Download Multi-Step Equations and Inequalities

Multi-Step
Equations and
Inequalities
3A
Solving Multi-Step
Equations
3-1
Properties of Rational
Numbers
3-2 Simplifying Algebraic
Expressions
3-3 Solving Multi-Step
Equations
LAB Model Equations with
Variables on Both Sides
3-4 Solving Equations with
Variables On Both Sides
3B
Solving Inequalities
3-5
3-6
Inequalities
Solving Inequalities by
Adding or Subtracting
Solving Inequalities by
Multiplying or Dividing
Solving Two-Step
Inequalities
3-7
3-8
KEYWORD: MT8CA Ch3
Equations and inequalities
can describe how fast
water is flowing in
streams and rivers.
Merced River,
Yosemite National Park
112
Chapter 3
Vocabulary
Choose the best term from the list to complete each sentence.
1. A letter that represents a value that can change is
called a(n) __?__.
algebraic expression
equation
numerical expression
2. A(n) __?__ has one or more variables.
3. A(n) __?__ is a mathematical sentence that uses an
equal sign to show that two expressions have the
same value.
variable
4. A mathematical phrase that contains operations and numbers is called
a(n) __?__. It does not have an equal sign.
Complete these exercises to review skills you will need for this chapter.
Operations with Fractions
Evaluate each expression.
2
1
5. 3 2
5 8
9. 6 1
5
13
19
6. 1
2
8
4
7 6
7. 8 1
1
11
121
10. 1
2
144
1
5
11. 6 8
9
9
8. 1
1
0
3
19
4
12. 2
5
0
Solve One-Step Equations
Use mental math to solve each equation.
13. x 7 21
14. p 3 22
15. 14 v 30
16. b 5 6
17. t 33 14
18. w 7 7
Connect Words and Equations
Write an equation to represent each situation.
19. The perimeter P of a rectangle is the sum of twice the length and twice
the width w.
20. The volume V of a rectangular prism is the product of its three
dimensions: length , width w, and height h.
21. The surface area S of a sphere is the product of 4π and the square of the
radius r.
22. The cost c of a telegram of 18 words is the cost f of the first 10 words plus
the cost a of each additional word.
Multi-Step Equations and Inequalities
113
The information below “unpacks” the standards. The academic vocabulary is
highlighted and defined to help you understand the language of the standards.
Refer to the lessons listed after each standard for help with the math terms and
phrases. The Chapter Concept shows how the standard is applied in this chapter.
California
Standard
AF1.1 Use variables and
appropriate operations to write an
expression, an equation, an
inequality, or a system of equations
or inequalities that represents a
verbal description (e.g., three less
than a number, half as large as area A).
Academic
Vocabulary
Chapter
Concept
variable a symbol, usually a letter, used to show an You rewrite a verbal statement
amount that can change
using mathematical symbols.
Example: x
verbal using words
Example: a number is greater
than –5
n > –5
(Lesson 3-5)
AF1.3 Simplify numerical
expressions by applying properties
of rational numbers (e.g., identity,
inverse, distributive, associative,
commutative) and justify the
process used.
property a characteristic of numbers, operations,
or equations
Example: One property of addition is that you
can add numbers in any order without changing
the sum.
You use mathematical
properties to simplify
expressions.
You give reasons for each step
when you simplify expressions.
justify give a reason for
(Lessons 3-1 and 3-2)
AF4.0 Students solve
simple linear equations and
inequalities over the rational
numbers.
solve find the value or values of an unknown
quantity that make one side of an equation equal
to the other side (make the equation true)
You find the values of a variable
that make an inequality true.
(Lessons 3-6 and 3-7)
AF4.1 Solve two-step
linear equations and inequalities in
one variable over the rational
numbers, interpret the solutions
in the context from which they
arose, and verify the reasonableness
of the results.
(Lesson 3-8)
114
Chapter 3
interpret to understand and explain the meaning of You understand and can
explain the meaning of
context in this case, a real-world situation
solutions to inequalities.
Reading Strategy: Read a Lesson for Understanding
Before you begin reading a lesson, find out which standard or standards are the main focus
of the lesson. These standards are located at the top of the first page of the lesson. Reading
with the standards in mind will help guide you through the lesson material. You can use the
following tips to help you follow the math as you read.
Lesson Features
Reading Tips
Identify the standard or standards
of the lesson. Then skim through
the lesson to get a sense of how the
standards are covered.
Work through each example. The
examples help to demonstrate
the standards.
Check your understanding of
the lesson by answering the
Think and Discuss questions.
Try This
Use Lesson 3-1 in your textbook to answer each question.
1. What is the standard of the lesson?
2. What questions or problems did you have when you read the lesson?
3. Write your own example problem similar to Example 1.
4. What skill is being practiced in the second Think and Discuss question?
Multi-Step Equations and Inequalities
115
3-1
California
Standards
AF1.3 Simplify numerical
expressions by applying
properties of rational numbers
(e.g., identity, inverse, distributive,
associative, commutative) and
justify the process used.
Properties of
Rational Numbers
Why learn this? You can use mental math and
properties of rational numbers to calculate costs when
shopping. (See Exercises 40 and 41.)
In Chapter 2, you performed operations with rational numbers. The
following properties are useful when you simplify expressions that
contain rational numbers.
PROPERTIES OF ADDITION AND MULTIPLICATION
Vocabulary
Commutative Property
Associative Property
Distributive Property
Words
Algebra
3 4 12 4 12 3
a b b a
2 17 17 2
ab ba
(6 8) 9 6 (8 9)
(a b) c a (b c)
Commutative Property
You can add numbers
in any order. You can
multiply numbers in
any order.
Associative Property
When you are only
adding or only
multiplying, changing
the grouping will not
affect the sum or
product.
EXAMPLE
Numbers
1
2 18 16 2 18 16
(a b) c a (b c)
Identifying Properties of Addition and Multiplication
Name the property that is illustrated in each equation.
3 (4 x) (3 4) x
3 (4 x) (3 4) x
The factors are grouped differently.
Associative Property of Multiplication
(9) 2 2 (9)
(9) 2 2 (9)
The order of the numbers changed.
Commutative Property of Addition
You can use the properties of rational numbers to rearrange or
regroup numbers in a way that helps you do math mentally.
116
Chapter 3 Multi-Step Equations and Inequalities
EXAMPLE
2
Using the Commutative and Associative Properties
Simplify each expression. Justify each step.
43 29 7
43 29 7 43 7 29
(43 7) 29
50 29 79
Compatible numbers
help you do math
mentally. Try to make
multiples of 5 or 10.
They are simpler to
use when multiplying.
Commutative Property of Addition
Associative Property of Addition
Add.
1
15 7 5
1
1
15 7 5 15 5 7
Commutative Property of Multiplication
15 5 7
Associative Property of Multiplication
3 7 21
Multiply.
1
The Distributive Property is also helpful when you do math mentally.
DISTRIBUTIVE PROPERTY
Numbers
7(6 12) 7 6 7 12
a(b c) a b a c
Algebra
5(7 3) 5 7 5 3
a(b c) a b a c
When you need to find the product of two numbers, write one of the
numbers as a sum or difference. Then use the Distributive Property to
help you find the product mentally.
EXAMPLE
3
Using the Distributive Property
Write each product using the Distributive Property. Then simplify.
Break the larger
factor into a sum
or difference that
contains a multiple
of 10.
5(43)
5(43) 5(40 3)
5 40 5 3
200 15 215
6(28)
6(28) 6(30 2)
6 30 6 2
180 12 168
Rewrite 43 as a sum.
Distributive Property
Multiply. Then add.
Rewrite 28 as a difference.
Distributive Property
Multiply. Then subtract.
Think and Discuss
1. Explain which property you would use to simplify 5.8 (0.2 4).
2. Describe two ways to use the Distributive Property to find 8 45.
3-1 Properties of Rational Numbers
117
3-1
California
Standards Practice
AF1.3
Exercises
KEYWORD: MT8CA 3-1
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
Name the property that is illustrated in each equation.
2. (5) 12 12 (5)
1. y 16 16 y
See Example 2
See Example 3
Simplify each expression. Justify each step.
3. 17 19 3
4. 51 48 9
5. 4 7 25
1
6. 3 8 9
7. 5 (13 2)
1
3
8. 4 3 4
Write each product using the Distributive Property. Then simplify.
9. 8(21)
10. 5(62)
11. 3(18)
12. 6(49)
13. 4(99)
14. (59)5
INDEPENDENT PRACTICE
See Example 1
Name the property that is illustrated in each equation.
15. 4x x 4
See Example 2
See Example 3
16. (7 1.5) 2 7 (1.5 2)
Simplify each expression. Justify each step.
17. 4 89 16
18. (0.5 9) 2
19. 2 13 50
20. 69 17 1
21. 8.8 (15 0.2)
1
22. 4 9 12
Write each product using the Distributive Property. Then simplify.
23. 7(19)
24. (53)4
25. 12(11)
26. (98)2
1
27. 2(42)
1
28. 3(87)
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP6.
Name the property that is illustrated in each equation.
29. 7(9 x) 7 9 7x
30. 16 0 16
31. (5 y) z 5 (y z)
32. 9 1 = 9
33. m 12n 12n m
34. 3(2 t) 3 2 3t
Simplify each expression. Justify each step.
35. 13 9 7 11
36. 4 3 25 2
38. Consumer Math Mikiko is buying five DVDs at
SaveMart. How can she use the Distributive Property
to find the total cost of the DVDs before tax?
39. Consumer Math Jerome is buying a DVD, a pair
of jeans, and a t-shirt at SaveMart. Show how he can
use properties of rational numbers to find the total
cost of the items before tax.
118
Chapter 3 Multi-Step Equations and Inequalities
1
2
1
3
37. 62 75 2 5
SaveMart Price List
Item
Price
DVD
$18
Jeans
$23
T-Shirt
$12
Write an example of each property using rational numbers.
40. Distributive Property
41. Associative Property of Multiplication
42. Commutative Property of Addition
Complete each equation. Then name the property that is illustrated in each.
(4 7) 8 4 8 7
45. (x y) 12 x (
44. 4.8 6 6 46. 9(8 z) 9 12)
47. Weather Leann wants to know the
total amount of rainfall in Berkeley,
California, from 2002 through 2005.
Explain how she can use mental math
and properties of rational numbers to
calculate this amount.
48. Reasoning Make a conjecture: Is
division of rational numbers
commutative? Explain your thinking.
9z
Annual Rainfall, Berkeley, CA
28
Total rainfall (in.)
43.
26
24
22
26
24
M708CS_C03_L01_301_A
21
19
20
18
0
2002
2003
2004
2005
Year
49. What’s the Error? A student writes,
“You can use the Associative Property
of Addition to change the order of two numbers before you add them.”
