinomial so that it is a perfect square. Then factor the trinomial.

The first three problems have been done for you so you may observe the pattern.

Procedure for Completing the Square

Start Add… The Result Factored Form

9

1

Section 9.2 203

How to Solve a Quadratic Equation by Completing the Square

1) Collect all terms on the left side of the equation. The constant term should be on the right side. 2) Divide both sides of the equation through by the leading coefficient to create a leading coefficient 1.

3) Complete the square by adding

to both sides of the equation, where is the coefficient of .

4) Factor the perfect square trinomial on the left side of the equation. Simplify the right side. 5) Solve by the Square Root Property from the previous section.

Example 2: Solve each equation by completing the square.

a)

b)

204 Section 9.2

c)

d)

e)

Section 9.3 205

9.3 Solving Quadratic Equations by the Quadratic Formula

We can solve any quadratic equation by completing the square, but the method is tedious. In this section, we learn a formula which can be used to solve any general quadratic equation:

Example 1 Identify the values of , , and in the following quadratic equations. a) ______ ______ ______ b) ______ ______ ______ c) ______ ______ ______ d) ______ ______ ______

Notice that the is under both the – and the . When using this formula:

1. First simplify the numerator . 2. If possible, factor out the GCF from the numerator. Then you may divide

out the common factor.

Correct: Wrong:

The Quadratic Formula The solutions of the quadratic equation are

Factor out 2 in numerator. Then we can cancel the 2’s.

We are NOT allowed to cancel these 2’s because 2 is not a factor of the numerator.

206 Section 9.3

Example 2 Use the quadratic formula to solve each equation. Simplify all radicals, and write all answers in lowest terms.

a)

b)

c)

Section 9.3 207

d)

e)

f)

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Class Notes for Math 251 - Saddleback College · Class Notes for Math 251 Beginning Algebra ... The textbook for Math 251 is Beginning Algebra, 11th Edition, by Lial, ... Section