Embed Size (px)
Citation preview
CK-12 Algebra I - MiddleSchool Math Packet
Eve RawleyAnne Gloag
Andrew GloagMelissa Kramer
Say Thanks to the AuthorsClick http://www.ck12.org/saythanks
(No sign in required)
www.ck12.org
To access a customizable version of this book, as well as otherinteractive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission toreduce the cost of textbook materials for the K-12 market both inthe U.S. and worldwide. Using an open-source, collaborative, andweb-based compilation model, CK-12 pioneers and promotes thecreation and distribution of high-quality, adaptive online textbooksthat can be mixed, modified and printed (i.e., the FlexBook®textbooks).
Copyright © 2015 CK-12 Foundation, www.ck12.org
The names “CK-12” and “CK12” and associated logos and theterms “FlexBook®” and “FlexBook Platform®” (collectively“CK-12 Marks”) are trademarks and service marks of CK-12Foundation and are protected by federal, state, and internationallaws.
Any form of reproduction of this book in any format or medium,in whole or in sections must include the referral attribution linkhttp://www.ck12.org/saythanks (placed in a visible location) inaddition to the following terms.
Except as otherwise noted, all CK-12 Content (including CK-12Curriculum Material) is made available to Users in accordancewith the Creative Commons Attribution-Non-Commercial 3.0Unported (CC BY-NC 3.0) License (http://creativecommons.org/licenses/by-nc/3.0/), as amended and updated by Creative Com-mons from time to time (the “CC License”), which is incorporatedherein by this reference.
Complete terms can be found at http://www.ck12.org/about/terms-of-use.
Printed: June 9, 2015
AUTHORSEve RawleyAnne GloagAndrew GloagMelissa Kramer
SOURCEAnne Gloag
iii
Contents www.ck12.org
Contents
1 Equations and Functions 11.1 Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Properties of Real Numbers 172.1 Properties of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Addition of Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Addition of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4 Subtraction of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5 Multiplication of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.6 Expressions and the Distributive Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Linear Equations 463.1 Two-Step Equations and Properties of Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Equations with Ratios and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Graphs of Linear Equations and Functions 624.1 Graphs in the Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Graphs of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
iv
www.ck12.org Chapter 1. Equations and Functions
CHAPTER 1 Equations and FunctionsChapter Outline
1.1 ORDER OF OPERATIONS
1.2 EQUATIONS AND INEQUALITIES
1
1.1. Order of Operations www.ck12.org
1.1 Order of Operations
Learning Objectives
• Evaluate algebraic expressions with grouping symbols.• Evaluate algebraic expressions with fraction bars.• Evaluate algebraic expressions using a graphing calculator.
Introduction
Look at and evaluate the following expression:
2+4×7−1 =?
How many different ways can we interpret this problem, and how many different answers could someone possiblyfind for it?
The simplest way to evaluate the expression is simply to start at the left and work your way across:
2+4×7−1
= 6×7−1
= 42−1
= 41
This is the answer you would get if you entered the expression into an ordinary calculator. But if you entered theexpression into a scientific calculator or a graphing calculator you would probably get 29 as the answer.
In mathematics, the order in which we perform the various operations (such as adding, multiplying, etc.) isimportant. In the expression above, the operation of multiplication takes precedence over addition, so we evaluateit first. Let’s re-write the expression, but put the multiplication in brackets to show that it is to be evaluated first.
2+(4×7)−1 =?
First evaluate the brackets: 4×7 = 28. Our expression becomes:
2+(28)−1 =?
When we have only addition and subtraction, we start at the left and work across:
2+28−1
= 30−1
= 29
2
www.ck12.org Chapter 1. Equations and Functions
Algebra students often use the word “GEMDAS” to help remember the order in which we evaluate the mathematicalexpressions: Grouping Symbols, Exponents, Multiplication, Division, Addition and Subtraction.
Order of Operations
1. Evaluate expressions within Grouping Symbols (all brackets [ ], parenthesis (), division bars ——–, radicals√) first.
2. Evaluate all Exponents (terms such as 32 or x3) next.3. Multiplication and Division is next - work from left to right completing both multiplication and division in
the order that they appear.4. Finally, evaluate Addition and Subtraction - work from left to right completing both addition and subtraction
in the order that they appear.
Evaluate Algebraic Expressions with Grouping Symbols
The first step in the order of operations includes all grouping symbols in this step—not just parentheses (), but alsosquare brackets [ ] and curly braces { }, division bars ——-, and radicals
√to name just a few. When multiple
grouping symbols are present, work from the inside out.
Example 1
Evaluate the following:
a) 4−7−11+2
b) 4− (7−11)+2
c) 4− [7− (11+2)]
Each of these expressions has the same numbers and the same mathematical operations, in the same order. Theplacement of the various grouping symbols means, however, that we must evaluate everything in a different ordereach time. Let’s look at how we evaluate each of these examples.
a) This expression doesn’t have parentheses, exponents, multiplication, or division. GEMDAS states that we treataddition and subtraction as they appear, starting at the left and working right (it’s NOT addition then subtraction).
4−7−11+2 =−3−11+2
=−14+2
=−12
b) This expression has parentheses, so we first evaluate 7−11 =−4. Remember that when we subtract a negative itis equivalent to adding a positive:
4− (7−11)+2 = 4− (−4)+2
= 8+2
= 10
c) An expression can contain any number of sets of parentheses. Sometimes expressions will have sets of parenthesesinside other sets of parentheses. When faced with nested parentheses, start at the innermost parentheses and workoutward.
3
1.1. Order of Operations www.ck12.org
Brackets may also be used to group expressions which already contain parentheses. This expression has bothbrackets and parentheses. We start with the innermost group: 11+ 2 = 13. Then we complete the operation inthe brackets.
4− [7− (11+2)] = 4− [7− (13)]
= 4− [−6]
= 10
Example 2
Evaluate the following:
a) 3×5−7÷2
b) 3× (5−7)÷2
c) (3×5)− (7÷2)
a) There are no grouping symbols. GEMDAS dictates that we multiply and divide first, working from left to right:3×5 = 15 and 7÷2 = 3.5. (NOTE: It’s not multiplication then division.) Next we subtract:
3×5−7÷2 = 15−3.5
= 11.5
b) First, we evaluate the expression inside the parentheses: 5−7 =−2. Then work from left to right:
3× (5−7)÷2 = 3× (−2)÷2
= (−6)÷2
=−3
c) First, we evaluate the expressions inside parentheses: 3×5 = 15 and 7÷2 = 3.5. Then work from left to right:
(3×5)− (7÷2) = 15−3.5
= 11.5
Note that adding parentheses didn’t change the expression in part c, but did make it easier to read. Parentheses canbe used to change the order of operations in an expression, but they can also be used simply to make it easier tounderstand.
We can also use the order of operations to simplify an expression that has variables in it, after we substitute specificvalues for those variables.
Example 3
Use the order of operations to evaluate the following:
a) 2− (3x+2) when x = 2
b) 3y2 +2y+1 when y =−3
c) 2− (t−7)2× (u3− v) when t = 19, u = 4, and v = 2
4
www.ck12.org Chapter 1. Equations and Functions
a) The first step is to substitute the value for x into the expression. We can put it in parentheses to clarify the resultingexpression.
2− (3(2)+2)
(Note: 3(2) is the same as 3×2.)
Follow GEMDAS - first parentheses. Inside parentheses follow GEMDAS again.
2− (3×2+2) = 2− (6+2) Inside the parentheses, we multiply first.
2−8 =−6 Next we add inside the parentheses, and finally we subtract.
b) The first step is to substitute the value for y into the expression.
3× (−3)2 +2× (−3)−1
Follow GEMDAS: we cannot simplify the expressions in parentheses, so exponents come next.
3× (−3)2 +2× (−3)−1 Evaluate exponents: (−3)2 = 9
= 3×9+2× (−3)−1 Evaluate multiplication: 3×9 = 27; 2×−3 =−6
= 27+(−6)−1 Add and subtract in order from left to right.
= 27−6−1
= 20
c) The first step is to substitute the values for t, u, and v into the expression.
2− (19−7)2× (43−2)
Follow GEMDAS:
2− (19−7)2× (43−2) Evaluate parentheses: (19−7) = 12; (43−2) = (64−2) = 62
= 2−122×62 Evaluate exponents: 122 = 144
= 2−144×62 Multiply: 144×62 = 8928
= 2−8928 Subtract.
=−8926
In parts (b) and (c) we left the parentheses around the negative numbers to clarify the problem. They did not affectthe order of operations, but they did help avoid confusion when we were multiplying negative numbers.
Part (c) in the last example shows another interesting point. When we have an expression inside the parentheses, weuse GEMDAS to determine the order in which we evaluate the contents.
5
1.1. Order of Operations www.ck12.org
Evaluate Algebraic Expressions with Fraction Bars
Fraction bars count as grouping symbols for G EMDAS, so we evaluate them in the first step of solving anexpression. All numerators and all denominators can be treated as if they have invisible parentheses around them.When real parentheses are also present, remember that the innermost grouping symbols come first. If, for example,parentheses appear on a numerator, they would take precedence over the fraction bar. If the parentheses appearoutside of the fraction, then the fraction bar takes precedence.
Example 4
Use the order of operations to evaluate the following expressions:
a) z+34 −1 when z = 2
b)(a+2
b+4 −1)+b when a = 3 and b = 1
c) 2×(
w+(x−2z)(y+2)2 −1
)when w = 11,x = 3,y = 1, and z =−2
a) We substitute the value for z into the expression.
2+34−1
Although this expression has no parentheses, the fraction bar is also a grouping symbol—it has the same effect as aset of parentheses. We can write in the “invisible parentheses” for clarity:
(2+3)4−1
Using GEMDAS, we first evaluate the numerator:
54−1
We can convert 54 to a mixed number:
54= 1
14
Then evaluate the expression:
54−1 = 1
14−1 =
14
b) We substitute the values for a and b into the expression:
(3+21+4
−1)+1
This expression has nested parentheses (remember the effect of the fraction bar). The innermost grouping symbol isprovided by the fraction bar. We evaluate the numerator (3+2) and denominator (1+4) first.
6
www.ck12.org Chapter 1. Equations and Functions
(3+21+4
−1)+1 =
(55−1)+1 Next we evaluate the inside of the parentheses. First we divide.
= (1−1)+1 Next we subtract.
= 0+1 = 1
c) We substitute the values for w,x,y, and z into the expression:
2×(
11+(3−2(−2))(1+2)2 −1
)This complicated expression has several layers of nested parentheses. One method for ensuring that we start withthe innermost parentheses is to use more than one type of parentheses. Working from the outside, we can leave theoutermost brackets as parentheses (). Next will be the “invisible brackets” from the fraction bar; we will write theseas [ ]. The third level of nested parentheses will be the { }. We will leave negative numbers in round brackets.
2×
[11+{3−2(−2)}][{1+2}2
] −1
Start with the innermost grouping sign: {} .
{1+2}= 3; {3−2(−2)}= 3+4 = 7
= 2([11+7][32]
−1)
Next, evaluate the square brackets.
= 2(
189−1)
Next, evaluate the round brackets. Start with division.
= 2(2−1) Finally, do the addition and subtraction.
= 2(1) = 2
Additional Resources
For more practice, you can play an algebra game involving order of operations online at http://www.funbrain.com/algebra/index.html .
