Download Principles of Algebra Principles of Algebra

Principles of
Algebra
1A
Expressions and
Integers
1-1
Evaluating Algebraic
Expressions
Writing Algebraic
Expressions
Integers and Absolute
Value
Adding Integers
Subtracting Integers
Multiplying and Dividing
Integers
1-2
1-3
1-4
1-5
1-6
1B
Solving Equations
1-7
Solving Equations by
Adding or Subtracting
Solving Equations by
Multiplying or Dividing
Model Two-Step Equations
Solving Two-Step
Equations
1-8
LAB
1-9
KEYWORD: MT8CA Ch1
Firefighters can use
algebra to find out how
fast a fire is spreading
and to create a plan to
stop it.
2
Chapter 1
Vocabulary
addition
Choose the best term from the list to complete each sentence.
1. __?__is the __?__ of addition.
2. The expressions 3 4 and 4 3 are equal by the __?__.
Associative Property
Commutative
Property
3. The expressions 1 (2 3) and (1 2) 3 are equal by
the __?__.
division
4. Multiplication and __?__ are opposite operations.
opposite operation
5. __?__ and __?__ are commutative.
subtraction
multiplication
Complete these exercises to review skills you will need for this chapter.
Whole Number Operations
Simplify each expression.
6. 8 116 43
7. 2431 187
8. 204 38
9. 6447 21
Compare and Order Whole Numbers
Order each sequence of numbers from least to greatest.
10. 1050; 11,500; 105; 150 11. 503; 53; 5300; 5030 12. 44,400; 40,040; 40,400; 44,040
Inverse Operations
Rewrite each expression using the inverse operation.
13. 72 18 90
14. 12 9 108
15. 100 34 66
16. 56 8 7
Order of Operations
Simplify each expression.
17. 2 3 4
18. 50 2 5
19. 6 3 3 3
20. (5 2)(5 2)
21. 5 6 2
22. 16 4 2 3
23. (8 3)(8 3)
24. 12 3 2 5
Evaluate Expressions
Determine whether the given expressions are equal.
25. (4 7) 2
and 4 (7 2)
26. (2 4) 2
and 2 (4 2)
27. 2 (3 3)
and (2 3) 3
28. 5 (50 44)
and 5 50 44
29. 9 (4 2)
and (9 4) 2
30. 2 3 2 4
and 2 (3 4)
31. (16 4) 4
32. 5 (2 3)
and 16 (4 4)
and (5 2) 3
Principles of Algebra
3
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
NS2.5 Understand the meaning of
absolute
value of a number; interpret
the
the absolute value as the distance of the
number from zero on a number line; and
determine the absolute value of real
numbers.
Academic
Vocabulary
Chapter
Concept
absolute value the distance a number
is from zero on a number line
You find the absolute value of a
number by determining how
far it is from zero on a number
line.
Example: Since 3 is 3 units
from 0 on a number line, the
absolute value of 3 is 3.
interpret to understand and explain
the meaning of
(Lessons 1-3 and 1-5)
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).
variable a symbol, usually a letter,
used to show an amount that can
change
Example: x
verbal using words
You rewrite a verbal statement
using mathematical symbols.
Example: three less than a
number
n3
(Lesson 1-2)
AF1.2 Use the correct order of operations
to evaluate algebraic expressions such as
3(2x 5)2.
(Lesson 1-1)
AF1.4 Use algebraic terminology (e.g.,
variable, equation, term, coefficient, inequality,
expression, constant) correctly.
evaluate find the value
Example: x 3 for x 2
x3
23
5
terminology words used to represent
certain items or ideas
You find the value of
expressions by performing
basic arithmetic operations in a
specific order.
You learn the names of
different mathematical
concepts.
(Lessons 1-1 and 1-7)
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.
(Lesson 1-9)
context the facts or circumstances that
surround a situation
You perform more than one
operation with integers to find
reasonableness makes sense given the the value of a variable that
makes an equation true. You
facts or circumstances
also decide if answers make
sense for the problems.