What is the student’s error?
50. Write About It A case of cat food has 24 cans. Explain how to use mental
math and the Distributive Property to find the number of cans in 5 cases.
51. Challenge Simplify the expression 1213 16 14. Justify each step.
NS1.2,
NS1.5,
AF1.3
52. Multiple Choice The equation 3 (5 x) 3 (x 5) is an example
of which property?
A
Associative Property of Addition
C
Commutative Property of Multiplication
B
Commutative Property of Addition
D
Distributive Property
53. Multiple Choice Which is an example of the Associative Property of Multiplication?
6
A
(4) y y (4)
C
1
1
(18 6) 18
3
3
B
2(9 1) 2 9 2 1
D
3 5 3 2 3(5 2)
Write each decimal as a fraction in simplest form. (Lesson 2-1)
54. 0.68
56. 2.01
55. 1.4
57. 0.04
Divide. Write each answer in simplest form. (Lesson 2-5)
1
3
58. 2 8
2
2
59. 43 29
5
60. 8 2
4
1
61. 25 1
0
3-1 Properties of Rational Numbers
119
3-2
California
Standards
AF1.3 Simplify numerical
expressions by applying
properties of rational numbers
(e.g., identity, inverse, distributive,
associative, commutative) and
justify the process used.
Simplifying Algebraic
Expressions
Who uses this? Consumers can simplify algebraic
expressions to find the total cost of tickets. (See Exercise 52.)
In the expression below, 7x, 5, 3y, and 2x are terms. A term can be a
number, a variable, or a product of numbers and variables. Terms in
an expression are separated by plus or minus signs.
Constant
Vocabulary
term
like terms
coefficient
constant
equivalent expressions
Coefficients
Like terms , such as 7x and 2x, can be grouped together because they
have the same variable raised to the same power. Often, like terms have
different coefficients. A coefficient is a number that is multiplied by a
variable in an algebraic expression. A constant is a number that does
not change. Constants, such as 4, 0.75, and 11, are also like terms.
When you combine like terms, you change the way an expression
looks but not the value of the expression. Equivalent expressions
have the same value for all values of the variables.
EXAMPLE
1
Combining Like Terms in One-Variable Expressions
Combine like terms.
When you rearrange
terms, move the
operation in front of
each term with that
term.
7x 2x
7x 2x
Identify like terms.
9x
Combine coefficients: 7 2 9.
5m 2m 8 3m 6
120
5m 2m 8 3m 6
Identify like terms.
5m 2m 3m 8 6
Commutative Property
0m 14
Combine coefficients.
14
Simplify.
Chapter 3 Multi-Step Equations and Inequalities
EXAMPLE
2
Combining Like Terms in Two-Variable Expressions
Combine like terms.
7a 4a 3b 5
7a 4a 3b 5
11a 3b 5
Identify like terms.
Combine coefficients: 7 4 11.
k 3n 2n 4k
1k 3n 2n 4k
Identify like terms; the coefficient
of k is 1 because 1k k.
1k 4k 3n 2n
5k n
Commutative Property
Combine coefficients: 1 4 5;
3 2 1.
3f – 9g 15
3f 9g 15
No like terms
3f 9g 15
To simplify an expression, perform all possible operations, including
combining like terms. You may need to use the Associative,
Commutative, or Distributive Properties.
EXAMPLE
3
Using the Distributive Property to Simplify
Simplify 6(y 8) 5y.
6(y 8) 5y
6(y) 6(8) 5y
Distributive Property
6y 48 5y
Multiply.
6y 48 5y
Identify like terms.
6y 5y 48
Commutative Property
1y 48
Combine coefficients: 6 5 1.
y 48
1y y
Think and Discuss
1. Describe the first step in simplifying the expression
2 8(3y 5) y.
2. Tell how many sets of like terms are in the expression in
Example 1B. What are they?
3-2 Simplifying Algebraic Expressions
121
3-2
California
Standards Practice
AF1.3
Exercises
KEYWORD: MT8CA 3-2
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
See Example 2
Combine like terms.
1. 9x 4x
2. 2z 5 3z
3. 6f 3 4f 5 10f
4. 9g 8g
5. 7p 9 p
6. 3x 5 x 3 4x
7. 6x 4y x 4y
8. 4x 5y y 3x
9. 5x 3y 4x 2y
11. 7g 5h 12
12. 3h 4m 7h 4m
13. 4(r 3) 3r
14. 7(3 x) 2x
15. 7(t 8) 5t
16. 3(2 p) 4p
17. 2(5y 4) 9
18. 7(5 2m) m
10. 6p 3p 7z 3z
See Example 3
Simplify.
INDEPENDENT PRACTICE
See Example 1
See Example 2
See Example 3
Combine like terms.
19. 7y 6y
20. 4z 5 2z
21. 3a 6 2a 9 5a
22. 5z z
23. 9x 3 4x
24. 9b 6 3b 3 b
25. 3z 4z b 5
26. 5a a 4z 3z
27. 9x 8y 2x 8 4y
28. 6x 2 3x 6q
29. 7d d 3e 12
30. 16a 7c 5 7a c
31. 5(y 2) y
32. 2(3y 7) 6y
33. 3(x 6) 8x
34. 3(4y 5) 8
35. 6(2x 8) 9x
36. 4(4x 4) 3x
Simplify.
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP6.
37. Geometry A rectangle has length 5x and width x. Write and simplify an
expression for the perimeter of the rectangle.
38. Hobbies Charlie has x state quarters. Ty has 3 more quarters than
Charlie has. Vinnie has 2 times as many quarters as Ty has. Write and
simplify an expression to show how many state quarters they have in all.
39. Reasoning Determine whether the expression r 17m 8 is equivalent
to 3(2r 4m) 5(m 3 r) 7. Use properties to justify your answer.
Simplify each expression. Justify each step.
122
40. 6(4 7k) 16 14
41. 5d 7 4d 2d 6
42. 7x 2(y 3x)
43. 3r 6r 2 5r 9r
44. 6y 3 7y 10 3z
45. 2(k 5) 3 k
Chapter 3 Multi-Step Equations and Inequalities
Write and simplify an expression for each situation.
46. Business A promoter charges $7 for each adult ticket, plus an additional
$2 per ticket for tax and handling. What is the total cost of x tickets?
47. Sports Write an expression for the total number of medals won in the
2004 Summer Olympics by the countries shown below.
United States
Great Britain
Brazil
Lithuania
35 Gold
39 Silver
29 Bronze
9 Gold
9 Silver
12 Bronze
4 Gold
3 Silver
3 Bronze
1 Gold
2 Silver
0 Bronze
Write an algebraic expression for each verbal description. Then simplify the
expression.
48. four times the sum of m and p, decreased by six times m
49. y squared minus twice the sum of x and y squared
50. the product of three and r, increased by the sum of nine, 2r, and one
51. What’s the Error? A student said that 3x 4y can be simplified to 7xy
by combining like terms. What error did the student make?
52. Write About It Write an expression that can be simplified by combining
like terms. Then write an expression that cannot be simplified, and
explain why it is already in simplest form.
53. Challenge Simplify the expression 36x 4 9x 5x 1
.
0
1
1
2
AF1.3,
AF4.1
54. Multiple Choice Which expression is equivalent to p 3 5t 4p?
A
5p 2t
B
7p 5t
C
5(p t) 3
D
3 5t 4p
55. Gridded Response Simplify 3(2x 7) 10x. What is the coefficient of x?
Solve. (Lesson 2-8)
x 14
8
56. 3
a 35
57. 5 9
1
58. 4w 7 10.7
Complete each equation. Then name the property that is illustrated in each.
(Lesson 3-1)
59.
(x 3) 2 x 2 3
60. 4.8 6 6 61. 8(5 9) (8 )9
3-2 Simplifying Algebraic Expressions
123
3-3
Solving Multi-Step
Equations
Why learn this? You can solve problems about average
speed by solving multi-step equations. (See Example 3.)
California
Standards
Extension of
AF4.1 Solve
two-step linear equations and
inequalities in one variable over
the rational numbers, interpret the
solution or solutions in the context
from which they arose, and verify the
reasonableness of the results.
EXAMPLE
A multi-step equation requires more than two steps to solve. To solve
a multi-step equation, you may have to simplify the equation first by
combining like terms.
1
Solving Equations That Contain Like Terms
Solve 3x 5 6x 7 25.
3x 5 6x 7 25
3x 6x 5 7 25
Commutative Property of Addition
9x 2 25
Combine like terms.
2 2
Since 2 is subtracted from 9x, add 2 to
9x
27
both sides.
9x
27
9
9
x 3
Since x is multiplied by 9, divide both
sides by 9.
If an equation contains fractions, it may help to multiply both sides of the
equation by the least common denominator (LCD) to clear the fractions
before you isolate the variable.
EXAMPLE
2
Solving Equations That Contain Fractions
Solve.
3y
5
1
7
7
7
3y
5
1
7 7 7 7 7
73y 75 71
7
7
7
1 5
1
1 3y
1
771 7 71 771
3y 5 1
5 5
3y
6
3y
6
3
3
y 2
124
Chapter 3 Multi-Step Equations and Inequalities
Multiply both sides by 7.
Distributive Property
Simplify.
Since 5 is added to 3y, subtract 5 from
both sides.
Since y is multiplied by 3, divide both
sides by 3.
Solve.
5p
p
1
11
2
6
6
3
5p
p
1
11
6 6 3 2 6 6
5p
p
1
6 6 6 3 6 1 6 1
2
6
1 11
3 1
2 p
1 5p
6 6 6 3 6 2 6 6
1
1
1
1
The least common
denominator (LCD) is
the smallest number
that each of the
denominators will
divide into evenly.
5p 2p 3 11
7p 3 11
3 3
7p
14
See Skills Bank p. SB9.
7p
14
7
7
p2
EXAMPLE
3
Multiply both sides by 6, the
LCD of the fractions.
Distributive Property
Simplify.
Combine like terms.
Since 3 is subtracted from 7p,
add 3 to both sides.
Since p is multiplied by 7,
divide both sides by 7.
Travel Application
On the first day of her vacation, Carly drove m miles in 4 hours. On
the second day, she drove twice as far in 7 hours. If her average
speed for the two days was 62.8 mi/h, how far did she drive on the
first day? Round your answer to the nearest tenth of a mile.
Carly’s average speed is her total distance for the two days divided by
the total time.
total distance
average speed
total time
m 2m
62.8
47
3m
62.8
11
3m
11 11 11(62.8)
Substitute m 2m for total distance
and 4 7 for total time.
Simplify.
Multiply both sides by 11.
3m 690.8
3m
690.8
3
3
Divide both sides by 3.
m 230.27
Carly drove approximately 230.3 miles on the first day.