Review Questions
1. Use the order of operations to evaluate the following expressions.
a. 8− (19− (2+5)−7)b. 2+7×11−12÷3c. (3+7)÷ (7−12)d. 2·(3+(2−1))
4−(6+2) − (3−5)
e. 4+7(3)9−4 + 12−3·2
2f. (4−1)2 +32 ·2g. (22+5)2
52−42 ÷ (2+1)
2. Evaluate the following expressions involving variables.
a. jkj+k when j = 6 and k = 12
b. 2y2 when x = 1 and y = 5
7
1.1. Order of Operations www.ck12.org
c. 3x2 +2x+1 when x = 5d. (y2− x)2 when x = 2 and y = 1e. x+y2
y−x when x = 2 and y = 3
3. Evaluate the following expressions involving variables.
a. 4x9x2−3x+1 when x = 2
b. z2
x+y +x2
x−y when x = 1, y =−2, and z = 4
c. 4xyzy2−x2 when x = 3, y = 2, and z = 5
d. x2−z2
xz−2x(z−x) when x =−1 and z = 3
4. Insert parentheses in each expression to make a true equation.
a. 5−2×6−5+2 = 5b. 12÷4+10−3×3+7 = 11c. 22−32−5×3−5 = 30d. 12−8−4×5 =−8
5. Evaluate each expression using a graphing calculator.
a. x2 +2x− xy when x = 250 and y =−120b. (xy− y4)2 when x = 0.02 and y =−0.025c. x+y−z
xy+yz+xz when x = 12 , y = 3
2 , and z =−1
d. (x+y)2
4x2−y2 when x = 3 and y =−5
e. (x−y)3
x3−y + (x+y)2
x+y4 when x = 4 and y =−2
8
www.ck12.org Chapter 1. Equations and Functions
1.2 Equations and Inequalities
Learning Objectives
• Write equations and inequalities.• Check solutions to equations.• Check solutions to inequalities.• Solve real-world problems using an equation.
Introduction
In algebra, an equation is a mathematical expression that contains an equals sign. It tells us that two expressionsrepresent the same number. For example, y = 12x is an equation. An inequality is a mathematical expressionthat contains inequality signs. For example, y ≤ 12x is an inequality. Inequalities are used to tell us that anexpression is either larger or smaller than another expression. Equations and inequalities can contain both variablesand constants.
Variables are usually given a letter and they are used to represent unknown values. These quantities can changebecause they depend on other numbers in the problem.
Constants are quantities that remain unchanged. Ordinary numbers like 2, −3, 34 , and π are constants.
Equations and inequalities are used as a shorthand notation for situations that involve numerical data. They are veryuseful because most problems require several steps to arrive at a solution, and it becomes tedious to repeatedly writeout the situation in words.
Write Equations and Inequalities
Here are some examples of equations:
3x−2 = 5 x+9 = 2x+5x3= 15 x2 +1 = 10
To write an inequality, we use the following symbols:
>greater than
≥ greater than or equal to
<less than
≤ less than or equal to
6= not equal to
Here are some examples of inequalities:
3x < 5 4− x≤ 2x x2 +2x−1 > 03x4≥ x
2−3
9
1.2. Equations and Inequalities www.ck12.org
The most important skill in algebra is the ability to translate a word problem into the correct equation or inequalityso you can find the solution easily. The first two steps are defining the variables and translating the word probleminto a mathematical equation.
Defining the variables means that we assign letters to any unknown quantities in the problem.
Translating means that we change the word expression into a mathematical expression containing variables andmathematical operations with an equal sign or an inequality sign.
Example 1
Define the variables and translate the following expressions into equations.
a) A number plus 12 is 20.
b) 9 less than twice a number is 33.
c) $20 was one quarter of the money spent on the pizza.
Solution
a) Define
Let n = the number we are seeking.
Translate
A number plus 12 is 20.
n+12 = 20
b) Define
Let n = the number we are seeking.
Translate
9 less than twice a number is 33.
This means that twice the number, minus 9, is 33.
2n−9 = 33
c) Define
Let m = the money spent on the pizza.
Translate
$20 was one quarter of the money spent on the pizza.
20 =14
m
Often word problems need to be reworded before you can write an equation.
Example 2
Find the solution to the following problems.
a) Shyam worked for two hours and packed 24 boxes. How much time did he spend on packing one box?
10
www.ck12.org Chapter 1. Equations and Functions
b) After a 20% discount, a book costs $12. How much was the book before the discount?
Solution
a) Define
Let t = time it takes to pack one box.
Translate
Shyam worked for two hours and packed 24 boxes. This means that two hours is 24 times the time it takes to packone box.
2 = 24t
Solve
t =224
=1
12hours
112×60 minutes = 5 minutes
Answer
Shyam takes 5 minutes to pack a box.
b) Define
Let p = the price of the book before the discount.
Translate
After a 20% discount, the book costs $12. This means that the price minus 20% of the price is $12.
p−0.20p = 12
Solve
p−0.20p = 0.8p, so 0.8p = 12
p =120.8
= 15
Answer
The price of the book before the discount was $15.
Check
If the original price was $15, then the book was discounted by 20% of $15, or $3. $15− 3 = $12. The answerchecks out.
Example 3
Define the variables and translate the following expressions into inequalities.
a) The sum of 5 and a number is less than or equal to 2.
11
1.2. Equations and Inequalities www.ck12.org
b) The distance from San Diego to Los Angeles is less than 150 miles.
c) Diego needs to earn more than an 82 on his test to receive a B in his algebra class.
d) A child needs to be 42 inches or more to go on the roller coaster.
Solution
a) Define
Let n = the unknown number.
Translate
5+n≤ 2
b) Define
Let d = the distance from San Diego to Los Angeles in miles.
Translate
d < 150
c) Define
Let x = Diego’s test grade.
Translate
x > 82
d) Define
Let h = the height of child in inches.
Translate:
h≥ 42
Check Solutions to Equations
You will often need to check solutions to equations in order to check your work. In a math class, checking that youarrived at the correct solution is very good practice. We check the solution to an equation by replacing the variablein an equation with the value of the solution. A solution should result in a true statement when plugged into theequation.
Example 4
Check that the given number is a solution to the corresponding equation.
a) y =−1; 3y+5 =−2y
b) z = 3; z2 +2z = 8
c) x =−12 ; 3x+1 = x
12
www.ck12.org Chapter 1. Equations and Functions
Solution
Replace the variable in each equation with the given value.
a)
3(−1)+5 =−2(−1)
−3+5 = 2
2 = 2
This is a true statement. This means that y =−1 is a solution to 3y+5 =−2y.
b)
32 +2(3) = 8
9+6 = 8
15 = 8
This is not a true statement. This means that z = 3 is not a solution to z2 +2z = 8 .
c)
3(−1
2
)+1 =−1
2(−3
2
)+1 =−1
2
−12=−1
2
This is a true statement. This means that x =−12 is a solution to 3x+1 = x.
Check Solutions to Inequalities
To check the solution to an inequality, we replace the variable in the inequality with the value of the solution. Asolution to an inequality produces a true statement when substituted into the inequality.
Example 5
Check that the given number is a solution to the corresponding inequality.
a) a = 10; 20a≤ 250
b) b =−0.5; 3−bb >−4
c) x = 34 ; 4x+5≤ 8
Solution
Replace the variable in each inequality with the given value.
a)
20(10)≤ 250
200≤ 250
This statement is true. This means that a = 10 is a solution to the inequality 20a≤ 250.
Note that a = 10 is not the only solution to this inequality. If we divide both sides of the inequality by 20, we canwrite it as a≤ 12.5. This means that any number less than or equal to 12.5 is also a solution to the inequality.
13
1.2. Equations and Inequalities www.ck12.org
b)
3− (−0.5)(−0.5)
>−4
3+0.5−0.5
>−4
−3.50.5
>−4
−7 >−4
This statement is false. This means that b =−0.5 is not a solution to the inequality 3−bb >−4 .
c)
4(
34
)+5≥ 8
3+5≥ 8
8≥ 8
This statement is true. It is true because this inequality includes an equals sign; since 8 is equal to itself, it is also“greater than or equal to” itself. This means that x = 3
4 is a solution to the inequality 4x+5≤ 8.
Solve Real-World Problems Using an Equation
Let’s use what we have learned about defining variables, writing equations and writing inequalities to solve somereal-world problems.
Example 6
Tomatoes cost $0.50 each and avocados cost $2.00 each. Anne buys six more tomatoes than avocados. Her total billis $8. How many tomatoes and how many avocados did Anne buy?
Solution
Define
Let a = the number of avocados Anne buys.
Translate
Anne buys six more tomatoes than avocados. This means that a+6 = the number of tomatoes.
Tomatoes cost $0.50 each and avocados cost $2.00 each. Her total bill is $8. This means that .50 times the numberof tomatoes plus 2 times the number of avocados equals 8.
0.5(a+6)+2a = 8
0.5a+0.5 ·6+2a = 8
2.5a+3 = 8
2.5a = 5
a = 2
Remember that a= the number of avocados, so Anne buys two avocados. The number of tomatoes is a+6= 2+6=8.
14
www.ck12.org Chapter 1. Equations and Functions
Answer
Anne bought 2 avocados and 8 tomatoes.
Check
If Anne bought two avocados and eight tomatoes, the total cost is: (2× 2)+ (8× 0.5) = 4+ 4 = 8. The answerchecks out.
Example 7
To organize a picnic Peter needs at least two times as many hamburgers as hot dogs. He has 24 hot dogs. What isthe possible number of hamburgers Peter has?
Solution
Define
Let x = number of hamburgers
Translate
Peter needs at least two times as many hamburgers as hot dogs. He has 24 hot dogs.
This means that twice the number of hot dogs is less than or equal to the number of hamburgers.
2×24≤ x, or 48≤ x
Answer
Peter needs at least 48 hamburgers.
Check
48 hamburgers is twice the number of hot dogs. So more than 48 hamburgers is more than twice the number of hotdogs. The answer checks out.
Additional Resources
For more practice solving inequalities, check out http://www.aaastudy.com/equ725x7.htm .
Review Questions
1. Define the variables and translate the following expressions into equations.
a. Peter’s Lawn Mowing Service charges $10 per job and $0.20 per square yard. Peter earns $25 for a job.b. Renting the ice-skating rink for a birthday party costs $200 plus $4 per person. The rental costs $324 in
total.c. Renting a car costs $55 per day plus $0.45 per mile. The cost of the rental is $100.d. Nadia gave Peter 4 more blocks than he already had. He already had 7 blocks.
2. Define the variables and translate the following expressions into inequalities.
a. A bus can seat 65 passengers or fewer.b. The sum of two consecutive integers is less than 54.c. The product of a number and 3 is greater than 30.d. An amount of money is invested at 5% annual interest. The interest earned at the end of the year is
greater than or equal to $250.e. You buy hamburgers at a fast food restaurant. A hamburger costs $0.49. You have at most $3 to spend.
Write an inequality for the number of hamburgers you can buy.
15
1.2. Equations and Inequalities www.ck12.org
f. Mariel needs at least 7 extra credit points to improve her grade in English class. Additional book reportsare worth 2 extra credit points each. Write an inequality for the number of book reports Mariel needs todo.
3. Check whether the given number is a solution to the corresponding equation.
a. a =−3; 4a+3 =−9b. x = 4
3 ; 34 x+ 1
2 = 32
c. y = 2; 2.5y−10.0 =−5.0d. z =−5; 2(5−2z) = 20−2(z−1)
4. Check whether the given number is a solution to the corresponding inequality.
a. x = 12; 2(x+6)≤ 8xb. z =−9; 1.4z+5.2 > 0.4zc. y = 40; −5
2 y+ 12 <−18
d. t = 0.4; 80≥ 10(3t +2)
16
www.ck12.org Chapter 2. Properties of Real Numbers
CHAPTER 2Properties of Real Numbers
Chapter Outline2.1 PROPERTIES OF RATIONAL NUMBERS
2.2 ADDITION OF INTEGERS
2.3 ADDITION OF RATIONAL NUMBERS
2.4 SUBTRACTION OF RATIONAL NUMBERS
2.5 MULTIPLICATION OF RATIONAL NUMBERS
2.6 EXPRESSIONS AND THE DISTRIBUTIVE PROPERTY
Introduction
Integers and Rational Numbers
Integers and rational numbers are important in daily life. The price per square yard of carpet is a rational number.The number of frogs in a pond is expressed using an integer. The organization of real numbers can be drawn as ahierarchy. Look at the hierarchy below.
17
www.ck12.org
Real numbers are all around us. The majority of numbers calculated are considered real numbers. This chapterdefines a real number and explains important properties and rules that apply to real numbers. The most genericnumber is the real number; it can be a combination of negative, positive, decimal, fractional, or non-repeatingdecimal values. Real numbers have two major categories: rational numbers and irrational numbers. Irrationalnumbers are non-repeating, non-terminating decimals such as π or
√2. In this chapter, you will learn how to
perform operations and solve problems involving rational numbers and integers, which are a special kind of rationalnumber.