Standards NS1.1, NS1.2, and AF1.3 are also covered in this chapter. To see these standards unpacked, go to Chapter 2, p. 62
(NS1.2), Chapter 3, p. 114 (AF1.3), and Chapter 4, p. 166 (NS1.1).
4
Chapter 1 Principles of Algebra
Reading Strategy: Use Your Book for Success
Understanding how your textbook is organized will help you locate and use
helpful information.
As you read through an example problem, pay attention to the margin notes ,
such as Helpful Hints, Reading Math notes, and Caution notes. These
notes will help you understand concepts and avoid common mistakes.
Read (ⴚ4)3 as “ⴚ4
to the 3rd power” or
“ⴚ4 cubed.”
The glossary is found
in the back of your
textbook. Use it to
find definitions and
examples of unfamiliar
words or properties.
The index is located
at the end of your
textbook. Use it to
find the page where
a particular concept
is taught.
The Skills Bank is
found in the back of
your textbook. These
pages review concepts
from previous math
courses.
Try This
Use your textbook for the following problems.
1. Use the glossary to find the definition of absolute value.
2. Where can you review the order of operations?
3. On what page can you find answers to exercises in Chapter 2?
4. Use the index to find the page numbers where algebraic expressions,
monomials, and volume of prisms are explained.
Principles of Algebra
5
Evaluating Algebraic
Expressions
1-1
California
Standards
AF1.2 Use the correct order of
operations to evaluate algebraic
expressions such as 3(2x 5)2.
AF1.4 Use algebraic terminology
(e.g., variable, equation, term,
coefficient, inequality, expression,
constant) correctly.
Vocabulary
expression
variable
numerical expression
algebraic expression
evaluate
Why learn this? You can evaluate an expression to
convert a temperature from degrees Celsius to degrees
Fahrenheit. (See Example 3.)
An expression is a mathematical phrase that contains operations,
numbers, and/or variables. A variable is a letter that represents a
value that can change or vary. There are two types of expressions:
numerical and algebraic.
A numerical expression
does not contain variables.
An algebraic expression
contains one or more variables.
Numerical Expressions
Algebraic Expressions
32
4(5)
x2
4n
27 18
3
4
pr
x
4
To evaluate an algebraic expression, substitute a given number for
the variable. Then use the order of operations to find the value of the
resulting numerical expression.
EXAMPLE
1
Evaluating Algebraic Expressions with One Variable
Evaluate each expression for the given value of the variable.
x 5 for x 11
11 5
Substitute 11 for x.
16
Add.
Order of Operations
PEMDAS:
1. Parentheses
2. Exponents
3. Multiply and
Divide from left
to right.
4. Add and Subtract
from left to right.
See Skills Bank p. SB14.
6
2a 3 for a 4
2(4) 3
Substitute 4 for a.
83
Multiply.
11
Add.
4(3 n) 2 for n 0, 1, 2
n
Substitute
Parentheses
Multiply
Subtract
0
4(3 0) 2
4(3) 2
12 2
10
1
4(3 1) 2
4(4) 2
16 2
14
2
4(3 2) 2
4(5) 2
20 2
18
Chapter 1 Principles of Algebra
EXAMPLE
2
Evaluating Algebraic Expressions with Two Variables
Evaluate each expression for the given values of the variables.
5x 2y for x 13 and y 11
5(13) 2(11)
Substitute 13 for x and 11 for y.
65 22
Multiply.
87
Add.
p(24 q) for p 8 and q 17
8(24 17)
Substitute 8 for p and 17 for q.
8(7)
Simplify within parentheses.
56
Multiply.
EXAMPLE
3
Science Application
If c is a temperature in degrees Celsius, then 1.8c 32 can be
used to find the temperature in degrees Fahrenheit. Convert
each temperature from degrees Celsius to degrees Fahrenheit.
freezing point of water: 0°C
1.8c 32
1.8(0) 32
Substitute 0 for c.