Think and Discuss
1. List the steps required to solve 3x 4 2x 7.
2. Tell how you would clear the fractions in 34x 23x 58 1.
3-3 Solving Multi-Step Equations
125
3-3
California
Standards Practice
Extension of
AF4.1;
AF4.2
Exercises
KEYWORD: MT8CA 3-3
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
See Example 2
Solve.
1. 7d 12 2d 3 18
2. 3y 4y 6 20
3. 10e 2e 9 39
4. 4c 5 14c 67
5. 5h 6 8h 3h 76
6. 7x 2x 3 32
4x
3
1
1
1
7. 1
3
3
3
8. 2 6 3 2
y
2p
4
6
9. 5 5 5
See Example 3
5y
1
1
15
1
10. 8z 4 4
11. Travel Barry’s family drove 843 mi to see his grandparents. On the first
day, they drove 483 mi. On the second day, how long did it take to reach
Barry’s grandparents’ house if they averaged 60 mi/h?
INDEPENDENT PRACTICE
See Example 1
See Example 2
Solve.
12. 5n 3n n 5 26
13. 81 7k 19 3k
14. 36 4c 3c 22
15. 12 5w 4w 15
16. 37 15a 5a 3
17. 30 7y 35 6y
p
3
1
18. 8 8 38
4g
See Example 3
g
7h
4h
18
19. 1
1
1
2
2
2
20. 1
8 1
1
6
6
6
7
3m
m
1
21. 1
6 3 4
2
4
2b
6b
22. 1
13 26
3
3x
21x
1
23. 4 32 18
3
3
24. Recreation Lydia rode 243 miles in a three-day bike trip. On the first
day, Lydia rode 67 miles. On the second day, she rode 92 miles. How many
miles per hour did she average on the third day if she rode for 7 hours?
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP6.
Solve and check.
5n
1
3
25. 8 2 4
26. 4n 11 7n 13
27. 7b 2 12b 63
x
2
5
28. 2 3 6
29. 2x 7 3x 10
7
3r
4
30. 4 5 1
0
31. 4y 3 9y 32
32. 7n 10 9n 13
33. Finance Alessia is paid 1.4 times her normal hourly rate for each hour
she works over 30 hours in a week. Last week she worked 35 hours and
earned $436.60. What is her normal hourly rate?
126
Chapter 3 Multi-Step Equations and Inequalities
34. Geometry The obtuse angle of an isosceles triangle measures 120°.
Write and solve an equation to find the measure of the base angles.
(Hint: An isosceles triangle has two congruent angles. An obtuse angle
measures more than 90° but less than 180°.)
Sports
35. Reasoning The sum of two consecutive numbers
is 63. What are the two numbers? Explain your solution.
You can estimate
the weight in
pounds of a fish
that is L inches
long and G inches
around at the
thickest part by
using the formula
LG 2
.
W
800
36. Sports The average weight of the top 5 fish
caught at a fishing tournament was
12.3 pounds. The weights of the second-,
third-, fourth-, and fifth-place fish are shown
in the table. What was the weight of the
heaviest fish?
F 32
1.8
37. Science The formula K 273 is
used to convert a temperature from degrees
Fahrenheit to kelvins. Water boils at 373
kelvins. Use the formula to find the boiling
point of water in degrees Fahrenheit.
Winning Entries
Caught by
Weight (lb)
Wayne S.
Carla P.
12.8
Deb N.
12.6
Virgil W.
11.8
Brian B.
9.7
38. What’s the Error? A student’s work in solving an equation is shown.
What error has the student made, and what is the correct answer?
1
x 5x 13
5
x 5x 65
6x 65
65
x 6
39. Write About It Compare the steps you would use to solve the equations
4x 8 16 and 4(x 2) 16.
40. Challenge Solve the following equation.
1
1
4
4 3x 4 3x
1 6
3
AF1.3,
AF4.1, Ext. of
AF4.1
41. Multiple Choice Solve 4k 7 3 5k 59.
A
k6
B
k 6.6
C
k7
D
k 11.8
42. Gridded Response Antonio’s first four test grades were 85, 92, 91, and
80. What must he score on the next test to have an 88 test average?
Solve. (Lesson 1-9)
43. 5n 6 21
x
44. 17y 31 3
30
45. 41 11
47. 6t 3k 15
48. 5a 3 b 1
Combine like terms. (Lesson 3-2)
46. 9m 8 4m 7 5m
3-3 Solving Multi-Step Equations
127
Model Equations with
Variables on Both Sides
3-4
Use with Lesson 3-4
KEYWORD: MT8CA Lab3
KEY
REMEMBER
Algebra tiles
Adding or subtracting zero does not
change the value of an expression.
+
x
−
+
1
−
–x
1
–1
+ − 0
+
California
Standards
Extension of
AF4.1 Solve two-step linear
− 0
To solve an equation with the same variable on both sides
of the equal sign, you must first add or subtract to
eliminate the variable term from one side of the equation.
equations and inequalities in one variable over
the rational numbers, interpret the solution or
solutions in the context from which they arose, and
verify the reasonableness of the results.
Activity
1 Model and solve the equation x 2 2x 4.
− +
+ + − −
− −
+
+ − +
+
x 2 2x 4
+ − +
+
+ + + − −
− −
Add x to both sides.
+ +
+ +
+ +
Remove zero.
+ +
+ +
+ +
+ + + − −
− −
+ + + − − + +
− − + +
Add 4 to both sides.
+ + + − − + +
− − + +
Remove zero.
+ +
+ +
+ +
+ + +
+
+ +
Divide each side into 3 equal
groups. 31 of each side
is the solution.
2
x
Think and Discuss
1. How would you check the solution to x 2 2x 4 using algebra tiles?
2. Why must you isolate the variable terms by having them on only one side
of the equation?
Try This
Model and solve each equation.
1. x 3 x 3
128
2. 3x 3x 18
3. 6 3x 4x 8
Chapter 3 Multi-Step Equations and Inequalities
4. 3x 3x 2 x 17
3-4
Solving Equations with
Variables on Both Sides
Who uses this? Consumers can use equations with
variables on both sides to compare costs. (See Example 3.)
California
Standards
Extension of
AF4.1 Solve
two-step linear equations and
inequalities in one variable over
the rational numbers, interpret the
solution or solutions in the context
from which they arose, and verify the
reasonableness of the results.
Also covered: AF1.1
The fees for two dogsitting services are
shown at right. To find
the number of hours for
which the costs will be
the same for both
services, you can write
and solve an equation
with variables on both
sides of the equal sign.
To solve an equation like this, first use inverse
operations to “collect” variable terms on one side
of the equation.
EXAMPLE
1
Solving Equations with Variables on Both Sides
Solve.
You can always check
your solution by
substituting the
value back into the
original equation.
3a 2a 3
3a 2a 3
2a
2a
a
3
To collect the variable terms on one side,
subtract 2a from both sides.
Check
3a 2a 3
?
3(3) 2(3) 3
?
963
?
99✔
Substitute 3 for a in the original equation.
3v 8 7 8v
3v 8 7 8v
3v
3v
8 7 5v
7 7
15 5v
To collect the variable terms on one side,
subtract 3v from both sides.
15
5v
5
5
3 v
Since 7 is added to 5v, subtract 7 from
both sides.
Since v is multiplied by 5, divide both
sides by 5.
3-4 Solving Equations with Variables on Both Sides
129
Solve.
g7g3
g7 g3
g
g
7
3
If the variables in
an equation are
eliminated and the
resulting statement
is false, the equation
has no solution.
To collect the variable terms on one
side, subtract g from both sides.
There is no solution. There is no number that can be substituted
for the variable g to make the equation true.
To solve more complicated equations, you may need to first simplify
by combining like terms or clearing fractions. Then add or subtract to
collect variable terms on one side of the equation. Finally, use
properties of equality to isolate the variable.
EXAMPLE
2
Solving Multi-Step Equations with Variables on Both Sides
Solve.
2c 4 3c 9 c 5
2c 4 3c 9 c 5
c 4 4 c
c
c
4 4 2c
4 4
8
2c
8
2c
2
2
Combine like terms.
To collect the variable terms on one
side, add c to both sides.
Since 4 is add to 2c, add 4 to
both sides.
Since c is multiplied by 2, divide both
sides by 2.
4c
2w
5w
1
11
w 3
6
4
9
2w
5w
1
11
w 3
6
4
9
2w
5w
1
11
36 3 6 4 36 w 9
Multiply both sides by
36, the LCD.
12
9
4
2w 6
5w
1
11
3631 3661 3641 36(w) 3691 Distributive Property
24w 30w 9 6w 9 6w
9
44
35 36w 44
36w 44
6w
42w 44
44
42w
35
42w
42
42
5
6 w
130
Chapter 3 Multi-Step Equations and Inequalities
Combine like terms.
Add 6w to both sides.
Subtract 44 from
both sides.
Divide both sides by 42.
A system of equations is a set of two or more equations that
contain two or more variables. To solve a system of two
equations, you can reduce the system to one equation that has
only one variable.
EXAMPLE
3
Business Application
Happy Paws charges a flat fee of $19.00 plus $1.50 per hour to
keep a dog. Woof Watchers charges a flat fee of $15.00 plus $2.75
per hour. Find the number of hours for which you would pay the
same amount for both services. What is the cost?
Write an equation for each service. Let c represent the total cost and
h represent the number of hours.
total cost
is
flat fee
plus
cost
per hour
Happy Paws:
c
19.00
1.5
h
Woof Watchers:
c
15.00
2.75
h
Now write an equation showing that the costs are equal.
19.00 1.5h 15.00 2.75h
1.5h 1.5h To collect the variable terms on one
19.00
15.00 1.25h side, subtract 1.5h from both sides.
15.00
15.00
Subtract 15.00 from both sides.
4.00
1.25h
4.00
1.25h
1.25
1.25
Divide both sides by 1.25.
3.2 h
The two services cost the same when used for 3.2 hours.
To find the cost, substitute 3.2 for h in either equation.
Happy Paws:
Woof Watchers:
c 19.00 1.5h
c 15.00 2.75h
c 19.00 1.5(3.2)
c 15.00 2.75(3.2)
c 19.00 4.8
c 15.00 8.8
c 23.8
c 23.8
The cost for 3.2 hours at either service is $23.80.
Think and Discuss
1. Explain how you would solve the equation 3x 4 2x 6x 2 5x 2. What do you think the solution means?
2. Give a series of steps that you can use to solve any equation with
variables on both sides of the equal sign.
3-4 Solving Equations with Variables on Both Sides
131
3-4
California
Standards Practice
AF1.1, Extension of
AF4.1;
MG2.1
Exercises
KEYWORD: MT8CA 3-4
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
See Example 2
Solve.