18
www.ck12.org Chapter 2. Properties of Real Numbers
2.1 Properties of Rational Numbers
Here you’ll learn how to distinguish between rational numbers written as proper and improper fractions. You’ll alsolearn how to simplify these fractions and decide if one is less than or greater than another.
Suppose that you and two of your friends went out to lunch and that you ate 35 of a personal pan pizza, one of your
friends ate 58 of a personal pan pizza, and your other friend ate 4
3 of a personal pan pizza. Could you tell which ofyour fractions of pizza were proper and which were improper? Also, could you put the fractions of pizza in orderfrom smallest to largest? In this Concept, you will answer questions such as these for any group of fractions, alsoknown as rational numbers.
Guidance
Rational numbers include fractions, integers and whole numbers. The definition below shows that all rationalnumbers can be written in the form of a fraction:
Definition: A rational number is a number that can be written in the form ab , where a and b are integers and b 6= 0.
An integer, such as the number 3, is also a rational number because it can written as 31 .
A Review of Fractions
You can think of a rational number as a fraction of a cake. If you cut the cake into b slices, your share is a of thoseslices. For example, when we see the rational number 1
2 , we imagine cutting the cake into two parts. Our share isone of those parts. Visually, the rational number 1
2 looks like this.
There are three main types of fractions:
• Proper fractions are rational numbers where the numerator is less than the denominator. A proper fractionrepresents a number less than one. With a proper fraction you always end up with less than a whole cake!
• Improper fractions are rational numbers where the numerator is greater than the denominator. Improperfractions can be rewritten as a mixed number –an integer plus a proper fraction. An improper fractionrepresents a number greater than one.
• Equivalent fractions are two fractions that give the same numerical value when evaluated. For example, lookat a visual representation of the rational number 2
4 .
19
2.1. Properties of Rational Numbers www.ck12.org
The visual of 12 is equivalent to the visual of 2
4 . We can write out the prime factors of both the numerator and thedenominator and cancel matching factors that appear in both the numerator and denominator.(2
4
)=(
�2·1�2·2·1
)We then re-multiply the remaining factors.
(24
)=(1
2
)Therefore, 1
2 = 24 . This process is called reducing the fraction, or writing the fraction in lowest terms. Reducing a
fraction does not change the value of the fraction; rather, it simplifies the way we write it. When we have canceledall common factors, we have a fraction in its simplest form.
Example A
Classify and simplify the following rational numbers.
a)(3
7
)b)(9
3
)c)(50
60
)Solution:
a) Because both 3 and 7 are prime numbers, 37 is a proper fraction written in its simplest form.
b) The numerator is larger than the denominator; therefore, this is an improper fraction.
93=
3×33
=31= 3
c) This is a proper fraction; 5060 = 5×2×5
6×2×5 = 56
Ordering Rational Numbers
To order rational numbers is to arrange them according to a set of directions, such as ascending (lowest to highest)or descending (highest to lowest). Ordering rational numbers is useful when determining which unit cost is thecheapest.
Example B
Cans of tomato sauce come in three sizes: 8 ounces, 16 ounces, and 32 ounces. The costs for each size are $0.59,$0.99, and $1.29, respectively. Find the unit cost and order the rational numbers in ascending order.
Solution: Use proportions to find the cost per ounce: $0.598 = $0.07375
ounce ; $0.9916 = $0.061875
ounce ; $1.2932 = $0.0403125
ounce . Arrangingthe rational numbers in ascending order, we have: 0.0403125, 0.061875, 0.07375.
Example C
Which is greater, 37 or 4
9 ?
20
www.ck12.org Chapter 2. Properties of Real Numbers
Solution: Begin by creating a common denominator for these two fractions. Which number is evenly divisible by 7and 9? 7×9 = 63, so the common denominator is 63.
3×97×9
=2763
4×79×7
=2863
Because 28 > 27, 49 > 3
7 .
Video Review
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79454
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79455
–>
Guided Example
For the fractions 59 and 10
20 :
a.) Determine whether these are proper or improper fractions.
b.) Simplify if necessary.
c.) Order the two fractions.
Solution:
a.) For both fractions, the numerator is smaller than the denominator, so they are both proper fractions.
b.) For 59 , since 5 is prime and 9 is not a multiple of 5, this fraction cannot be simplified.
For 1020 , both 10 and 20 are multiples of 10, so we rewrite them as
1·102·10 = 1
2 .
c.) To order the fractions, we’ll use the simplified versions:5·29·2 = 10
18 and 1·92·9 = 9
18 .
Since9
18 < 1018
then12 < 5
9 .
21
2.1. Properties of Rational Numbers www.ck12.org
Explore More
Sample explanations for some of the practice exercises below are available by viewing the following video. Notethat there is not always a match between the number of the practice exercise in the video and the number of thepractice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Integers and Rational Numbers (13:00)
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/57
1. What are the three types of fractions?
In 2–4, determine what fraction of the whole each shaded region represents.
2.
3.
4.
In 5–8, place the following sets of rational numbers in order from least to greatest.
5. 12 ,
13 ,
14
6. 1112 ,
1211 ,
1310
7. 3960 ,
4980 ,
59100
8. 711 ,
813 ,
1219
In 9–14, find the simplest form of the following rational numbers.
22
www.ck12.org Chapter 2. Properties of Real Numbers
9. 2244
10. 927
11. 1218
12. 315420
13. 19101
14. 9911
Mixed Review
15. Evaluate the following expression: 56 d +7a2; use a = (−1) and d = 24.
16. The length of a rectangle is one more inch than its width. If the perimeter is 22 inches, what are the dimensionsof the rectangle?
17. Determine if x =−2 is a solution to 4x+7≤ 15.18. Simplify: (7+3)÷2×32−5
(58−8) .
23
2.2. Addition of Integers www.ck12.org
2.2 Addition of Integers
Here you’ll learn the Commutative Property of Addition, Associative Property of Addition, and Identity Property ofAddition so that you can effectively add integers.
Suppose you took a job as a peer tutor and the person you were tutoring asked you to give examples of theCommutative Property of Addition, Associative Property of Addition, and Identity Property of Addition. Couldyou do it? Would you be able to distinguish between the different properties? After completing this Concept, you’llhave a firm understanding of these properties so that you can easily add integers.
Guidance
Addition of Integers
A football team gains 11 yards on one play, then loses 5 yards on the next play, and then loses 2 yards on the thirdplay. What is the total loss or gain of yardage?
A loss can be expressed as a negative integer. A gain can be expressed as a positive integer. To find the net gain orloss, the individual values must be added together. Therefore, the sum is 11+(−5)+(−2) = 4. The team has a netgain of 4 yards.
Addition can also be shown using a number line. If you need to add 2+3, start by making a point at the value of 2and move three integers to the right. The ending value represents the sum of the values.
Example A
Find the sum of −2+3 using a number line.
Solution: Begin by making a point at –2 and moving three units to the right. The final value is 1, so −2+3 = 1.
When the value that is being added is positive, we jump to the right. If the value is negative, we jump to the left (ina negative direction).
Example B
Find the sum of 2−3 using a number line.
24
www.ck12.org Chapter 2. Properties of Real Numbers
Solution: Begin by making a point at 2. The expression represents subtraction, so we will count three jumps to theleft.
The solution is: 2−3 =−1.
Algebraic Properties of Addition
In the previous Concept, you learned the Additive Inverse Property. This property states that the sum of a numberand its opposite is zero. Algebra has many other properties that help you manipulate and organize information.
The Commutative Property of Addition: For all real numbers a and b, a+b = b+a.
To commute means to change locations, so the Commutative Property of Addition allows you to rearrange the objectsin an addition problem.
The Associative Property of Addition: For all real numbers a, b, and c, (a+b)+ c = a+(b+ c).
To associate means to group together, so the Associative Property of Addition allows you to regroup the objects inan addition problem.
The Identity Property of Addition: For any real number a, a+0 = a.
This property allows you to use the fact that the sum of any number and zero is the original value.
These properties apply to all real numbers, but in this lesson we are applying them to integers, which are just aspecial kind of real number.
Example C
Nadia and Peter are building sand castles on the beach. Nadia built a castle two feet tall, stopped for ice-cream, andthen added one more foot to her castle. Peter built a castle one foot tall before stopping for a sandwich. After hissandwich, he built up his castle by two more feet. Whose castle is taller?
25
2.2. Addition of Integers www.ck12.org
Solution:
Nadia’s castle is (2+ 1) feet tall. Peter’s castle is (1+ 2) feet tall. According to the Commutative Property ofAddition, the two castles are the same height.
Video Review
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79452
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79453
–>
Guided Practice
Simplify the following using the properties of addition:
a) 9+(1+22)
b) 4,211+0
Solution:
a) It is easier to re-group 9+1, so by applying the Associative Property of Addition, (9+1)+22 = 10+22 = 32.
b) The Additive Identity Property states the sum of a number and zero is itself; therefore, 4,211+0 = 4,211.
Explore More
In exercises 1 and 2, write the sum represented by the moves on the number line.
1.
2.
26
www.ck12.org Chapter 2. Properties of Real Numbers
In 3–6, which property of addition does each situation involve?
3. Shari’s age minus the negative of Jerry’s age equals the sum of the two ages.4. Kerri has 16 apples and has added zero additional apples. Her current total is 16 apples.5. A blue whale dives 160 feet below the surface and then rises 8 feet. Write the addition problem and find the
sum.6. The temperature in Chicago, Illinois one morning was −8◦F . Over the next six hours the temperature rose 25
degrees Fahrenheit. What was the new temperature?
Mixed Review
7. Find the opposite of –72.8. Evaluate |16−29+78−114|.9. What is the domain and range of the following: {(2,−1),(3,0),(4,6),(1,−3)}?
10. Write a rule for the following table:
TABLE 2.1:
Volume (in inches3) Mass (in grams)1 20.12 40.23 60.34 80.4
27
2.3. Addition of Rational Numbers www.ck12.org
2.3 Addition of Rational Numbers
Here you’ll learn how to add fractions (also known as rational numbers), even when one or more of the fractions isimproper and when the fractions have different denominators.
Suppose you drank 78 of a bottle of juice on Monday, 2
5 of a bottle on Tuesday, and 32 of a bottle on Wednesday. Can
you identify which of the fractions of a bottle of juice are improper, and can you write the fractions with a commondenominator? Also, can you add these fractions of a bottle of juice together? After completing this Concept, you’llbe able to perform these tasks so that you can figure out how many bottles of juice you drank in total over the threedays.
Guidance
Addition of Rational Numbers
To add rational numbers, we must first remember how to rewrite mixed numbers as improper fractions. Begin bymultiplying the denominator of the mixed number to the whole value. Add the numerator to this product. This valueis the numerator of the improper fraction. The denominator is the original.
Example A
Write 11 23 as an improper fraction:
Solution: 3×11 = 33+2 = 35. This is the numerator of the improper fraction.
1123=
353
Now that we know how to rewrite a mixed number as an improper fraction, we can begin to add rational numbers.There is one thing to remember when finding the sum or difference of rational numbers: The denominators must beequivalent.
The Addition Property of Fractions: For all real numbers a, b, and c, ac +
bc = a+b
c .
Example B
Add the following sets of fractions (rational numbers):
a.) 13 +
23
b.) 4 17 +
27
Solutions:
a.) Since the denominators are the same, we can go ahead and add the numerators:13 +
23 = 1+2
3 = 33 = 1
b.) 4 17 is a mixed fraction, and reads "4 and one seventh." This means we can think of it as 4+ 1
7 . Since the otherfraction also has 7 as a denominator, we can add the two fractions:
4 17 +
27 = 4+ 1
7 +27 = 4+ 1+2
7 = 4+ 37 = 4 3
7 .
28
www.ck12.org Chapter 2. Properties of Real Numbers
This mixed fraction can be turned into an improper fraction as follows:
4×7 = 2828+3
7 = 317
Algebraic Properties of Addition
Since these properties apply to all real numbers, they apply to fractions or rational numbers as well. Let’s reviewthem here:
The Commutative Property of Addition: For all real numbers a,and b, a+b = b+a.
To commute means to change locations, so the Commutative Property of Addition allows you to rearrange the objectsin an addition problem.
The Associative Property of Addition: For all real numbers a, b, and c, (a+b)+ c = a+(b+ c).
To associate means to group together, so the Associative Property of Addition allows you to regroup the objects inan addition problem.
The Identity Property of Addition: For any real number a, a+0 = a.
Another way you sometimes see a rational number is as a decimal number, such as 2.5, 30.01, or 2.9999. We willpractice some of the above properties on rational numbers in their different forms.
Example C
To convince ourselves that the algebraic properties are true, in this exercise we will check whether the followingequations are equal:
a.) Commutative Property: 2.5+3.5 = 3.5+2.5
b.) Associate Property: 19 +(2
9 +59
)=(1
9 +29
)+ 5
9
Solutions:
a.) We will check each side separately to see if they equal the same thing.
2.5+3.5 = 6
3.5+2.5 = 6
So we conclude that the equality is satisfied.
b.) We check each side of the equation here as well.19 +(2
9 +59
)= 1
9 +79 = 8
9(19 +
29
)+ 5
9 = 39 +
59 = 8
9
So we conclude that the equality is satisfied.
Common Denominators
In order to add two fractions, they must have a common denominator. This means that they must have the samenumber in the denominator. If two fractions to be added do not have common denominators, either one or both of thefractions can be changed so that they do have common denominators. In general, when two fractions have differentdenominators, use the pattern below.ab +
cd = a
b ·dd +
cd ·
bb = ad
bd +bcbd = ad+bc
bd
To understand how this works, see the next example.
29
2.3. Addition of Rational Numbers www.ck12.org
Example D
Find common denominators for the follow sums, and then add them.
1. f2. 1
5 +310
Solutions:
1. Follow the pattern; that is, multiply the top and bottom of each fraction by the denominator of the other fraction:211 +
13 = 2
11 ·33 +
13 ·
1111 = 6
33 +1133 = 6+11
33 = 1733
2. In this example, our denominator, 10, is a multiple of the other denominator, 5. There is no need to change thefraction with a denominator of 10. Simply multiply the top and bottom of the first fraction in order to make itsdenominator 10:15 +
310 = 1
5 ·22 +
310 = 2
10 +310 = 2+3
10 = 510 = 1
2
Video Review
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79456
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79457
–>
Guided Practice
Perform the addition of rational numbers and give your final answer as an mixed fraction.
5 13 +2 3
4 .
Solution:
We can break up the mixed fractions:
5 13 +2 3
4 = 5+ 13 +2+ 3
4 .
Using the Commutative Property we can rearrange and simplify by adding integers:
5+ 13 +2+ 3
4 = 5+2+ 13 +
34 = 7+ 1
3 +34 .
Now we just need to add the fractions. Since they do not have common denominators, we have to give them commondenominators. The denominators do not share any factors, so we need to multiply them by each other:
30
www.ck12.org Chapter 2. Properties of Real Numbers
13 +
34 = 1×4
3×4 +3×34×3 = 4
12 +9
12 = 4+912 = 13
12 .
Now we know what the sum of the fractions is:
7+ 13 +
34 = 7+ 13
12 .
Since our answer needs to be a mixed fraction, we will turn the improper fraction into a mixed fraction. Since 12goes into 13 one time with a remainder of 1, we get:
7+ 1312 = 7+1 1
12 = 8 112 .
The answer is:
5 13 +2 3
4 = 8 112 .
Explore More
Sample explanations for some of the practice exercises below are available by viewing the following video. Notethat there is not always a match between the number of the practice exercise in the video and the number of thepractice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Addition of Rational Numbers (7:40)
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/58
Find the sum. Write the answer in its simplest form.
1. 37 +
27
2. 310 +
15
3. 516 +
512
4. 38 +
916
5. 825 +
710
6. 16 +
14
7. 715 +
29
8. 519 +
227
9. −2.6+11.1910. −8+1311. −7.1+(−5.63)12. 9.99+(−0.01)13. 4 7
8 +1 12
14. −3 13 +(−2 3
4
)In 15–17, which property of addition does each situation involve?
15. Whichever order your groceries are scanned at the store, the total will be the same.16. Suppose you go buy a DVD for $8.00, another for $29.99, and a third for $14.99. You can add (8+29.99)+
14.99 or you can add 8+(29.99+14.99) to obtain the total.17. Nadia, Peter, and Ian are pooling their money to buy a gallon of ice cream. Nadia is the oldest and gets the
greatest allowance. She contributes half of the cost. Ian is next oldest and contributes one third of the cost.
31
2.3. Addition of Rational Numbers www.ck12.org
Peter, the youngest, gets the smallest allowance and contributes one fourth of the cost. They figure that thiswill be enough money. When they get to the check-out, they realize that they forgot about sales tax and worrythere will not be enough money. Amazingly, they have exactly the right amount of money. What fraction ofthe cost of the ice cream was added as tax?
In 18–24, evaluate each expression for v = 5.8.
18. 9.1+ v19. v+(−v)20. −v+4.1221. −23.14+−v22. 7.86+(−v)23. −v+3.524. −v+ v
32
www.ck12.org Chapter 2. Properties of Real Numbers
2.4 Subtraction of Rational Numbers
Here you’ll learn how to subtract a negative number by using the Opposite-Opposite Property. You’ll also learn howto find a common denominator when subtracting fractions and how to find the difference in a dependent variablerelative to the difference in the independent variable.
Suppose that you know that two points on a line are (2,−4) and (3,−2). How would you go about finding the"steepness" of the line? As you’ll learn in a future concept, it is the difference of the dependent variable divided bythe difference of the independent variable. Do you know how to do this? After completing this Concept, you will.
Guidance
Subtraction of Rational Numbers
In the previous two concepts, you have learned how to find the opposite of a rational number and them together. Youcan use these two ideas to subtract rational numbers as well. Suppose you want to find the difference of 9 and 12.Symbolically, it would be 9−12. Begin by placing a dot at nine and move to the left 12 units.
9−12 =−3
Rule: To subtract a number, add its opposite.
3−5 = 3+(−5) =−2 9−16 = 9+(−16) =−7
A special case of this rule can be written when trying to subtract a negative number.
The Opposite-Opposite Property: For any real numbers a and b, a− (−b) = a+b.
Example A
Simplify −6− (−13).
Solution: Using the Opposite-Opposite Property, the double negative is rewritten as a positive.
−6− (−13) =−6+13 = 7
Example B
Simplify 56 −(− 1
18
).
33
2.4. Subtraction of Rational Numbers www.ck12.org
Solution: Begin by using the Opposite-Opposite Property.
56+
118
Next, create a common denominator: 5×36×3 +
118 = 15
18 +118 .
Add the fractions: 1518 +
118 = 16
18 .
Reduce: 2×2×2×23×3×2 = 8
9 .
Evaluating Change Using a Variable Expression
You have learned how to graph a function by using an algebraic expression to generate a table of values. Using thetable of values you can find the difference of the dependent values relative to the difference of the two independentvalues.
Example C
In a previous concept, you wrote an expression to represent the pattern of the total cost to the number of CDspurchased. The table is repeated below:
Number of CDs 2 4 6 8 10
Cost ($) 24 48 72 96 120
To determine the average cost of the CDs, you must find the difference between the dependent values and divide itby the difference in the independent values.
Solution: We begin by finding the difference between the cost of two values. For example, the change in costbetween 4 CDs and 8 CDs.
96−48 = 48
Next, we find the difference between the number of CDs.
8−4 = 4
Finally, we divide.484
= 12
This tells us that the cost for each CD was 12 dollars.
Video Review
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79470
34
www.ck12.org Chapter 2. Properties of Real Numbers
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79471
–>
Guided Practice
For the equation y = 3x−5, find the difference between the dependent variable from x = 2 and x = 4. Then, find thesteepness of the line.
Solution: First we find the value of the equation, or the value of y for each x value:
y = 3x−5 = 3(2)−5 = 6−5 = 1
y = 3x−5 = 3(4)−5 = 12−5 = 7
Now, we calculate the difference in the dependent values:
7−1 = 6
Finally, to find the steepness (also called slope in later concepts) we divide by the difference between the independentvalues:
4−2 = 262 = 3
The steepness is 3. Notice that this is the value in front of x in the equation. We will learn more about that in futureconcepts.
Explore More
Sample explanations for some of the practice exercises below are available by viewing the following video. Notethat there is not always a match between the number of the practice exercise in the video and the number of thepractice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Subtraction of Rational Numbers (10:22)
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/59
In 1–20, subtract the following rational numbers. Be sure that your answer is in the simplest form.
1. 9−142. 2−7
35
2.4. Subtraction of Rational Numbers www.ck12.org
3. 21−84. 8− (−14)5. −11− (−50)6. 5
12 −918
7. 5.4−1.018. 2
3 −14
9. 34 −
13
10. 14 −(−2
3
)11. 15
11 −97
12. 213 −
111
13. −78 −(−8
3
)14. 7
27 −939
15. 611 −
322
16. −3.1−21.4917. 13
64 −740
18. 1170 −
1130
19. −68− (−22)20. 1
3 −12
21. Determine the steepness of the line between (1, 9) and (5, –14).22. Consider the equation y = 3x+2. Determine the steepness in the line between x = 3 and x = 7.23. Consider the equation y = 2
3 x+ 12 . Determine the steepness in the line between x = 1 and x = 2.
24. True or false? If the statement is false, explain your reasoning. The difference of two numbers is less thaneach number.
25. True or false? If the statement is false, explain your reasoning. A number minus its opposite is twice thenumber.
26. KMN stock began the day with a price of $4.83 per share. At the closing bell, the price dropped $0.97 pershare. What was the closing price of KMN stock?
In 27–32, evaluate the expression. Assume a = 2, b =−3, and c =−1.5.
27. (a−b)+ c28. |b+ c|−a29. a− (b+ c)30. |b|+|c|+a31. 7b+4a32. (c−a)−b
Mixed Review
33. Graph the following ordered pairs: {(0,0),(4,4),(7,1),(3,8)}. Is the relation a function?34. Evaluate the following expression when m =
(−2
3
): 23+m
4 .35. Translate the following into an algebraic equation: Ricky has twelve more dollars than Stacy. Stacy has 5 less
dollars than Aaron. The total of the friends’ money is $62.36. Simplify 1
3 +75 .
37. Simplify 214 −
23 .
36
www.ck12.org Chapter 2. Properties of Real Numbers
2.5 Multiplication of Rational Numbers
Here you’ll learn how to use the Multiplication Property of –1, the Multiplicative Identity Property, and the ZeroProperty of Multiplication when multiplying rational numbers in the form of integers and fractions.
Suppose you wrote a computer program that multiplies 15 by a random number. What if the random number were
–1? What if it were 0? What if it were 1? In fact, the random number doesn’t even have to be an integer. What if itwere 3
4 ? What if it were−27 ? In this Concept, you’ll learn about the Multiplication Property of –1, the Multiplicative
Identity Property, and the Zero Property of Multiplication so that you can answer these questions.
Guidance
When you began learning how to multiply whole numbers, you replaced repeated addition with the multiplicationsign (×). For example:
6+6+6+6+6 = 5×6 = 30
Multiplying rational numbers is performed the same way. We will start with the Multiplication Property of –1.
The Multiplication Property of –1: For any real number a,(−1)×a =−a.
This can be summarized by saying, "A number times a negative is the opposite of the number."
Example A
Evaluate −1 ·9,876.
Solution:
Using the Multiplication Property of −1: −1 ·9,876 =−9,876.
This property can also be used when the values are negative, as shown in Example B.
Example B
Evaluate −1 ·−322.
Solution:
Using the Multiplication Property of −1: −1 ·−322 = 322.
A basic algebraic property is the Multiplicative Identity. Similar to the Additive Identity, this property states that anyvalue multiplied by 1 will result in the original value.
The Multiplicative Identity Property: For any real number a, (1)×a = a.
A third property of multiplication is the Multiplication Property of Zero. This property states that any valuemultiplied by zero will result in zero.
The Zero Property of Multiplication: For any real number a, (0)×a = 0.
Multiplication of fractions can also be shown visually, as you can see in the example below.
37
2.5. Multiplication of Rational Numbers www.ck12.org
Example C
Find 13 ·
25 , drawing one model to represent the first fraction and a second model to represent the second fraction.