1. 6x 3 x 8
2. 5a 5 7 2a
3. 2x 7 10x 9
4. 4y 2 6y 6
5. 13x 15 11x 25
6. 5t 5 5t 7
7. 5x 2 3x 17 12x 23
n
3n
6 5 2n 18
8. 4 1
2
5
11d
9. 1
1
3 3d 7 4d
2
2
See Example 3
10. 4(x 5) 2 x 3
11. Business A long-distance phone company charges $0.027 per minute
and a $2 monthly fee. Another long-distance phone company charges
$0.035 per minute with no monthly fee. Find the number of minutes
for which the charges for both companies would be the same. What is
the cost?
INDEPENDENT PRACTICE
See Example 1
See Example 2
See Example 3
Solve.
12. 3n 16 7n
13. 8x 3 11 6x
14. 5n 3 14 6n
15. 3(2x 11) 6x 33
16. 6x 3 x 8
17. 7y 8 5y 4
3p
7p
3
1
p
1
18. 8 16 4 4 1
2
6
19. 4(x 5) 5 6x 7.4 4x
1
20. 2(2n 6) 5n 12 n
a
9
4
20a
21. 2
5.5 2a 1
1
1
6
3
3
3
22. Business Al’s Rentals charges $25 per hour to rent a Windsurfer™ and a
wet suit. Wendy’s charges $20 per hour plus $15 extra for a wet suit. Find
the number of hours for which the total charges for both would be the
same. What is the cost?
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP6.
Solve and check.
23. 3y 1 13 4y
24. 4n 8 9n 7
25. 5n 20n 5(n 20)
26. 3(4x 2) 12x
27. 100(x 3) 450 50x 28. 0.2p 1.2 1.2 0.2p
29. Find two consecutive whole numbers such that 34 of the first number is
5 more than 12 the second number. (Hint: Let n represent the first number.
Then n 1 represents the next consecutive whole number.)
132
Chapter 3 Multi-Step Equations and Inequalities
Write an equation to represent each relationship. Then solve the equation.
30. Six plus the product of 3 and a number is the same as the product of 9 and
the number.
31. A number decreased by 25 is the same as 10 minus 4 times the number.
32. Eight less than 2 times a number is the same as the number increased by 24.
Science
The figures in each pair have the same perimeter. Find each perimeter.
x 15
33.
34.
x
x6
x
x 45
x 40
x
x2
x4
x 25
35. Science An atom of chlorine (Cl) has 6 more protons than an atom of
sodium (Na). The atomic number of chlorine is 5 less than twice the
atomic number of sodium. The atomic number of an element is equal
to the number of protons per atom.
Sodium and
chlorine bond
together to form
sodium chloride,
or salt. The atomic
structure of sodium
chloride causes it
to form cubes.
a. How many protons are in an atom of chlorine?
b. What is the atomic number of sodium?
36. Choose a Strategy Solve the following equation for t. How can you
determine the solution once you have combined like terms?
3(t 24) 7t 4(t 18)
37. Write About It Two cars are traveling in the same direction. The first
car is going 45 mi/h, and the second car is going 60 mi/h. The first car left
2 hours before the second car. Explain how you could solve an equation
to find how long it will take the second car to catch up to the first car.
x2
6
x1
38. Challenge Solve the equation 8 7 2.
NS1.1,
AF4.1
39. Multiple Choice Find three consecutive integers (x, x 1, and x 2) so
that the sum of the first two integers is 10 more than the third integer.
A
7, 6, 5
B
4, 5, 6
C
11, 12, 13
D
35, 36, 37
C
w 1
D
w 5
40. Multiple Choice Solve 6w 15 9w.
A
w3
B
w0
Solve. (Lesson 1-9)
41. 6x 3 15
n
42. 7 2 1
3
43. 72 5g 12
y
44. 4 7 7
Compare. Write , , or . (Lesson 2-2)
5
45. 9
13
21
13
46. 1
1
8
7
1
47. 7
1
8
2
48. 3
14
2
1
3-4 Solving Equations with Variables on Both Sides
133
Quiz for Lessons 3-1 Through 3-4
3-1
Properties of Rational Numbers
Name the property that is illustrated in each equation.
1
1
2. m n n m
1. 3 3 7 3 3 7
Simplify each expression. Justify each step.
4. 20 19 5
5. 35.5 12.7 4.5
1
1
1
3. 2 x 6 2 x 2 6
1
6. 4 11 16
Write each product using the Distributive Property. Then simplify.
7. 3(57)
3-2
8. (42)7
9. 5(95)
Simplifying Algebraic Expressions
Simplify.
10. 5x 3x
11. 6p 6 p
12. 2t 3 t 4 5t
13. 3x 4y x 2y
14. 2(r 1) r
15. 4n 2m 8n 2m
3-3
Solving Multi-Step Equations
Solve.
16. 2c 6c 8 32
3x
2
10
17. 7 7 7
t
t
7
18. 4 3 1
2
4m
m
7
19. 3 6 2
3
1
20. 4b 5b 11
r
r
21. 3 7 5 3
22. Marlene drove 540 miles to visit a friend. She drove 3 hours and
stopped for gas. She then drove 4 hours and stopped for lunch. How
many more hours did she drive if her average speed for the trip was
60 miles per hour?
3-4
Solving Equations with Variables on Both Sides
Solve.
23. 4x 11 x 2
24. q 5 2q 7
25. 6n 21 4n 57
26. 2m 6 2m 1
27. 4a 2a 11 6a
7
1
5
y 4 2y 3
28. 1
2
29. The rectangle and the triangle
have the same perimeter.
Find the perimeter of
each figure.
x
x9
x2
x7
x7
134
Chapter 3 Multi-Step Equations and Inequalities
California
Standards
MR1.1
Analyze problems by
identifying relationships,
distinguishing relevant information,
identifying missing information,
sequencing and prioritizing information,
and observing patterns.
Also covered: AF1.1, Extension of
Make a Plan
AF4.1
• Write an equation
Several steps may be needed to solve a problem.
It often helps to write an equation that represents the steps.
Example:
Juan’s first 3 exam scores are 85, 93, and 87. What does he need to
score on his next exam to average 90 for the 4 exams?
Let x be the score on his next exam. The
average of the exam scores is the sum of the
4 scores, divided by 4. This amount must
equal 90.
Write the equation in words:
Exam 1 Exam 2 Exam 3 Exam 4
Number of exams
90
85 93 87 x
90
4
265 x
90
4
265 x
4 4 4(90)
265 x 360
265
265
x
95
Juan needs a 95 on his next exam.
Read each problem and write an equation that could be used to
solve it.
1 The average of two numbers is 34. The first
number is three times the second number.
What are the two numbers?
2 Nancy spends 13 of her monthly salary on
rent, 0.1 on her car payment, 112 on food, and
20% on other bills. She has $680 left for other
expenses. What is Nancy’s monthly salary?
3 A vendor at a concert sells new and used
CDs. The new CDs cost 2.5 times as much as
the old CDs. If 4 used CDs and 9 new CDs
cost $159, what is the price of each item?
4 Amanda and Rick have the same amount
to spend on school supplies. Amanda buys
4 notebooks and has $8.60 left. Rick buys
7 notebooks and has $7.55 left. How much
does each notebook cost?
135
3-5
Inequalities
Why learn this? You can show the maximum capacity
of an elevator using an inequality.
California
Standards
AF1.1 Use variables and
appropriate operations to write
an expression, an equation, an
inequality, or a system of
equations or inequalities that
represents a verbal description
(e.g., three less than a number, half as
large as area A).
An inequality compares two expressions using , , , or .
Symbol
Vocabulary
inequality
algebraic inequality
solution set
EXAMPLE
Meaning
Word Phrases
Is less than
Fewer than, below
Is greater than
More than, above
Is less than or equal to
At most, no more than
Is greater than or equal to
At least, no less than
An inequality that contains a variable is an algebraic inequality .
1
Translating Word Phrases into Inequalities
Write an inequality for each situation.
The capacity of an elevator is at most 12 people.
Let c the capacity of the elevator.
“At most” means less than or equal to.
c 12
There are more than 1000 books in the library.
Let b the number of books in the library.
b 1000
“More than” means greater than.
EXAMPLE
2
Writing Inequalities
Write an inequality for each statement.
A number x plus 14 is greater than or equal to 30.
A number x
plus
14
is greater than or equal to
30
x
14
30
x 14 30
A number n decreased by 3 is less than 21.
A number n
decreased by
3
is less than
21
n
3
21
n 3 21
136
Chapter 3 Multi-Step Equations and Inequalities
A solution of an inequality is any value of the variable that makes the
inequality true. All of the solutions of an inequality are called the
solution set . You can graph the solution set on a number line. The
symbols and indicate an open circle.
This open circle shows that 5 is not a solution.
0
1
2
3
4
5
6
7
8
9 10
The symbols and indicate a closed circle.
This closed circle shows that 3 is a solution.
–3 –2 –1 0
EXAMPLE
3
1
2
3
4
5
6
7
Graphing Inequalities
Graph each inequality.
x 4
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
1
12 m
112
m is the same
as m 112.
⫺2 ⫺1
0
1
2
3
4
5
Draw an open circle at 4.
The solutions are all values of
x greater than 4, so shade
to the right of 4.
Draw a closed circle at 112. The
solutions are 112 and all values
of m less than 112, so shade to
the left of 112.
A compound inequality is the result of combining two inequalities. The
words and and or are used to describe how the two parts are related.
EXAMPLE
4
Writing Compound Inequalities
Write a compound inequality for each statement.
The compound
inequality in
Example 4B can also
be written with the
variable between the
two endpoints.
6 n 9.5
A number t is either less than 2 or greater than or equal to 1.
t 2 or t 1
A number n is both greater than or equal to 6 and less than 9.5.
n 6 and n 9.5
Think and Discuss
1. Explain how to write “x is no less than 16” as an inequality.
2. Compare the graphs of the inequalities x 3 and x 3.
3-5 Inequalities
137
3-5
California
Standards Practice
Exercises
AF1.1
KEYWORD: MT8CA 3-5
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
Write an inequality for each situation.
1. There are no more than 60 people in the theater.
2. The temperature of the water is above 72°F.
See Example 2
Write an inequality for each statement.
3. A number m increased by 7 is at least 15.
4. Twice a number x is less than 18.
See Example 3
Graph each inequality.
5. x 2
See Example 4
6. w 1
7. 2.5 y
1
8. m 32
Write a compound inequality for each statement.
9. A number s is either less than 5 or greater than or equal to 3.
10. A number t is both greater than 10 and less than 1.
INDEPENDENT PRACTICE
See Example 1
Write an inequality for each situation.
11. Fewer than 10 students rode their bikes to the game.
12. No more than 18 people may ride the roller coaster at one time.
See Example 2
Write an inequality for each statement.
13. A number x decreased by 11 is less than 35.
1
14. Three times a number n is greater than 43.
15. A number y divided by 7 is at most 10.
See Example 3
See Example 4
Graph each inequality.
16. m 3
17. s 1.5
18. 2 x
20. b 1
21. x 0
22. n 2
1
19. 4 y
1
23. 22 c
Write a compound inequality for each statement.
24. A number x is both less than 1.5 and greater than or equal to 0.
1
25. A number c is either greater than or equal to 2 or less than or equal to 7.
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP7.
138
26. Suly earned 87 points on her first test and p points on her second test. She
needs a total of at least 140 points on the two tests to pass the class. Write
an inequality for this situation.
Chapter 3 Multi-Step Equations and Inequalities
Write an inequality for each statement.
27. A number w multiplied by 5 is no less than 60.
28. The sum of 10 and a number g is greater than 4.8.
2
1
29. A number m decreased by 25 is at most 35.
Write an inequality shown by each graph.
30.
31.
4 2
0
2
4
6
8
4 2
0
2
4
6
8
32.
0
33.
2
4
6
8
10 12
4 2
0
2
4
6
8
34. Business A cafe sells fruit smoothies for $3.50 each. The manager of the
cafe wants the total daily revenue from the smoothies to be at least $175.
Assume the cafe sells n smoothies per day. Write an inequality that
represents the manager’s goal.
35. Astronomy The diameter of Jupiter, the largest planet in the Solar
System, is 89,000 miles. Let d be the diameter of any planet in the Solar
System. Write an inequality for d.
36. What’s the Error? A student was asked to
graph the inequality 2 n. Explain the
student’s error in the graph at right.
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
37. Write About It In mathematics, the conventional way to write an
inequality is with the variable on the left, such as x 5. Explain how to
rewrite the inequality 4 x in the conventional way.
38. Challenge Write an inequality for the statement “14 less than twice a
number x is greater than three times the number.”
NS1.1,
NS1.2, AF1.1
39. Multiple Choice Which inequality represents the statement, “A number
z decreased by 9 is no more than 20”?
A
z 9 20
B
z 9 20
9 z 20
C
D
9 z 20
D
x 4
40. Multiple Choice Which inequality is shown by the graph?
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
A
x 4
B
x 4
0
C
1
2
x 4
Write each set of integers in order from least to greatest. (Lesson 1-3)
41. 19, 25, 12
42. 4, 0, 4, 3
43. 5, 9, 7, 11
44. 2, 6, 5, 0
Add or subtract. Write each answer in simplest form. (Lesson 2-3)
17
6
45. 121
121
46. 7 7
1
8
29
13
47. 1
1
2
2
29
45
48. 5 5
3-5 Inequalities
139
Solving Inequalities by
Adding or Subtracting
3-6
Why learn this? You can solve an inequality to find
the amount of a nutrient that you should consume.
(See Example 2.)
California
Standards
AF4.0 Students solve
simple linear equations and
inequalities over the rational
numbers.
When you add or subtract the same number on both sides of an
inequality, the resulting inequality will still be true.
2 5
7 7
5 12
You can use this idea to solve inequalities. You find solution sets
of inequalities the same way you find solutions of equations,
by isolating the variable.
EXAMPLE
1
Solving Inequalities by Adding or Subtracting
Solve and graph.
x 7 10
x 7 10
7
7
x
17
Since 7 is added to x, subtract 7 from both sides.
–21 –20 –19 –18 –17 –16 –15 –14 –13 –12 –11
When checking your
solution, choose
numbers that are
easy to work with.
Remember to
substitute the
numbers into the
original inequality.
Check
According to the graph, 20 should be a solution and 3 should
not be a solution.
x 7 10
x 7 10
?
?
20 7 10
Substitute
3 7 10
Substitute
?
?
20 for x.
3 for x.
13 10 ✔
10 10 ✘
So 20 is a solution.
t 11 22
t 11 22
11 11
t
11
Since 11 is subtracted from t, add 11 to both sides.
–15 –13 –11
140
So 3 is not a solution.
–9
Chapter 3 Multi-Step Equations and Inequalities
–7
–5
–3
–1
1
3
5
Solve and graph.
z 6 3
z 6 3
6 6
z
9
Since 6 is added to z, subtract 6 from both sides.
–10 –9
1
1
1
1
1
24
–8
–7
–6
–5 –4
–3
–2
–1
0
1
2
44 n 24
44 n 24
24
1
62 n
1
–1
EXAMPLE
2
1
1
Since 24 is subtracted from n, add 24 to both sides.
0
1
2
3
4
5
6
7
8
9
10 11
Nutrition Application
Manganese is a mineral that is found in nuts and grains. It is
recommended that men consume at least 2.3 mg of manganese each
day. Eric has consumed 0.9 mg today. Write and solve an inequality
to find how many additional milligrams he should consume.
Let m the number of additional milligrams of manganese.
0.9 milligrams
plus
additional milligrams
is at least
2.3 milligrams
0.9
m
2.3
0.9 m 2.3
0.9
0.9
m 1.4
Since 0.9 is added to m,
subtract 0.9 from both sides.
Eric should consume at least 1.4 additional milligrams of manganese.
Check
0.9 m 2.3
?
0.9 2 2.3
?
2.9 2.3 ✔
2 is greater
than 1.4.
Substitute
2 for m.
0.9 m 2.3
?
0.9 1 2.3
?
1.9 2.3 ✘
1 is less
than 1.4.
Substitute
1 for m.
Think and Discuss
1. Explain how you know whether to use addition or subtraction to
solve an inequality.
2. Describe how to check whether 11 is a solution of 6 t 4.
3-6 Solving Inequalities by Adding or Subtracting
141
3-6
California
Standards Practice
NS1.2, AF1.1,
AF4.0
Exercises
KEYWORD: MT8CA 3-6
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
See Example 2
Solve and graph.
1. x 3 4
2. 4 b 20
3. 6 f 30
4. z 8 13
5. 2.1 k 7.2
6. x 3 2
1
7. A measuring cup can hold no more than 16 fluid ounces of liquid. Rosa
pours 612 fluid ounces of water into the cup. Write and solve an inequality
to determine how many additional fluid ounces of water she can add.
8. Paul’s car can go at most 375 miles on one tank of gas. Paul fills the tank
and then drives 167 miles. Write and solve an inequality to find out how
many more miles Paul can drive before he will have to refill the tank.
INDEPENDENT PRACTICE
See Example 1
Solve and graph.
9. 7 x 49
12. 0.6 y 0.72
See Example 2
10. 1 t 4
1
11. 3 x 12
2
13. c 53 83
14. 2 a (5)
15. Consumer Math A clothes store gives customers a free gift if they spend
at least $50 in the store. Stacey plans to buy a pair of jeans that cost
$21.75. Write and solve an inequality to show how much more she must
spend in order to get the free gift.
16. Consumer Math Latrell’s cell-phone plan allows him to talk for no more
than 500 minutes per month. He has already used 288 minutes this
month. Write and solve an inequality to determine how many more
minutes he can talk on the phone this month.
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP7.
Solve and graph.
2
17. z 0.75 0.75
18. 7 x 3
19. 7 y 8.8
20. m (12) 6
4
1
21. 5 k 5
22. 39.5 15.5 g
You can use set-builder notation to write the solution of an inequality. For
example, {x : x 5} means the set of all real numbers x such that x is less
than 5. Solve each inequality and write the solution using set-builder notation.
23. x 12 8
24. z 4 16
25. 3.5 b 7
26. Reasoning When a number is added to 15, the result is greater than
12. What are the possible values of the number? Graph them on a
number line.
142
Chapter 3 Multi-Step Equations and Inequalities
27. Business Toshi Business Solutions will make a profit for the current year
if their total sales are greater than their operating costs. Their accountants
estimate that the company will have operating costs of $201,522 for the
entire year. So far this year, the company has sales of $98,200.
Califor nia
Language Arts
a. Write and solve an inequality to find out how much more money Toshi
must earn in sales for the remainder of the year to show a profit.
b. Check your answer. Then explain why your answer is reasonable.
Californian John
Steinbeck, author
of The Grapes of
Wrath and Of
Mice and Men,
won the Nobel
Prize for
Literature in
1962.
Great Novels
189
494
Title
28. Language Arts Danielle is
reading one of the novels in the
graph. She has already read 65
pages. Write and solve two different
inequalities to find out how many
pages she has left to read. (Hint:
Write one inequality based on the
minimum number of pages and one
inequality based on the maximum
number of pages.)
359
275
29. Reasoning Substitute the values 1,
2, 3, 4, 5, and 6 for x in 10 x 6.
Use the results to make a conjecture
about the solution of the inequality.
0
75
150
225
300
375
450
525
Number of pages
30. Write a Problem Write a word problem that can be answered by solving
the inequality x 40 75.
31. Write About It Explain how to check the solution of an inequality.
32. Challenge The inequality y 3 is missing a
number. The solution of the inequality is shown on
the number line. What is the missing number?
⫺2 ⫺1 0 1 2 3 4 5
NS1.2, AF1.1,
AF4.0
33. Multiple Choice Solve m 5 8 for m.
A
m 13
B
m 3
C
m3
D
m 13
34. Multiple Choice In the inequality 200 80 x, x is the length of a movie
in minutes. Which phrase most accurately describes the length of the movie?
A
At least 120 minutes
C
At most 120 minutes
B
More than 120 minutes
D
Less than 120 minutes
Add or subtract. (Lesson 2-6)
1
3
35. 4 7
7
5
36. 2
8
0
9
5
37. 1
6
0
4
1
38. 37 25
Write an inequality for each statement. (Lesson 3-5)
39. A number t increased by 2 is less than 8.
40. Twice a number w is no more than 12.
3-6 Solving Inequalities by Adding or Subtracting
143
3-7
Solving Inequalities by
Multiplying or Dividing
Why learn this? You can solve an inequality to determine
how many representatives voted on a bill. (See Exercise 18.)
California
Standards
AF4.0 Students solve
simple linear equations and
inequalities over the rational
numbers.
Also covered: AF1.1
When you multiply (or divide) both sides of an inequality by a
negative number, you must reverse the inequality symbol to make
the statement true.
–b
–a
0
ab
a
b Multiply both sides by 1.
a b Use the number line to
EXAMPLE
1
a
b
b a
b
a Multiply both sides by 1.
b a Use the number line to
determine the direction
determine the direction
of the inequality symbol.
of the inequality symbol.