Solution:
By placing one model (divided in thirds horizontally) on top of the other (divided in fifths vertically), you divide onewhole rectangle into smaller parts.
The product of the two fractions is the shaded regionstotal regions .
13· 2
5=
215
Video Review
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79466
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79467
–>
38
www.ck12.org Chapter 2. Properties of Real Numbers
Guided Practice
Simplify 37 ·
45 .
Solution: By drawing visual representations, you can see that
37· 4
5=
1235
Explore More
Multiply the following rational numbers.
1. 12 ·
34
2. −7.85 ·−2.33. 2
5 ·59
4. 13 ·
27 ·
25
5. 4.5 ·−36. 1
2 ·23 ·
34 ·
45
7. 512 ×
910
8. 275 ·0
9. 23 ×
14
10. −11.1(4.1)
Multiply the following by negative one.
11. 79.512. π
13. (x+1)14. |x|15. 2516. –10517. x2
18. (3+ x)19. (3− x)
Quick Quiz
1. Order from least to greatest:(5
6 ,2326 ,
3132 ,
314
).
2. Simplify 59 ×
274 .
3. Simplify |−5+11|−|9−37|.4. Add 21
5 and 78 .
39
2.6. Expressions and the Distributive Property www.ck12.org
2.6 Expressions and the Distributive Property
Here you’ll learn to simplify an expression in the form M(N +K) or M(N−K) by using the Distributive Property.
Suppose a piece of gift-wrapping paper had a length of 5 feet and a width of x feet. When wrapping a present, youcut off a vertical strip of 1 foot so that the width of the piece of gift-wrapping paper decreased by 1 foot. Whatwould be the area of the remaining paper in square feet? Could you write this expression in simplified form? In thisConcept, you’ll learn to use the Distributive property so that you can simplify expressions like the one representingthis area.
Guidance
At the end of the school year, an elementary school teacher makes a little gift bag for each of his students. Each bagcontains one class photograph, two party favors, and five pieces of candy. The teacher will distribute the bags amonghis 28 students. How many of each item does the teacher need?
Example A
You could begin to solve this problem by deciding your variables.
Let p = photograph, f = f avors, and c = candy.
Next you can write an expression to represent the situation: p+2 f +5c.
There are 28 students in class, so the teacher needs to repeat the bag 28 times. An easier way to write this is28 · (p+2 f +5c).
We can omit the multiplication symbol and write 28(p+2 f +5c).
Therefore, the teacher needs 28p+28(2 f )+28(5c) or 28p+56 f +140c.
The teacher needs 28 photographs, 56 favors, and 140 pieces of candy to complete the end-of-year gift bags.
When you multiply an algebraic expression by another expression, you apply the Distributive Property.
The Distributive Property: For any real numbers M, N, and K:
40
www.ck12.org Chapter 2. Properties of Real Numbers
M(N +K) = MN +MK
M(N−K) = MN−MK
Example B
Determine the value of 11(2+6) using both Order of Operations and the Distributive Property.
Solution: Using the Order of Operations: 11(2+6) = 11(8) = 88.
Using the Distributive Property: 11(2+6) = 11(2)+11(6) = 22+66 = 88.
Regardless of the method, the answer is the same.
Example C
Simplify 7(3x−5).
Solution 1: Think of this expression as seven groups of (3x−5). You could write this expression seven times andadd all the like terms.
(3x−5)+(3x−5)+(3x−5)+(3x−5)+(3x−5)+(3x−5)+(3x−5) = 21x−35
Solution 2: Apply the Distributive Property.
7(3x−5) = 7(3x)+7(−5) = 21x−35
Video Review
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79460
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79461
–>
Guided Practice
Simplify 27(3y2−11).
Solution: Apply the Distributive Property.
41
2.6. Expressions and the Distributive Property www.ck12.org
27(3y2 +−11) =
27(3y2)+
27(−11) =
27(3y2)
1+
27(−11)
1=
2×3y2
7×1+
2× (−11)7×1
=
6y2
7+−22
7=
6y2
7− 22
7
Explore More
Use the Distributive Property to simplify the following expressions.
1. (x+4)−2(x+5)2. 1
2(4z+6)3. (4+5)− (5+2)4. (x+2+7)5. 0.25(6q+32)6. y(x+7)7. −4.2(h−11)8. 13x(3y+ z)9. 1
2(x− y)−410. 0.6(0.2x+0.7)11. (2− j)(−6)12. 4(m+7)−6(4−m)13. −5(y−11)+2y
Mixed Review
14. Translate the following into an inequality: Jacob wants to go to Chicago for his class trip. He needs at least$244 for the bus, hotel stay, and spending money. He already has $104. How much more does he need to payfor his trip?
15. Underline the math verb(s) in this sentence: 6 times a number is 4 less than 16.16. Draw a picture to represent 3 3
4 .17. Determine the change in y in the equation y = 1
6 x−4 between x = 3 and x = 9.
Summary
This chapter talks about the properties of rational numbers and how to add, subtract, multiply, and divide them.It also shows how to add integers and covers additive inverses, absolute values, reciprocals, and the distributiveproperty. Square roots and irrational numbers are discussed as well, and how to classify and order real numbers isexplained in detail. Finally, many real-world problems are given, and the chapter concludes by going over the guessand check and working backward problem-solving strategies.
Properties of Real Numbers Review
Compare the real numbers.
1. 7 and –11
42
www.ck12.org Chapter 2. Properties of Real Numbers
2. 45 and 11
163. 10
15 and 23
4. 0.985 and 3132
5. –16.12 and −3009
Order the real numbers from least to greatest.
6. 811 ,
710 ,
59
7. 27 ,
111 ,
813 ,
47 ,
89
Graph these values on the same number line.
8. 3 13
9. –1.87510. 7
811. 0.16̄12. −55
5
Simplify by applying the Distributive Property.
13. 6n(−2+5n)−n(−3n−8)14. 7x+2(−6x+2)15. −7x(x+5)+3(4x−8)16. −3(−6r−5)−2r(1+6r)17. 1+3(p+8)18. 3(1−5k)−1
Approximate the square root to the nearest hundredth.
19.√
2620.√
33021.√
62522.√
12123.√
22524.√
1125.√
8
Rewrite the square root without using a calculator.
26.√
5027.√
828.√
8029.√
32
Simplify by combining like terms.
30. 8+b+1−7b31. 9n+9n+1732. 7h−3+333. 9x+11− x−3+5x+2
43
2.6. Expressions and the Distributive Property www.ck12.org
Evaluate.
34. 85 −
43
35. 43 −
12
36. 16 +1 5
637. −5
4 ×13
38. 49 ×
74
39. −1 57 ×−2 1
240. 1
9 ÷−1 13
41. −32 ÷
−107
42. −3 710 ÷2 1
443. 1 1
5 −(−3 3
4
)44. 4 2
3 +3 23
45. 5.4+(−9.7)46. (−7.1)+(−0.4)47. (−4.79)+(−3.63)48. (−8.1)− (−8.9)49. 1.58− (−13.6)50. (−13.6)+12− (−15.5)51. (−5.6)− (−12.6)+(−6.6)52. 19.4+24.253. 8.7+3.8+12.354. 9.8−9.455. 2.2−7.3
List all the categories that apply to the following numbers.
56. 10.957. −9
1058. 3π
59. π
2 −π
260. –2161. 8
Which property has been applied?
62. 6.78+(−6.78) = 063. 9.8+11.2+1.2 = 9.8+1.2+11.264. 3a+(4a+8) = (3a+4a)+865. 4
3 −(−5
6
)= 4
3 +56
66. (1) j = j67. 8(11)
(18
)= 8
(18
)(11)
Solve the real-world situation.
68. Carol has 18 feet of fencing and purchased an additional 132 inches. How much fencing does Carol have?69. Ulrich is making cookies for a fundraiser. Each cookie requires 3
8 -pound of dough. He has 12 pounds ofcookie dough. How many cookies can Ulrich make?
70. Herrick bought 11 DVDs at $19.99 each. Use the Distributive Property to show how Herrick can mentallycalculate the amount of money he will need.
44
www.ck12.org Chapter 2. Properties of Real Numbers
71. Bagger 288 is a trench digger, which moves at 38 miles/hour. How long will it take to dig a trench 14 miles
long?72. Georgia started with a given amount of money, a. She spent $4.80 on a large latte, $1.20 on an English muffin,
$68.48 on a new shirt, and $32.45 for a present. She now has $0.16. How much money, a, did Georgia havein the beginning?
73. The formula for the area of a square is A = s2. A square garden has an area of 145 meters2. Find the exactlength of the garden.
Properties of Real Numbers Test
Simplify by using the Distributive Property.
1. −3+7(3a−2)2. 8(3+2q)+5(q+3)
Simplify.
3. 8p−5p4. 9z+33−2z−155. −9
5 ÷26. 1 6
7 ×5 34
7. 12 −3 2
38. 3
14 +158
9. 3.5−5−10.410. 1
6 − (−6.5)× 65
Find the exact value of the square root without a calculator.
11.√
12512.√
1813. How is the multiplicative inverse different from the additive inverse?14. A square plot of land has an area of 168 miles2. To the nearest tenth, what is the length of the land?15. Troy plans to equally divide 228 candies among 16 people. Can this be done? Explain your answer.16. Laura withdrew $15 from the ATM, wrote a check for $46.78, and deposited her paycheck of $678.12. After
her deposit she had $1123.45 in her account. How much money did Laura begin with?17. Will the area of a circle always be an irrational number? Explain your reasoning.18. When would you use the Commutative Property of Multiplication? Give an example to help illustrate your
explanation.
Texas Instruments Resources
In the CK-12 Texas Instruments Algebra I FlexBook® resource, there are graphing calculator activities designedto supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9612 .
45
www.ck12.org
CHAPTER 3 Linear EquationsChapter Outline
3.1 TWO-STEP EQUATIONS AND PROPERTIES OF EQUALITY
3.2 EQUATIONS WITH RATIOS AND PROPORTIONS
Introduction
Aside from simplifying algebraic expressions and graphing functions, solving equations is one of the most importantconcepts in mathematics. To successfully manipulate an equation, you must understand and be able to apply the rulesof mathematics.
Mathematical equations are used in many different career fields. Medical researchers use equations to determinethe length of time it takes for a drug to circulate throughout the body, botanists use equations to determine theamount of time it takes a Sequoia tree to reach a particular height, and environmental scientists can use equations toapproximate the number of years it will take to repopulate the bison species.
46
www.ck12.org Chapter 3. Linear Equations
In this chapter, you will learn how to manipulate linear equations to solve for a particular variable. You already havesome experience solving equations. This chapter is designed to help formalize the mental math you use to answerquestions in daily life.
47
3.1. Two-Step Equations and Properties of Equality www.ck12.org
3.1 Two-Step Equations and Properties ofEquality
Here you’ll learn how to use the Addition Property and Multiplication Property to solve equations that you havewritten in two steps.
Suppose you are saving up to buy a leather jacket that costs $250. You currently have $90 put away, and you’resaving at a rate of $40 per month. Can you write an equation to represent this situation and then solve it in two stepsto find the number of months needed to reach your goal? In this Concept, you’ll learn how to solve problems justlike this.
Guidance
Suppose Shaun weighs 146 pounds and wants to lose enough weight to wrestle in the 130-pound class. Hisnutritionist designed a diet for Shaun so he will lose about 2 pounds per week. How many weeks will it takeShaun to weigh enough to wrestle in his class?
This is an example that can be solved by working backward. In fact, you may have already found the answer byusing this method. The solution is 8 weeks.
By translating this situation into an algebraic sentence, we can begin the process of solving equations. To solve anequation means to “undo” all the operations of the sentence, leaving a value for the variable.
Translate Shaun’s situation into an equation.
−2w+146 = 130
This sentence has two operations: addition and multiplication. To find the value of the variable, we must use bothproperties of equality: the Addition Property of Equality and the Multiplication Property of Equality.
Procedure to Solve Equations of the Form ax+b = some number:
1. Use the Addition Property of Equality to get the variable term ax alone on one side of the equation:
ax = some number
2. Use the Multiplication Property of Equality to get the variable x alone on one side of the equation:
x = some number
Example A
Solve Shaun’s problem.
Solution: −2w+146 = 130
Apply the Addition Property of Equality: −2w+146−146 = 130−146.
Simplify: −2w =−16.
48
www.ck12.org Chapter 3. Linear Equations
Apply the Multiplication Property of Equality: −2w÷−2 =−16÷−2.
The solution is w = 8.
It will take 8 weeks for Shaun to weigh 130 pounds.
Solving Equations by Combining Like Terms
Michigan has a 6% sales tax. Suppose you made a purchase and paid $95.12, including tax. How much was thepurchase before tax?
Begin by determining the noun that is unknown and choose a letter as its representation.
The purchase price is unknown so this is our variable. Call it p. Now translate the sentence into an algebraicequation.
price+(0.06)price = total amount
p+0.06p = 95.12
To solve this equation, you must know how to combine like terms.
Like terms are expressions that have identical variable parts.
According to this definition, you can only combine like terms if they are identical. Combining like terms onlyapplies to addition and subtraction! This is not a true statement when referring to multiplication and division.
The numerical part of an algebraic term is called the coefficient. To combine like terms, you add (or subtract) thecoefficients of the identical variable parts.
Example B
Identify the like terms, and then combine.
10b+7bc+4c+(−8b)
Solution: Like terms have identical variable parts. The only terms having identical variable parts are 10b and −8b.To combine these like terms, add them together.
10b+7bc+4c+−8b = 2b+7bc+4c
You will now apply this concept to the Michigan sales tax situation.
Example C
What was the purchase amount from this section’s opening scenario?
Solution: p+0.06p = 95.12
Combine the like terms: p+0.06p = 1.06p, since p = 1p.
Simplify: 1.06p = 95.12.
Apply the Multiplication Property of Equality: 1.06p÷1.06 = 95.12÷1.06.
Simplify: p = 89.74.
49
3.1. Two-Step Equations and Properties of Equality www.ck12.org
The price before tax was $89.74.
The next several examples show how algebraic equations can be created to represent real-world situations.
Video Review
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79409
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79410
–>
Guided Practice
1. An emergency plumber charges $65 as a call-out fee plus an additional $75 per hour. He arrives at a house at9:30 and works to repair a water tank. If the total repair bill is $196.25, at what time was the repair completed?
2. To determine the temperature in Fahrenheit, multiply the Celsius temperature by 1.8 and then add 32. Determinethe Celsius temperature if it is 89◦F .
Solutions:
1. Translate the sentence into an equation. The number of hours it took to complete the job is unknown, so call it h.
Write the equation: 65+75(h) = 196.25.
Apply the Addition Property and simplify.
65+75(h)−65 = 196.25−65
75(h) = 131.25
50
www.ck12.org Chapter 3. Linear Equations
Apply the Multiplication Property of Equality: 75(h)÷75 = 131.25÷75.
Simplify: h = 1.75.
The plumber worked for 1.75 hours, or 1 hour, 45 minutes. Since he started at 9:30, the repair was completed at11:15.
2. Translate the sentence into an equation. The temperature in Celsius is unknown; call it C.
Write the equation: 1.8C+32 = 89.
Apply the Addition Property and simplify.
1.8C+32−32 = 89−32
1.8C = 57
Apply the Multiplication Property of Equality: 1.8C÷1.8 = 57÷1.8.
Simplify: C = 31.67.
If the temperature is 89◦F , then it is 31.67◦C.
Explore More
Sample explanations for some of the practice exercises below are available by viewing the following video. Notethat there is not always a match between the number of the practice exercise in the video and the number of thepractice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Two-Step Equations (13:50)
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/66
1. Define like terms. Give an example of a pair of like terms and a pair of unlike terms.2. Define coefficient.
In 3 –7, combine the like terms.
3. −7x+39x4. 3x2 +21x+5x+10x2
5. 6xy+7y+5x+9xy6. 10ab+9−2ab7. −7mn−2mn2−2mn+88. Explain the procedure used to solve −5y−9 = 74.
Solve and check your solution.
9. 1.3x−0.7x = 1210. 6x−1.3 = 3.2
51
3.1. Two-Step Equations and Properties of Equality www.ck12.org
11. 5x− (3x+2) = 112. 4(x+3) = 113. 5q−7 = 2
314. 3
5 x+ 52 = 2
315. s− 3s
8 = 56
16. 0.1y+11 = 017. 5q−7
12 = 23
18. 5(q−7)12 = 2
319. 33t−99 = 020. 5p−2 = 3221. 14x+9x = 16122. 3m−1+4m = 523. 8x+3 = 1124. 24 = 2x+625. 66 = 2
3 k26. 5
8 = 12(a+2)
27. 16 =−3d−528. Jayden purchased a new pair of shoes. Including a 7% sales tax, he paid $84.68. How much did his shoes cost
before sales tax?29. A mechanic charges $98 for parts and $60 per hour for labor. Your bill totals $498.00, including parts and
labor. How many hours did the mechanic work?30. An electric guitar and amp set costs $1195.00. You are going to pay $250 as a down payment and pay the rest
in 5 equal installments. How much should you pay each month?31. Jade is stranded downtown with only $10 to get home. Taxis cost $0.75 per mile, but there is an additional
$2.35 hire charge. Write a formula and use it to calculate how many miles she can travel with her money.Determine how many miles she can ride.
32. Jasmin’s dad is planning a surprise birthday party for her. He will hire a bouncy castle and provide party foodfor all the guests. The bouncy castle costs $150 dollars for the afternoon, and the food will cost $3.00 perperson. Andrew, Jasmin’s dad, has a budget of $300. Write an equation to help him determine the maximumnumber of guests he can invite.
Mixed Review
33. Trish showed her work solving the following equation. What did she do incorrectly?
−2c = 36
c = 18
34. Write an expression for the following situation: Yoshi had d dollars, spent $65, and earned $12. He had $96left.
35. Find the domain of the following graph.36. Does the graph represent a function? Explain your answer.
52
www.ck12.org Chapter 3. Linear Equations
37. Find the difference: 12 −
159 .
38. What is the additive identity?39. Find the opposite of –4.1398.
53
3.2. Equations with Ratios and Proportions www.ck12.org
3.2 Equations with Ratios and Proportions
Here you’ll learn what ratios and proportions are and how to solve problems by using them.
Suppose that in 5 minutes, you can type 150 words. If you continued at this rate, do you know how many words youcould type in 12 minutes? In order to answer this question, you could use ratios or a proportion. In this Concept,you’ll learn how to solve equations with ratios and proportions so that you’ll be able to answer questions like thiswhen they arise.
Guidance
Ratios and proportions have a fundamental place in mathematics. They are used in geometry, size changes, andtrigonometry. This Concept expands upon the idea of fractions to include ratios and proportions.
A ratio is a fraction comparing two things with the same units.
A rate is a fraction comparing two things with different units.
You have experienced rates many times: 65 mi/hour, $1.99/pound, and $3.79/yd2. You have also experienced ratios.A “student to teacher” ratio shows approximately how many students one teacher is responsible for in a school.
Example A
The State Dining Room in the White House measures approximately 48 feet long by 36 feet wide. Compare the lengthof the room to the width, and express your answer as a ratio.
Solution:
48 f eet36 f eet
=43
The length of the State Dining Room is 43 the width.
A proportion is a statement in which two fractions are equal: ab = c
d .
Example B
Is 23 = 6
12 a proportion?
Solution: Find the least common multiple of 3 and 12 to create a common denominator.
23=
8126= 6
12
This is NOT a proportion because these two fractions are not equal.
A ratio can also be written using a colon instead of the fraction bar.ab = c
d can also be read, “a is to b as c is to d” or a : b = c : d.
54
www.ck12.org Chapter 3. Linear Equations
The values of a and d are called the extremes of the proportion and the values of b and c are called the means. Tosolve a proportion, you can use the cross products.
The Cross Products of a Proportion:
If ab = c
d , then ad = bc.
Example C
Solve a9 = 7
6 .
Solution: Apply the Cross Products of a Proportion.
6a = 7(9)
6a = 63
Solve for a.
a = 10.5
Video Review
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79415
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79416
–>
Guided Practice
Consider the following situation: A train travels at a steady speed. It covers 15 miles in 20 minutes. How far will ittravel in 7 hours, assuming it continues at the same rate?
55
3.2. Equations with Ratios and Proportions www.ck12.org
Solution:
This is an example of a problem that can be solved using several methods, including proportions. To solve using aproportion, you need to translate the statement into an algebraic sentence. The key to writing correct proportions isto keep the units the same in each fraction.
milestime
=milestime
milestime
6= timemiles
Before we substitute in miles and time into our proportion, we need to have our units consistent. We are given thetrain’s rate in minutes, and asked how far the train will go in 7 hours, so we need to figure out how many minutesare an equivalent amount of time to 7 hours. Since there are 60 minutes in each hour:
7×60 = 420
So 7 hours is equivalent to 420 minutes.
milestime
=milestime
1520
=miles420
Now we solve for the number of miles using the Cross Product of Proportions:
15×420 = miles×20
15×420× 120
= miles×20× 120
15× 42020
= miles
15×21 = miles
315 = miles
The train traveled 315 miles in 7 hours.
56
www.ck12.org Chapter 3. Linear Equations
Explore More
Sample explanations for some of the practice exercises below are available by viewing the following video. Notethat there is not always a match between the number of the practice exercise in the video and the number of thepractice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Ratio and Proportion (10:25)
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/69
Write the following comparisons as ratios. Simplify fractions where possible.
1. $150 to $32. 150 boys to 175 girls3. 200 minutes to 1 hour4. 10 days to 2 weeks
In 5 –10, write the ratios as a unit rate.
5. 54 hot dogs to 12 minutes6. 5000 lbs to 250 in2
7. 20 computers to 80 students8. 180 students to 6 teachers9. 12 meters to 4 floors
10. 18 minutes to 15 appointments11. Give an example of a proportion that uses the numbers 5, 1, 6, and 3012. In the following proportion, identify the means and the extremes: 5
12 = 3584
In 13 –23, solve the proportion.
13. 136 = 5
x14. 1.25
7 = 3.6x
15. 619 = x
1116. 1
x = 0.015
17. 3004 = x
9918. 2.75
9 = x( 2
9)19. 1.3
4 = x1.3
20. 0.11.01 = 1.9
x21. 5p
12 = 311
22. −9x = 4
1123. n+1
11 =−224. A restaurant serves 100 people per day and takes in $908. If the restaurant were to serve 250 people per day,
what might the cash collected be?25. The highest mountain in Canada is Mount Yukon. It is 298
67 the size of Ben Nevis, the highest peak in Scotland.Mount Elbert in Colorado is the highest peak in the Rocky Mountains. Mount Elbert is 220
67 the height of BenNevis and 44
48 the size of Mont Blanc in France. Mont Blanc is 4800 meters high. How high is Mount Yukon?
57
3.2. Equations with Ratios and Proportions www.ck12.org
26. At a large high school, it is estimated that two out of every three students have a cell phone, and one in fiveof all students have a cell phone that is one year old or less. Out of the students who own a cell phone, whatproportion own a phone that is more than one year old?
27. The price of a Harry Potter Book on Amazon.com is $10.00. The same book is also available used for $6.50.Find two ways to compare these prices.
28. To prepare for school, you purchased 10 notebooks for $8.79. How many notebooks can you buy for $5.80?29. It takes 1 cup mix and 3
4 cup water to make 6 pancakes. How much water and mix is needed to make 21pancakes?
30. Ammonia is a compound consisting of a 1:3 ratio of nitrogen and hydrogen atoms. If a sample contains 1,983hydrogen atoms, how many nitrogen atoms are present?
31. The Eagles have won 5 out of their last 9 games. If this trend continues, how many games will they have wonin the 63-game season?
Mixed Review
32. Solve 1516 ÷
58 .
33. Evaluate |9−108|.34. Simplify: 8(8−3x)−2(1+8x).35. Solve for n : 7(n+7) =−7.36. Solve for x : −22 =−3+ x.37. Solve for u : 18 = 2u.38. Simplify: −1
7 −(−1 1
3
).