Solving Inequalities by Multiplying or Dividing
Solve and graph.
h
24 5
h
5 24 5 5
120 h, or h 120
When graphing an
inequality on a
number line, an
open circle means
that the point is not
part of the solution
and a closed circle
means that the point
is part of the solution.
Multiply both sides by 5.
115 116 117 118 119 120 121 122
Check
According to the graph, 119 should be a solution and 121 should
not be a solution.
h
h
24 5
24 5
? 119
24 5
?
24 23.8 ✔
? 121
24 5
?
24 24.2 ✘
Substitute
119 for h.
So 119 is a solution.
Substitute
121 for h.
So 121 is not a solution.
7x 42
42
7x
7
7
Divide both sides by 7; changes to .
x 6
12 11 10
144
9
Chapter 3 Multi-Step Equations and Inequalities
8
7
6
5
4
EXAMPLE
2
PROBLEM SOLVING APPLICATION
If all the sheets of paper used by personal computer printers each
year were laid end to end, they would circle Earth more than 800
times. Earth’s circumference is about 25,120 mi (1,591,603,200 in.),
and one letter-size sheet of paper is 11 in. long. About how many
sheets of paper are used each year?
1 Understand the Problem
The answer is the number of sheets of paper used by personal
computer printers in one year. List the important information:
• The amount of paper would circle the earth more than
800 times.
• Once around Earth is approximately 1.6 billion in.
• One sheet of paper is 11 in. long.
Show the relationship of the information:
the number of
the length
sheets of paper
of one sheet
800 the distance
around Earth
2 Make a Plan
Use the relationship to write an inequality. Let x represent the
number of sheets of paper.
x
3 Solve
11x 800 1.6
11x 1280
1280
11x
11
11
11 in.
800 1.6 billion in.
Simplify.
Divide both sides by 11.
x 116.36
More than 116 billion sheets of paper are used by personal
computer printers in one year.
4 Look Back
1,600,0
00,000
To circle Earth once takes 145,454,545
11
sheets of paper; to circle it 800 times would take
800 145,454,545 116,363,636,000 sheets.
Think and Discuss
1. Give all the symbols that make 5 3
15 true. Explain.
2. Explain how you would solve the inequality 4x 24.
3-7 Solving Inequalities by Multiplying or Dividing
145
3-7
Exercises
California
Standards Practice
AF1.1,
AF4.0
KEYWORD: MT8CA 3-7
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
Solve and graph.
r
1. 3 6
a
5. 10 4
See Example 2
j
2. 4w 12
3. 20 6
4. 6r 30
6. 36 2m
r
21
7. 3
8. 20 5x
9. The owner of a sandwich shop is selling the special of the week for $5.90.
At this price, he makes a profit of $3.85 on each sandwich sold. To make a
total profit of at least $400 from the special, what is the least number of
sandwiches he must sell?
INDEPENDENT PRACTICE
See Example 1
See Example 2
Solve and graph.
x
p
10. 16 2r
11. 15 5
12. 18w 54
13. 11 7
t
14. 9 4
15. 9h 108
a
14
16. 7
17. 16q 64
18. Social Studies A bill in the U.S. House of Representatives passed
because at least 23 of the members present voted in favor of it. If the bill
received 284 votes, at least how many members of the House of
Representatives were present for the vote?
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP7.
Solve and graph.
x
p
19. 18 3r
20. 27 3
21. 17w 51
22. 101 7
t
5
23. 19
24. 3h 108
a
25. 1
12
0
26. 6q 72
Write and solve an algebraic inequality.
27. Nine times a number is less than 99.
28. The quotient of a number and 6 is at least 8.
29. The product of 7 and a number is no more than 63.
30. The quotient of some number and 3 is greater than 18.
Write and solve an algebraic inequality. Then explain the solution.
31. A school receives a shipment of books. There are 60 cartons, and each
carton weighs 42 pounds. The school’s elevator can hold 2200 pounds.
What is the greatest number of cartons that can be carried on the elevator
at one time if no people ride with them?
32. Each evening, Marisol spends at least twice as much time reading as
she spends doing homework. If Marisol works on her homework for
40 minutes, how much time can she spend reading?
146
Chapter 3 Multi-Step Equations and Inequalities
Choose the graph that represents each inequality.
33. 2y 14
A.
B.
9 8 7 6 5 4 3 2 1
12 11 10 9 8 7 6 5
C.
5
6
7
8
9 10 11 12 13
h
34. 6 5
A.
28 29 30 31 32 33 34 35 36
B.
25 26 27 28 29 30 31 32 33
C.
25 26 27 28 29 30 31 32 33
35. What’s the Error? A student solved x 3 12 and got an answer of
x 36. What error did the student make?
36. Write About It The expressions no more than, at most, and less than
or equal to all indicate the same relationship between values. Write a
problem that uses this relationship. Write the problem using each of
the three expressions.
37. Challenge Angel weighs 5 times as much as his dog. When they stand
on a scale together, the scale gives a reading of less than 163 pounds. If
both their weights are whole numbers, what is the most each can weigh?
NS1.2, AF1.1,
AF4.0
38. Multiple Choice Which inequality is shown by the graph?
5 4 3 2 1
A
w 3
B
w 3
0
1
C
2
3
w 3
3 w
D
39. Gridded Response In order to have the $200 he needs for a bike, Kevin
plans to put money away each week for the next 15 weeks. What is the
minimum amount in dollars that Kevin will need to average each week in
order to reach his goal?
Multiply. Write each answer in simplest form. (Lesson 2-4)
40. 71
4
3
9
25
11
41. 1
5 121 42. 78 9
1 4
43. 513
2
44. Frank needs to earn at least $350. He earns $15 for each hour h that he
babysits. Write an inequality that represents Frank’s goal. (Lesson 3-5)
3-7 Solving Inequalities by Multiplying or Dividing
147
3-8
Solving Two-Step
Inequalities
Why learn this? Drama club members can use two-step
inequalities to determine how many tickets they must sell
to a musical to break even. (See Example 3.)
California
Standards
AF4.1 Solve two-step linear
equations and inequalities in one
variable over the rational
numbers, interpret the solution or
solutions in the context from
which they arose, and verify the
reasonableness of the results.
EXAMPLE
When you solved two-step equations, you used the order of
operations in reverse to isolate the variable. You can use the
same process when solving two-step inequalities.
1
Solving Two-Step Inequalities
Solve and graph.
Math
Builders
For more on solving
two-step inequalities,
see the Step-by-Step
Solution Builder on
page MB4.
7y 4 24
7y 4 24
4
4
7y
28
Since 4 is subtracted from 7y, add 4 to
both sides.
7y
28
7
7
Since y is multiplied by 7, divide both sides by 7.
y 4
0
1
2
3
4
5
6
7
8
9 10
Check
According to the graph, 10 should be a solution and 0 should not
be a solution.
7y 4 24
7y 4 24
?
7(0) 4 24
So 10 is a solution.
So 0 is not a solution.
Substitute
?
10
for y.
66 24 ✔
If both sides of an
inequality are
multiplied or divided
by a negative
number, the
inequality symbol
must be reversed.
148
?
7(10) 4 24
2x 4 3
2x 4 3
4
4
2x
1
1
2x
2
2
1
x 2
2
0
Substitute
4 24 ✘ 0 for y.
?
Since 4 is added to 2x, subtract 4 from
both sides.
Since x is multiplied by 2, divide both sides
by 2. Change to .
2
4
Chapter 3 Multi-Step Equations and Inequalities
6
EXAMPLE
2
Solving Inequalities That Contain Fractions
7
3x
5
and graph the solution.
Solve 8 6 12
3x
5
24 8 6 3x
5
248 246
When an inequality
contains fractions,
you may want to
multiply both sides
by the LCD to clear
the fractions.
7
241
2
7
24 1
2
Multiply by the LCD, 24.
Distributive Property
9x 20 14
20
20
9x
6
Since 20 is added to 9x, subtract 20
from both sides.
6
9
9x
9
Since x is multiplied by 9, divide both
sides by 9. Change to .
6
x 9
2
x 3
1 23 1 13 1
EXAMPLE
3
23
Simplify.
13
0
1
3
2
3
1
1 13
1 23
School Application
The Drama Club is planning a spring musical.
Club members estimate that the entire
production will cost $1100.00. If they
have $610.75 left from fund-raising,
how many tickets must they sell to
at least break even?
In order to at least break even,
ticket sales plus the money in the
budget must be greater than or equal
to the cost of the production.
4.75t 610.75 1100.00
610.75
610.75
4.75t
489.25
4.75t
489.25
4.75
4.75
Subtract 610.75 from both sides.
Divide both sides by 4.75.
t 103
The drama club must sell at least 103 tickets in order to break even.
Think and Discuss
1. Compare solving a multi-step equation with solving a multi-step
inequality.
2. Describe two situations in which you would have to reverse the
inequality symbol when solving a multi-step inequality.
3-8 Solving Two-Step Inequalities
149
3-8
California
Standards Practice
AF4.1
Exercises
KEYWORD: MT8CA 3-8
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
See Example 2
Solve and graph.
1. 3k 5 11
2. 2z 29.5 10.5
3. 6y 12 36
4. 4x 6 14
5. 2y 2.5 16.5
6. 3k 2 13
x
1
2
7. 15 5 5
b
3
1
8. 1
5 2
0
h
5
9. 3 2 3
c
1
3
10. 8 2 4
See Example 3
1
d
1
11. 2 6 3
2
6m
12. 3 9
13. The chess club is selling caps to raise $425 for a trip. They have $175
already. If the club members sell caps for $12 each, at least how many
caps do they need to sell to make enough money for their trip?
INDEPENDENT PRACTICE
See Example 1
See Example 2
Solve and graph.
14. 8k 6 18
15. 5x 3 23
16. 3p 3 36
17. 13 11q 9
18. 3.6 7.2n 25.2
19. 7x 15 34
a
2
1
21. 9 3 3
22. 3 1
4
2
n
4
3
24. 7 1
7
4
r
1
1
25. 3 1
2
8
p
4
1
20. 1
5 3
5
2
1
5
23. 3 1
k 6
8
See Example 3
1
n
1
26. Josef is on the planning committee for the eighth-grade party. The food,
decoration, and entertainment costs a total of $350. The committee has
$75 already. If the committee sells the tickets for $5 each, at least how
many tickets must be sold to cover the remaining cost of the party?
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP7.
Solve and graph.
27. 3p 11 11
28. 9n 10 17
29. 3 5w 8
30. 6x 18 6
31. 12a 4 10
32. 4y 3 17
33. 3q 5q 12
3m
5
34. 4 8
35. 4b 3.2 7.6
36. 3k 6 4
90
5
37. 4 6 f
38. 9v 3
5
1
39. Reasoning What is the least whole number that is a solution
of 2r 4.4 8.6?
40. Entertainment A speech is being given in a gymnasium that can hold
no more than 650 people. A permanent bleacher will seat 136 people. The
event organizers are setting up 25 rows with an equal number of chairs. At
most, how many chairs can be in each row?
150
Chapter 3 Multi-Step Equations and Inequalities
41. Katie and April are making a string of beads for pi day (March 14).
The string already has 70 beads. If there are only 30 more days until pi
day, and they want to string 1000 beads by then, at least how many beads
do they have to string each day?
3
.1
415
9 26 5 3 5 89
3
79
6 4 3 3 8 3 2 7
4 62
95
8
02
3
2
8
8
4
42. Sports The Astros have won 35 and lost 52 baseball games. They
have 75 games remaining. At least how many of the remaining 75 games
must the Astros win to have a winning season? (Hint: A winning season
means they win more than 50% of their games.)
43. Economics Satellite TV customers can either purchase a dish and receiver
for $249 or pay a $50 fee and rent the equipment for $12 a month.
a. How much would it cost to rent the equipment for 9 months?
b. How many months would it take for the rental charges to
be more than the purchase price?
44. Write a Problem Write and solve an inequality using the following
shipping rates for orders from a mail-order catalog.
1
9
7
1
6
9
3
9
9
3
7
5
1
0
5
8
2
0
9
7
4
..
.
Mail-Order Shipping Rates
Merchandise
Amount
$0.01–
$25.00
$25.01
–50.00
$50.01
–75.00
$75.01
–125.00
$125.01
and over
Shipping Cost
$3.95
$5.95
$7.95
$9.95
$11.95
45. Write About It Describe two different ways to solve the inequality
3x 4 x.
x
x
1
46. Challenge Solve the inequality 5 6 1
.
5
AF1.3,
AF4.1
47. Multiple Choice Solve 3g 6 18.
A
g 21
B
g8
C
5x
1
g6
D
g4
2
48. Short Response Solve and graph 6 2 3.
Name the property that is illustrated in each equation. (Lesson 3-1)
49. 12y y 12
50. a (b c) (a b) c 51. x 13y 13y x
Simplify. (Lesson 3-2)
52. 5(x 1) 2x
53. 6(r 10) r
54. 3(8 n) 21
3-8 Solving Two-Step Inequalities
151
Quiz for Lessons 3-5 Through 3-8
3-5
Inequalities
Write an inequality for each statement.
1. A number n decreased by 15 is no more than 48.
2. The product of 7 and a number x is above 49.
Graph each inequality.
3. r 7
2
4. 4 a
6. h 2
5. c 3
Write a compound inequality for each statement.
7. A number m is both greater than 15 and less than or equal to 4.
8. A number d is either less than 12 or greater than 2 13.
3-6
Solving Inequalities by Adding or Subtracting
Solve and graph.
3
4
9. n 15 5
12. 19 t 13
3-7
10. 15 8 y
11. 101 x 89
13. 27 d 22
14. 5.3 n 2.7
Solving Inequalities by Multiplying or Dividing
Solve and graph.
k
y
15. 5x 15
16. 9 3
4
17. 4
18. 24 6m
19. n 10
h
20. 2 42
21. Rachael is serving lemonade from a pitcher that holds 60 ounces. What are the
possible numbers of 7-ounce juice glasses she can fill from one pitcher?
3-8
Solving Two-Step Inequalities
Solve and graph.
22. 2k 4 10
23. 0.5z 5.5 4.5
9
3x
3
5
24. 5 1
5
t
1
25. 3 9 2
1
3x
5
26. 3 4 6
m
3
2
27. 7 1
7
4
28. Jillian must average at least 90 on two quiz scores before she can move to the
next skill level. Jillian got a 92 on her first quiz. What scores could Jillian get on
her second quiz in order to move to the next skill level?
152
Chapter 3 Multi-Step Equations and Inequalities
Skate Away
Ms. Lucinda wants to treat her class of
30 students to a skating party to celebrate the end of
the school year.
Item
Cost
Rink rental
$50 plus
$25 per hour
Skate rental
(per person)
$1.50 plus
$0.50 per hour
Refreshments
(per person)
$3.50
1. Ms. Lucinda considers renting the rink at Skate Away.
How much would it cost to rent the rink for x hours?
2. Another rink, Skate Palace, charges $100 plus $15 per
hour to rent the rink. Write and solve an equation to find
the number of hours for which the cost of renting the rink
at Skate Palace is the same as the cost of renting the rink
at Skate Away.
3. Ms. Lucinda decides to take the class to Skate Away.
How much will it cost to rent skates for 30 students for
x hours? How much will it cost to buy refreshments for
30 students?
4. Ms. Lucinda has budgeted $400 for the party.
Write and solve an inequality to find the
maximum number
of hours the class
can have its party
at Skate Away. Be
sure to include the
cost of the rink, the
skates, and the
refreshments.
5. The final bill for the
party was $380.
How long did the
party last?
Concept Connection
153
Trans-Plants
Solve each equation below. Then use the values of the
variables to decode the answer to the question.
3a 17 25
24 6n 54
2b 25 5b 7 32
8.4o 6.8 14.2 6.3o
2.7c 4.5 3.6c 9
5
1
1
1
d d d d 6
12
12
6
3
4e 6e 5 15
4p p 8 2p 5
4 3r r 8
420 29f 73
2
5
1
3
s s 3
6
2
2
2(g 6) 20
4 15 4t 17
2h 7 3h 52
45 36u 66 23u 31
96i 245 53
6v 8 4 6v
3j 7 46
1
3
1
k k 2
4
2
30l 240 50l 160
3 67
4m 8
8
4w 3w 6w w 15 2w 3w
16 3q 3q 40
1
x 2x 3x 4x 5 75
2 2y
4y
8
5
11 25 4.5z
What happens to plants that live in a math classroom?
7, 9, 10, 11
16, 18, 10, 15
12, 4, 4, 14, 18, 10
18, 10, 10, 7, 12
24
24 Points
Points
This traditional Chinese game is played using
a deck of 52 cards numbered 1–13, with four
of each number. The cards are shuffled, and
four cards are placed face up in the center.
The winner is the first player who comes up
with an expression that equals 24, using
each of the numbers on the four cards once.
Complete rules and a set of game cards are available online.
154
Chapter 3 Multi-Step Equations and Inequalities
KEYWORD: MT8CA Games
Materials
•
•
•
•
magazine
glue
scissors
index cards
PROJECT
Picture Envelopes
A
Make these picture-perfect envelopes in which to
store your notes on the lessons of this chapter.
Directions
1 Flip through a magazine and carefully tear out
eight pages with full-page pictures that you like.
B
2 Lay one of the pages in front of you with the
picture face down. Fold the page into thirds as
shown, and then unfold the page. Figure A
3 Fold the sides in, about 1 inch, and then
unfold. Cut away the four rectangles at the
corners of the page. Figure B
C
4 Fold in the two middle flaps. Then fold up the
bottom and glue it onto the flaps. Figure C
5 Cut the corners of the top section at
an angle to make a flap. Figure D
D
6 Repeat the steps to make seven more
envelopes. Label them so that
there is one for each lesson of
the chapter.
Taking Note of the
Math
Use index cards to take notes
on the lessons of the chapter.
Store the cards in the
appropriate envelopes.
155
Vocabulary
algebraic inequality . . . . . . . . . . . . . . . . . . 136
equivalent expressions . . . . . . . . . . . . . . . 120
Associative Property . . . . . . . . . . . . . . . . . 116
inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 120
like terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Commutative Property . . . . . . . . . . . . . . . 116
solution set . . . . . . . . . . . . . . . . . . . . . . . . . 136
constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Distributive Property . . . . . . . . . . . . . . . . 117
Complete the sentences below with vocabulary words from the list above.
1. A(n) ___?____ is a statement that two quantities are not equal.
2. ___?____ states that two or more numbers can be added in any order or
multiplied in any order.
3. ___?____ in an expression are set apart by plus or minus signs.
3-1 Properties of Rational Numbers
EXAMPLE
■
AF1.3
(pp. 116–119)
EXERCISES
Name the property that is illustrated in
the equation.
10(x y) 10 x 10 y
Distributive
Property
Name the property that is illustrated in
each equation.
1
1
4. 22 5 5 22
5. x (8 y) (x 8) y
6. 5(3 n) 5 3 5n
7. 8 6 7 8 6 7
1
3-2 Simplifying Algebraic Expressions
EXAMPLE
■
Simplify.
3(z 6) 2z
3z 3(6) 2z
3z 18 2z
5z 18
(pp. 120–123)
EXERCISES
Simplify.
Distributive Property
3z and 2z are like terms.
Combine coefficients.
8. 5(3m 2) 4m
9. 12w 2(w 3)
10. 4x 3y 2x
11. 2t 2 4t 3t 3
156
1
Chapter 3 Multi-Step Equations and Inequalities
AF1.3
■
3-3 Solving Multi-Step Equations
(pp. 124–127)
EXAMPLE
EXERCISES
Ext. of
Solve.
Solve.
x
5x
1
3
6
9
3
2
x
5x
1
3 Multiply both sides
18 9 6 3 18 2 by 18.
x
5x
1
3
18 9 18 6 18 3 18 2 Distributive
Property
10x 3x 6 27
Simplify.
7x 6 27
Combine like terms.
6 6
Subtract 6 from
7x
21
both sides.
12. 3y 6 4y 7 8
7x
21
7
7
13. 5h 6 h 10 12
2t
1
1
14. 3 3 3
2r
4
2
15. 5 5 5
z
3z
1
1
16. 3 4 2 3
a
3a
7
17. 8 1
2 7
2
Divide both sides
by 7.
x3
3-4 Solving Equations with Variables on Both Sides
Solve.
Solve.
3x 5 5x 12 x 2
2x 5 10 x
2x
2x
5 10 3x
10 10
15 3x
18. 12s 8 2(5s 3)
15
3x
3
3
Combine like terms. 19.
Add 2x to
20.
both sides.
Add 10 to
both sides.
Divide both sides
by 3.
5x
3-5 Inequalities
■
AF4.1
c
5c
5c
13
3
8
6
4 5x 3 x
21. 4 2y 4y
22. 2n 8 2n 5
2z
3
3z
17
23. 3 2 2 3
AF1.1
(pp. 136–139)
EXERCISES
EXAMPLE
■
(pp. 129–133)
Ext. of
EXERCISES
EXAMPLE
■
AF4.1
Write an inequality for the situation.
Write an inquality for each situation.
The capacity of the elevator was at most
2000 pounds.
Let c capacity of elevator
c 2000 lb
“at most” means less
24. It is no more than a one mile walk from
home to the school.
Graph x 3.