39. Evaluate: 5× p6 |n| when n = 10 and p =−6.
40. Make a table when −4≤ x≤ 4 for f (x) = 18 x+2.
41. Write as an English phrase: y+11.
Summary
This chapter begins by talking about one-step equations and provides some real-world problems to solve by usingthese types of equations. It then moves on to two-step equations and multi-step equations, covering both thecombining of like terms and the Distributive Property. Next, solving equations with variables on both sides andwith ratios and proportions is discussed, as well as solving problems using scales and indirect measurement. Thechapter then gives instruction on percents, including the percent equation and the percent of change. Finally, theusing a formula problem-solving strategy is highlighted.
Linear Equations Review
Solve for the variable.
1. a+11.2 = 7.32. 9.045+ j = 273. 11 = b+ 5
74. −22 =−3+ k5. −9 = n−66. −6+ l =−277. s
2 =−188. 29 = e
279. u÷−66 = 11
10. −5 f =−110
58
www.ck12.org Chapter 3. Linear Equations
11. 76 =−19p12. −h =−913. q+1
11 =−214. −2−2m =−2215. −5+ d
6 =−716. 32 = 2b−3b+5b17. 9 = 4h+14h18. u−3u−2u = 14419. 2i+5−7i = 1520. −10 = t +15−4t21. 1
2 k−16+2 12 k = 0
22. −1543120 = 3
5 x+ 114
(−11
5 x+ 85
)23. −5.44x+5.11(7.3x+2) =−37.3997+6.8x24. −5(5r+7) = 25+5r25. −7p+37 = 2(−6p+1)26. 3(−5y−4) =−6y−3927. 5(a−7)+2(a−3(a−5)) = 0
Write the following comparisons as ratios. Simplify when appropriate.
28. 10 boys to 25 students29. 96 apples to 42 pears30. $600 to $90031. 45 miles to 3 hours
Write the following as a unit rate.
32. $4.99 for 16 ounces of turkey burger33. 40 computers to 460 students34. 18 teachers to 98 students35. 48 minutes to 15 appointments
Solve the proportion.
36. Solve for n : − 6n−7 =− 2
n+1 .37. Solve for x : −9
5 = x−7x+10 .
38. Solve for b : 5b12 = 3
11 .39. Solve for n : −12
n = 52n+6 .
Write each decimal as a percent.
40. 0.456741. 2.0142. 0.00543. 0.043
Write each percent as a decimal.
44. 23.5%45. 0.08%46. 0.025%
59
3.2. Equations with Ratios and Proportions www.ck12.org
47. 125.4%
Write each percent as a fraction.
48. 78%49. 11.2%50. 10.5%51. 33.3̄%
Solve using the percent equation.
52. 32.4 is 45% of what number?53. 58.7 is what percent of 1,000?54. What is 12% of 78?55. The original price is $44 and the mark-up is 20%. What is the new price?56. An item originally priced at $240 has a 15% discount. What is the new price?57. A pair of shoes originally priced at $89.99 is discounted to $74.99. What is the percent of mark-down?58. A salon’s haircut rose in price from $10 to $14. What is the percent of mark-up?59. The price of an item costing c dollars decreased by $48, resulting in a 30% mark-down. What was the original
price?60. The width of a rectangle is 15 units less than its length. The perimeter of the rectangle is 98 units. What is the
rectangle’s length?61. George took a cab from home to a job interview. The cab fare was $4.00 plus $0.25 per mile. His total fare
was $16.75. How many miles did he travel?62. The sum of twice a number and 38 is 124. What is the number?63. The perimeter of a square parking lot is 260 yards. What are the dimensions of the parking lot?64. A restaurant charges $3.79 for 1
8 of a pie. At this rate, how much does the restaurant charge for the entire pie?65. A 60-watt light bulb consumes 0.06 kilowatts/hour of energy. How long was the bulb left on if it consumed
5.56 kilowatts/hour of energy?66. A 6 1
2 -foot-tall car casts a 33.2-foot shadow. Next to the car is an elephant casting a 51.5-foot shadow. Howtall is the elephant?
67. Two cities are 87 miles apart. How far apart would they be on a map with a scale of 5 inches : 32 miles?
Linear Equations Test
1. School lunch rose from $1.60 to $2.35. What was the percent of mark-up?2. Solve for c : 3c
8 = 11.3. Write 6.35 as a percent.4. Write the following as a simplified ratio: 85 tomatoes to 6 plants.5. Yvonne made 12 more cupcakes than she did yesterday. She made a total of 68 cupcakes over the two days.
How many cupcakes did she make the second day?6. A swing set 8 feet tall casts a 4-foot-long shadow. How long is the shadow of a lawn gnome 4 feet tall?7. Solve the proportion: v
v−2 =−95 .
8. Find the distance between Owosso and Perry if they are 16 cm apart on a map with a scale of 21 km : 4 cm.9. Solve for j : −13
4 −32
(34 j− 4
5
)=−14
5 .10. Solve for m : 2m(2−4)+5m(−8) = 9.11. Job A pays $15 plus $2.00 per hour. Job B pays $3.75 per hour. When will the two jobs pay exactly the same?12. Solve for k : 9.0604+2.062k = 0.3(2.2k+5.9).13. Solve for a : −9−a = 15.14. 46 tons is 11% of what?15. 17% of what is 473 meters?
60
www.ck12.org Chapter 3. Linear Equations
16. Find the percent of change from 73 to 309.17. JoAnn wants to adjust a bread recipe by tripling its ingredients. If the recipe calls for 4 1
3 cups of pastry flour,how much should she use?
18. A sweater originally marked $80.00 went on sale for $45. What was the percent of change?
Texas Instruments Resources
In the CK-12 Texas Instruments Algebra I FlexBook® resource, there are graphing calculator activities designedto supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9613 .
61
www.ck12.org
CHAPTER 4 Graphs of Linear Equationsand Functions
Chapter Outline4.1 GRAPHS IN THE COORDINATE PLANE
4.2 GRAPHS OF LINEAR EQUATIONS
Introduction
The ability to graph linear equations and functions is important in mathematics. In fact, graphing equations andsolving equations are two of the most important concepts in mathematics. If you master these, all mathematicalsubjects will be much easier, even Calculus!
This chapter focuses on the visual representations of linear equations. You will learn how to graph lines fromequations and write functions of graphed lines. You will also learn how to find the slope of a line and how to use aslope to interpret a graph.
Weather, such as temperature and the distance of a thunderstorm can be predicted using linear equations. You willlearn about these applications and more in this chapter.
62
www.ck12.org Chapter 4. Graphs of Linear Equations and Functions
4.1 Graphs in the Coordinate Plane
Here you’ll learn how to find the coordinates of points that have been plotted on a Cartesian plane and how to plotpoints given the coordinates.
Suppose that every year you go to the doctor for a routine physical, and each time the doctor measures your restingheart rate in beats per minute. The doctor has kept this data, and in order to see how your resting heart rate haschanged over time, you want to plot the data on a coordinate plane. Could you do it? Or what if the doctor hasalready plotted the points for you? Could you look at the points and determine what your resting heart rate was eachyear? After completing this Concept, you’ll be able to perform tasks like these.
Guidance
The Coordinate Plane
In a previous Concept, you graphed ordered pairs. This Concept will expand upon your knowledge of graphingordered pairs to include vocabulary and the naming of specific items related to ordered pairs.
An ordered pair is also called a coordinate. The y−value is called the ordinate and the x−value is called the abscissa.
A two-dimensional (2-D) coordinate has the form (x,y).
The 2-D plane that is used to graph coordinates or equations is called a Cartesian plane or a coordinate plane. This2-D plane is named after its creator, Rene Descartes. The Cartesian plane is separated into four quadrants by twoaxes. The horizontal axis is called the x−axis and the vertical axis is called the y−axis. The quadrants are namedusing Roman numerals. The image below illustrates the quadrant names.
The first value of the ordered pair is the x−value. This value moves along the x−axis (horizontally). The secondvalue of the ordered pair is the y−value. This value moves along the y−axis (vertically). This ordered pair providesthe direction of the coordinate.
Multimedia link: For more information on the Cartesian plane and how to graph ordered pairs, visit Purple Math’s- http://www.purplemath.com/modules/plane.htm - website.
63
4.1. Graphs in the Coordinate Plane www.ck12.org
Example A
Find the coordinates of points Q and R.
Solution: In order to get to Q, we move three units to the right, in the positive x direction, then two units down, inthe negative y direction. The x−coordinate of Q is +3; the y−coordinate of Q is –2.
Q = (3,−2)
The coordinates of R are found in a similar way. The x−coordinate is +5 (five units in the positive x direction). They−coordinate is –2 (two units in the negative y direction).
R = (5,−2).
Words of Wisdom from the Graphing Plane
Not all axes will be labeled for you. There will be many times you are required to label your own axes. Someproblems may require you to graph only the first quadrant. Others need two or all four quadrants. The tic marks donot always count by ones. They can be marked in increments of 2, 5, or even 1
2 . The axes do not even need to havethe same increments! The Cartesian plane below shows an example of this.
The increments by which you count your axes should MAXIMIZE the clarity of the graph.
64
www.ck12.org Chapter 4. Graphs of Linear Equations and Functions
In previous Concepts, you learned the vocabulary words relation, function, domain, and range.
A relation is a set of ordered pairs.
A function is a relation in which every x−coordinate matches with exactly one y−coordinate.
The set of all possible x−coordinates is the domain.
The set of all possible y−coordinates is called the range.
Example B
Graphing Given Tables and Rules
If you kept track of the amount of money you earned for different hours of babysitting, you created a relation. Youcan graph the information in this table to visualize the relationship between these two variables.
65
4.1. Graphs in the Coordinate Plane www.ck12.org
Hours 4 5 10 12 16 18
Total $ 12 15 30 36 48 54
The domain of the situation would be all positive real numbers. You can babysit for a fractional amount of time butnot a negative amount of time. The domain would also be all positive real numbers. You can earn fractional money,but not negative money.
Example C
If you read a book and can read twenty pages an hour, there is a relationship between how many hours you read andhow many pages you read. You may even know that you could write the formula as either:
n = 20 ·h n = number of pages; h = time measured in hours. OR...
h =n
20
To graph this relation, you could make a chart. By picking values for the number of hours, you can determine thenumber of pages read. By graphing these coordinates, you can visualize the relation.
TABLE 4.1:
Hours Pages1 201.5 302 403.5 705 100
This relation appears to form a straight line. Therefore, the relationship between the total number of read pages andthe number of hours can be called linear. The study of linear relationships is the focus of the next series of lessons.
66
www.ck12.org Chapter 4. Graphs of Linear Equations and Functions
Video Review
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79606
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79607
–>
Guided Practice
Graph the following points, and give their coordinates:
1. 3 to the left, and 5 down.
2. 4 up, and 1 to the left.
3. 2 to the right, and 2 down.
Solutions:
The yellow point is the origin, from where you start. Following the instructions, arrive at the red point (-3,-5), bluepoint (-1,4), and green point (2,-2).
67
4.1. Graphs in the Coordinate Plane www.ck12.org
Explore More
Sample explanations for some of the practice exercises below are available by viewing the following video. Notethat there is not always a match between the number of the practice exercise in the video and the number of thepractice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: The Coordinate Plane (6:50)
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/74
In questions 1 –6, identify the coordinates of the given letter.
1. D2. A3. F4. E5. B6. C
68
www.ck12.org Chapter 4. Graphs of Linear Equations and Functions
Graph the following ordered pairs on one Cartesian plane. Identify the quadrant in which each ordered pair islocated.
7. (4, 2)8. (–3, 5.5)9. (4, –4)
10. (–2, –3)11.
(12 ,−
34
)12. (–0.75, 1)13.
(−2 1
2 ,−6)
14. (1.60, 4.25)
In 15 –22, using the directions given in each problem, find and graph the coordinates on a Cartesian plane. Thedirections tell how far the point should be from the origin.