–6
–5
–4
than or equal to
25. The cost of the trip will be at least $1500.
26. Fewer than 45 students are expected to
attend the workshop.
Graph each inequality.
–3
–2
–1
0
27. m 0
28. x 2
1
29. c 4
Study Guide: Review
157
3-6 Solving Inequalities by Adding or Subtracting
EXAMPLE
Solve and graph.
n 5 2
5 5
n
3
30. 13 r 17
0
1
Add 5 to both sides.
2
1
3
2
2
0
5
6
2
Subtract 4 3 from both
2
4 3
x
3
13
4
2
4 3
1
2
31. n 3 6 3 3
32. x 3.8 4
5 3 x 4 3
■
AF4.0
EXERCISES
Solve and graph.
■
(pp. 140–143)
1
3
sides.
2
3
4
3
1
33. 3 2 y
34. Ellory budgets at most $20 each week
for lunch. She has spent $17.75 so far
this week. Write and solve an inequality
to determine how much more Ellory can
spend and stay within her lunch budget.
5
3
3-7 Solving Inequalities by Multiplying or Dividing
EXAMPLE
■
EXERCISES
Solve and graph.
Solve and graph.
z
10
13
z
(13)10 Multiply both sides
(13)
13
by 13. Change
z 130
to .
m
35. 6 3
(pp. 144–147)
AF4.0
36. 4n 12
t
37. 8 2
38. 5p 15
b
39. 9 3
40. 6a 48
60
70
80
90
100 110 120 130 140 150 160
3-8 Solving Two-Step Inequalities
EXAMPLE
■
EXERCISES
Solve and graph 3x 3 9.
3x 3 9
3 3
3x
12
3x
12
3
3
x 4
6
158
4
(pp. 148–151)
2
Add 3 to both sides.
Solve and graph.
41. 5z 12 7
42. 2h 7 5
Divide both sides by 3.
Change to .
a
43. 10 3 2
x
44. 3 8 10
45. 5 3k 4
3
0
2
46. 2y 4 1
Chapter 3 Multi-Step Equations and Inequalities
AF4.1
Simplify each expression. Justify each step.
2
1. 12 7 3
2. 39 52 11
3. (25 9) 4
4. 2.1 (6.5 4.9)
Simplify.
5. 7x 5x
6. m 3m 3
8. 2y 2z 2
9. 3(s 2) s
7. 6n 1 n 5n
10. 10b 8(b 1)
Solve.
11. 10x 2x 16
3y 5y
12. 3 8
13. 4c 6 2c 24
2x
3
11
14. 5 5 5
2
1
15. 5b 4b 3
16. 15 6g 8 19
17. On her last three quizzes, Elise scored 84, 96, and 88. What grade must she
get on her next quiz to have an average of 90 for all four quizzes?
Solve.
18. 3x 13 x 1
19. q 7 2q 5
20. 8n 24 3n 59
21. m 5 m 3
22. 3a 9 3a 9
3z
17
2z
3
23. 2 3 3 2
24. The square and the equilateral triangle have the same
perimeter. Find the perimeter of each figure.
x2
x
Solve and graph.
h
25. 12 4
26. 36 6y
27. 56 7m
29. n 14 3
30. 8 22 p
31. 4 u 20
b
8
28. 4
32. 8 z 6
33. Glenda has a $40 gift certificate to a café that sells her favorite tuna
sandwich for $3.75 after tax. What are the possible numbers of tuna
sandwiches that Glenda can buy with her gift certificate?
Solve and graph.
34. 6m 4 2
35. 8 3p 14
36. 4z 4 8
x
1
2
37. 10 2 5
c
3
1
38. 4 8 2
2
1
d
39. 3 2 6
Chapter 3 Test
159
Gridded Response: Write Gridded Responses
When responding to a test item that requires you to place your answer in a grid,
examine the grid to be sure you know how to fill it in correctly. Grid formats may
vary from test to test. The grid in this book is used often, but it is not used on
every test that has gridded-response items.
Gridded Response: Divide. 3000 7.5
4 0 0
3000 10
3000 7.5 7.5 10
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
30,000
75
Divide.
400
Simplify.
• Write your answer in the answer boxes at the top of the grid.
• Put only one digit in each box. Do not leave a blank box in the
middle of an answer.
• Shade the bubble for each digit in the column beneath it.
1
1
1
2
1
4
3
x 2 2 3 2
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
2
Gridded Response: Solve x 2 3.
7 / 6
0
1
2
3
4
5
6
7
8
9
10
.
7.5 has 1 decimal place, so multiply by 10
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
x 6 6
7
1
x 6, 16 , or 1.16
1
Add 2 to both sides of the equation.
Find a common denominator.
Add.
• Mixed numbers and repeating decimals cannot be gridded, so
you must grid the answer as 76.
• Write your answer in the answer boxes at the top of the grid.
• Put only one digit or symbol in each box. On some grids, the
fraction bar and the decimal point have a designated box.
• Shade the bubble for each digit or symbol in the correct column.
160
Chapter 3 Multi-Step Equations and Inequalities
You cannot grid a negative number in a
gridded-response item because the grid
does not include the negative sign. If you
get a negative answer to a test item,
recalculate the problem because you
probably made a math error.
Item C
A student found 0.65 as
the answer to 5 (0.13).
Then the student filled in
the grid as shown.
Read each statement and then answer
the questions that follow.
Item A
A student correctly
evaluated an expression
and got 193 as a result.
Then the student filled in
the grid as shown.
9 /
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
2. Explain how to fill in the answer
correctly.
A student added 0.21 and
0.49 and got an answer
of 0.7. This answer is
displayed in the grid.
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
1 3
1. What error did the student make
when filling in the grid?
Item B
– 0 . 6 5
5. What error does the grid show?
6. Another student got an answer of
0.65. Explain why the student
knew this answer was wrong.
Item D
A student found that
x 512 was the solution to
the equation 2x 3 8.
Then the student filled in
the grid as shown.
5 1 / 2
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 . 7
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
7. What answer does the grid show?
8. Explain why you cannot fill in a
mixed number.
9. Write the answer 512 in two forms
that could be entered in the grid
correctly.
3. What errors did the student make
when filling in the grid?
4. Explain how to fill in the answer
correctly.
Strategies for Success
161
KEYWORD: MT8CA Practice
Cumulative Assessment, Chapters 1–3
Multiple Choice
1. A cell phone company charges $0.21
per minute for phone calls. Which
expression represents the cost of a
phone call of m minutes?
A
0.21m
C
0.21 m
B
0.21 m
D
0.21 m
2. Laurie had $88 in her bank account
on Sunday. The table below shows her
account activity for the past 5 days.
What is the balance in her account
on Friday?
Day
5. In order to apply for a driver’s permit
in Ohio, you have to be at least 16
years old. Which graph correctly
represents the possible ages of Ohioans
who can apply for a driver’s permit?
A
12 13 14 15 16 17 18 19
B
12 13 14 15 16 17 18 19
C
12 13 14 15 16 17 18 19
D
12 13 14 15 16 17 18 19
Deposit
Withdraw
Monday
$25
—
Tuesday
—
$58
A
2(x 3) 2 x 2 3
Wednesday
—
$45
B
2x 3 3 2x
Thursday
$32
—
C
Friday
$91
—
xyyx
D
2(xy) (2x)y
A
$91
C
$133
B
$103
D
$236
6. Which expression represents the
Distributive Property?
7. Which addition equation represents
the number line diagram below?
3. Which equation has a solution of
x 5?
A
2x 8 2
C
1
x 6 10
5
5 4 3 2 1
B
1
x 10 5
5
D
2x 10 5
A
B
4. You volunteer to bring 4 gallons
of juice for a class party. There are
28 students in the class. You plan to
give each student an equal amount
of juice. Which equation can you use
to determine the amount of juice per
student?
162
0
1
2
3
4
5
6
4 (2) 2
C
4 6 10
4 (6) 2
D
4 (6) 10
8. A snack package has 4 ounces of mixed
nuts, 112 ounces of wheat crackers,
534 ounces of pretzels, and 218 ounces
of popcorn. What is the total weight
of the snacks?
3
A
4x 28
C
28 x 4
A
138 ounces
B
x
4
28
D
28x 4
B
138 ounces
Chapter 3 Multi-Step Equations and Inequalities
1
5
C
128 ounces
D
98 ounces
3
9. Which inequality is the solution of
2 x 1?
3
2
1
A
x 6
B
x 6
1
1
C
x 16
D
x 16
1
10. Which value of x is the solution of the
equation 38 x 34 16?
9
A
x
22
B
x 9
5
5
C
x 19
D
x 29
4
11. Frank purchased x tickets for a concert.
Mark has 1 more ticket than Frank.
Karen has twice as many tickets as
Mark. Which expression represents how
many tickets they have in all?
A
4x 2
C
3x 2
B
3x 3
D
4x 3
When finding the solution of an
equation on a multiple-choice test,
work backward by substituting into
the equation the answer choices
provided.
Gridded Response
12. In 2004, the minimum wage for
workers was $5.85 per hour. To find the
amount of money someone can make
in x hours, use the equation y 5.85x.
How much money does a person who
works 5 hours earn?
13. Solve the equation 94x 31 for x.
14. The sum of two consecutive integers
(x, x 1) is 53. What is the smaller of
the two numbers?
15. In a local high school, 15 of the school’s
600 students earned National Merit
Scholarships. Write as a decimal the
number of scholarships earned per
total number of students.
Short Response
16. Alfred and Eugene each spent $62 on
campsite and gasoline expenses during
their camping trip. Each campsite they
used had the same per-night charge.
Alfred paid for 4 nights of campsites
and $30 of gasoline. Eugene paid for
2 nights of campsites and $46 of
gasoline. Write an equation that could
be used to determine the cost of one
night’s stay at a campsite. What was the
cost of one night’s stay at a campsite?
17. Omar opened a savings account with
a $125 deposit in June. Over the next
year, he withdrew $40.50 in September,
deposited $35.75 in November,
deposited $55 in February, and
withdrew $45.25 in May.
a. List the withdrawals and deposits in
order from least to greatest.
b. Over the 1 year period did the value
of the account rise or fall? What
value represents the total amount
of change in the value of the
account? Determine the final value
of the account.
Extended Response
18. Statement 1: Currently there are
8 more students in student council
than there are officers. There are
12 students total in student council.
Statement 2: In addition, there have to
be at least 4 officers in the council.
a.
Write an equation to represent
Statement 1 and an inequality to
represent Statement 2.
b.
Solve the equation, and plot the
solution to the equation on a
number line.
c.
Graph the solution set to the
inequality.
d.
Explain what the solution sets have
in common, and then explain how
they are different.
Cumulative Assessment, Chapters 1–3
163