15. Six left, four down16. One-half right, one-half up17. Three right, five down18. Nine left, seven up19. Four and one-half left, three up20. Eight right, two up21. One left, one down22. One right, three-quarter down23. Plot the vertices of triangle ABC : (0,0),(4,−3),(6,2).24. The following three points are three vertices of square ABCD. Plot them on a coordinate plane then determine
what the coordinates of the fourth point, D, would be. Plot that fourth point and label it. A(−4,−4) B(3,−4)C(3,3)25. Does the ordered pair (2, 0) lie in a quadrant? Explain your thinking.26. Why do you think (0, 0) is called the origin?27. Becky has a large bag of MMs that she knows she should share with Jaeyun. Jaeyun has a packet of Starburst
candy. Becky tells Jaeyun that for every Starburst he gives her, she will give him three MMs in return. If x isthe number of Starburst that Jaeyun gives Becky, and y is the number of MMs he gets in return, complete eachof the following.
a. Write an algebraic rule for y in terms of x.b. Make a table of values for y with x values of 0, 1, 2, 3, 4, 5.
69
4.1. Graphs in the Coordinate Plane www.ck12.org
c. Plot the function linking x and y on the following scale 0≤ x≤ 10,0≤ y≤ 10.
28. Consider the rule: y = 14 x+8. Make a table. Then graph the relation.
29. Ian has the following collection of data. Graph the ordered pairs and make a conclusion from the graph.
TABLE 4.2:
Year % of Men Employed in the United States1973 75.51980 72.01986 71.01992 69.81997 71.32002 69.72005 69.62007 69.82009 64.5
70
www.ck12.org Chapter 4. Graphs of Linear Equations and Functions
4.2 Graphs of Linear Equations
Here you’ll learn how to use coordinate pairs to graph linear equations.
Suppose that there is a linear relationship between your annual income and the amount you must pay in state incometax. What if you wanted to make a graph that showed how much tax you must pay based on your income? Could youdo it? After completing this Concept, you’ll be able to take a linear equation like the one representing the situationjust described and graph it on a coordinate plane.
Watch This
To see some simple examples of graphing linear equations by hand, view the video
Khan Academy graphing lines (9:49)
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/75
.
The narrator of the video models graphing linear equations by using a table of values to plot points and thenconnecting those points with a line. This process reinforces the procedure of graphing lines by hand.
Guidance
Previously, you learned how to solve equations in one variable. The answer was of the form variable = somenumber. In this Concept, you will learn how to solve equations with two variables. Below are several examples oftwo-variable equations:
p = 20(h)
m = 8.15(n)
y = 4x+7
You have seen each of these equations used in a previous Concept, and you will see how to graph them in thisConcept. Their solutions are not one value because there are two variables. The solutions to these equations arepairs of numbers. These pairs of numbers can be graphed in a Cartesian plane.
The solutions to an equation in two variables are sets of ordered pairs.
The solutions to a linear equation are the coordinates on the graphed line.
By making a table, you are finding the solutions to the equation with two variables.
71
4.2. Graphs of Linear Equations www.ck12.org
Example A
Graph p = 2(h).
Solution:
Make a table and then graph the points:
TABLE 4.3:
h p0 01 22 43 6
Example B
Graph y = 4x+7.
Solution:
Make a table and then graph the points.
TABLE 4.4:
x y-1 30 71 13
72
www.ck12.org Chapter 4. Graphs of Linear Equations and Functions
Example C
A taxi fare costs more the further you travel. Taxis usually charge a fee on top of the per-mile charge. In this case,the taxi charges $3 as a set fee and $0.80 per mile traveled. Find all the possible solutions to this equation.
Solution: Here is the equation linking the cost in dollars (y) to hire a taxi and the distance traveled in miles (x):y = 0.8x+3.
This is an equation in two variables. By creating a table, we can graph these ordered pairs to find the solutions.
TABLE 4.5:
x (miles) y (cost $)0 310 1120 1930 2740 35
73
4.2. Graphs of Linear Equations www.ck12.org
The solutions to the taxi problem are located on the green line graphed above. To find any cab ride cost, you justneed to find the y of the desired x.
Video Review
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79616
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/79617
–>
Guided Practice
Graph m = 8.15(n).
Solution:
To graph this linear equation, we will make a table of some points. We can plug in values for h to get values for m.For example:
m = 8.15(−1) =−8.15
74
www.ck12.org Chapter 4. Graphs of Linear Equations and Functions
TABLE 4.6:
n m−1 −8.150 01 8.152 16.3
Next, we will graph each point, and then connect the points with a line.
75
4.2. Graphs of Linear Equations www.ck12.org
Explore More
Sample explanations for some of the practice exercises below are available by viewing the following video. Notethat there is not always a match between the number of the practice exercise in the video and the number of thepractice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Graphs of Linear Equations (13:09)
MEDIAClick image to the left or use the URL below.URL: http://www.ck12.org/flx/render/embeddedobject/76
1. What are the solutions to an equation in two variables? How is this different from an equation in one variable?2. Think of a number. Triple it, and then subtract seven from your answer. Make a table of values and plot the
function that this sentence represents.
Graph the solutions to each linear equation by making a table and graphing the coordinates.
3. y = 2x+74. y = 0.7x−45. y = 6−1.25x
Mixed Review
6. Find the sum: 38 +
15 −
59 .
7. Solve for m : 0.05m+0.025(6000−m) = 512.8. Solve the proportion for u : 16
u−8 = 36u .
9. What does the Additive Identity Property allow you to do when solving an equation?10. Shari has 28 apples. Jordan takes 1
4 of the apples. Shari then gives away 3 apples. How many apples doesShari have?
11. The perimeter of a triangle is given by the formula Perimeter = a+b+ c, where a,b, and c are the lengths ofthe sides of a triangle. The perimeter of 4ABC is 34 inches. One side of the triangle is 12 inches. A secondside is 7 inches. How long is the remaining side of the triangle?
12. Evaluate y2−16+10y+2x2 , for x = 2 and y =−2.
Summary
This chapter deals with graphing, including plotting points on a coordinate plane and graphing linear equations. Aspart of this topic, the chapter discusses horizontal and vertical graphs and analyzing linear graphs. It then moves onto intercepts and slopes, showing how to find intercepts by substituting and by using the Cover-Up Method and whyslopes should be interpreted as rates of change. Slope-intercept form is also covered. Next, direct variation is talkedabout, and many real-world problems are solved. Finally, the chapter concludes by touching on function notationand linear functions, giving instruction on the graphs of linear functions and how to solve problems by making andreading a graph.
76
www.ck12.org Chapter 4. Graphs of Linear Equations and Functions
Graphing Linear Equations and Functions Review
Define the following words:
1. x− intercept2. y− intercept3. direct variation4. parallel lines5. rate of change
In 6 –11, identify the coordinates using the graph below.
6. D7. F8. A9. E
10. B11. C
In 12 –16, graph the following ordered pairs on one Cartesian Plane.
12.(1
2 ,−4)
13. (–6, 1)14. (0, –5)
77
4.2. Graphs of Linear Equations www.ck12.org
15. (8, 0)16.
(32 ,
84
)In 17 and 18, graph the function using the table.
17. .
TABLE 4.7:
x y–2 7–1 90 111 132 15
18.
d 0 1 2 3 4 5 6
t 0 75 150 225 300 375 450
In 19 –24, graph the following lines on one set of axes.
19. y =−32
20. x = 421. y = 522. x =−323. x = 024. y = 0
In 25 –28, find the intercepts of each equation.
25. y = 4x−526. 5x+5y = 2027. x+ y = 728. 8y−16x = 48
In 29 –34, graph each equation using its intercepts.
29. 3x+7y = 2130. 2y−5x = 1031. x− y = 432. 16x+8y = 1633. x+9y = 1834. 7+ y = 1
7 x
In 35 –44, find the slope between the sets of points.
35. (3, 20) and (19, 8)36. (12, 5) and (12, 0)
78
www.ck12.org Chapter 4. Graphs of Linear Equations and Functions
37.(−1
2 ,5)
and(3
2 ,3)
38. (8, 3) and (12, 3)39. (14, 17) and (–14, –22)40. (1, 4) and (18, 6)41. (10, 6) and (10, –6)42. (–3, 2) and (19, 5)43. (13, 9) and (2, 9)44. (10, –1) and (–10, 6)
In 45 –50, determine the rate of change.
45. Charlene reads 150 pages in 3 hours.46. Benoit cuts 65 onions in 1.5 hours.47. Brad drives 215 miles in 3.9 hours.48. Reece completes 65 jumping jacks in one minute.49. Harriet is charged $48.60 for 2,430 text messages.50. Samuel can eat 65 hotdogs in 22 minutes.
In 51 –55, identify the slope and the y−intercept of each equation.
51. x+ y = 352. 1
3 x = 7+ y53. y = 2
5 x+354. x = 455. y = 1
4
In 56 –60, graph each equation.
56. y = 56 x−1
57. y = x58. y =−2x+259. y =−3
8 x+560. y =−x+4
In 61 –63, decide whether the given lines are parallel.
61. 3x+6y = 8 and y = 2x−862. y = x+7 and y =−7− x63. 2x+4y = 16 and y = −1
2 x+6
In 64 –70, evaluate the function for the indicated value.
64. g(n) =−2|n−3|; Find g(7).65. h(a) = a2−4a; Find h(8).66. p(t) = 3t +1; Find p
(16
).
67. g(x) = 4|x|; Find g(−3).68. h(n) = 1
3 n−4; Find h(24).69. f (x) = x+8
6 ; Find f (20).70. r(c) = 0.06(c)+ c; Find r(26.99).71. The distance traveled by a semi-truck varies directly with the number of hours it has been traveling. If the
truck went 168 miles in 4 hours, how many miles will it go in 7 hours?
79
4.2. Graphs of Linear Equations www.ck12.org
72. The function for converting Fahrenheit to Celsius is given by C(F) = F−321.8 . What is the Celsius equivalent to
84◦F?73. Sheldon started with 32 cookies and is baking more at a rate of 12 cookies/30 minutes. After how many hours
will Sheldon have 176 cookies?74. Mixture A has a 12% concentration of acid. Mixture B has an 8% concentration of acid. How much of each
mixture do you need to obtain a 60-ounce solution with 12 ounces of acid?75. The amount of chlorine needed to treat a pool varies directly with its size. If a 5,000-gallon pool needs 5 units
of chlorine, how much is needed for a 7,500-gallon pool?76. The temperature (in Fahrenheit) outside can be predicted by crickets using the rule, “Count the number of
cricket chirps in 15 seconds and add 40.”
a. Convert this expression to a function. Call it T (c), where T = temperature and c = number of chirps in15 seconds.
b. Graph this function.c. How many chirps would you expect to hear in 15 seconds if the temperature were 67◦F?d. What does the y−intercept mean?e. Are there values for which this graph would not predict well? Why?
Graphing Linear Equations and Functions Test
Give the location of the following ordered pairs using the graph below.
1. A2. B3. C
4. Graph y = 73 x−4.
80
www.ck12.org Chapter 4. Graphs of Linear Equations and Functions
5. Find the slope between (–3, 5) and (–1.25, –2.25).6. Find the intercepts of 6x+9y = 54.7. In 2004, the high school graduation rate in the state of New Jersey was 86.3%. In 2008, the high school
graduation rate in New Jersey was 84.6%. Determine the average rate of change. Use this information tomake a conclusion regarding the graduation rate in New Jersey (source: http://nces.ed.gov/pubs2010/2010341.pdf ).
8. Identify the slope and y−intercept of 4x+7y = 28.9. Identify the slope and y−intercept of y = −3
5 x−8.10. Graph y = 9.11. Graph the line containing (3, 5) and (3, –7). What type of line have you created?12. The number of cups of milk is directly proportional to the number of quarts. If 26 quarts yields 104 cups of
milk, how many cups of milk is 2.75 quarts?13. Graph the direct variation situation using the table below:
0 1 2 3 4 5 6 7
0 2.25 4.5 6.75 9 11.25 13.5 15.75
14. h varies directly as m, and when m = 4,h = 27. Find h when m =−5.5.15. h(n) = 1
2 |6−n|+11; Find h(25).16. Are these lines parallel? y =−2x+1 and y = 2x−117. Mixture A has a 2% hydrogen solution and mixture B has a 1.5% hydrogen solution. How much of mixture B
needs to be added to 6 ounces of mixture A to obtain a value of 0.51?
Texas Instruments Resources
In the CK-12 Texas Instruments Algebra I FlexBook® resource, there are graphing calculator activities designedto supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9614 .
81