0 32
Multiply.
32
Add.
0°C 32°F
Water freezes at 32°F.
highest recorded temperature in the United States: 57°C
1.8c 32
1.8(57) 32
Substitute 57 for c.
102.6 32
Multiply.
134.6
Add.
57°C 134.6°F
The highest recorded temperature in the United States is 134.6°F.
Think and Discuss
1. Give an example of a numerical expression and of an algebraic
expression.
2. Tell how to evaluate an algebraic expression for a given value of a
variable.
3. Explain why you cannot find a numerical value for the expression
4x 5y for x 3.
1-1 Evaluating Algebraic Expressions
7
1-1
California
Standards Practice
AF1.2, AF1.4
Exercises
KEYWORD: MT8CA 1-1
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
Evaluate each expression for the given value of the variable.
1. x 4 for x 11
See Example 2
2. 2a 7 for a 7
Evaluate each expression for the given values of the variables.
4. 3x 2y for x 8 and y 10
See Example 3
3. 2(4 n) 5 for n 0
5. 5r (12 p) for p 15 and r 9
If c is the number of cups of water needed to make papier-mâché paste, then 41c
can be used to find the number of cups of flour needed. Find the number of cups
of flour needed for each number of cups of water.
6. 12 cups
7. 8 cups
8. 7 cups
9. 10 cups
INDEPENDENT PRACTICE
See Example 1
Evaluate each expression for the given value of the variable.
10. x 7 for x 23
See Example 2
11. 7t 2 for t 5
Evaluate each expression for the given values of the variables.
13. 4x 7y for x 9 and y 3
See Example 3
12. 4(3 k) 7 for k 0
14. 4m 2n for m 25 and n 2.5
If c is the number of cups, then 12c can be used to find the number of pints.
Find the number of pints for each of the following.
15. 26 cups
16. 12 cups
17. 20 cups
18. 34 cups
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP2.
Evaluate each expression for the given value of the variable.
19. 30 n for n 8
20. x 4.3 for x 6
21. 5t 5 for t 1
22. 11 6m for m 0
23. 3a 4 for a 8
24. 4g 5 for g 12
25. 4y 2 for y 3.5
26. 18 3y for y 6
27. 3(z 9) for z 6
Evaluate each expression for t 0, x 1.5, y 6, and z 23.
28. 3z 3y
29. yz
30. 1.4z y
31. 4.2y 3x
32. 4(y x)
33. 4(3 y)
34. 3(y 6) 8
35. 4(2 z) 5
36. 5(4 t) 6
37. y(3 t) 7
38. x y z
39. 10x z y
40. 4(z 5t) 3
41. 2y 6(x t)
42. 8txz
43. 2z 3xy
44. A rectangular shape has a length-to-width ratio of approximately 5 to 3.
A designer can use the expression 13(5w) to find the length of such a
rectangle with a given width w. Find the length of such a rectangle with
width 6 inches.
8
Chapter 1 Principles of Algebra
45. Finance A bank charges interest on money it loans. Interest is sometimes
a fixed amount of the loan. The expression a(1 i) gives the total amount
due for a loan of a dollars with interest rate i, where i is written as a
decimal. Find the amount due for a loan of $100 with an interest rate of
10%. (Hint: 10% 0.1)
Califor nia
Entertainment
46. Entertainment There are 24 frames, or still
shots, in one second of movie footage. To
determine the number of frames in a movie,
you can use the expression (24)(60)m, where
m is the running time in minutes. Using the
running time of E.T. the Extra-Terrestrial
shown at right, determine how many frames
are in the movie.
The photos above
were taken in
Palo Alto as part
of Eadweard
Muybridge’s 1878
study on horses
in motion.
Muybridge’s
multi-camera
techniques were
a landmark in
the history of
cinematography.
E.T. the Extra-Terrestrial (1982) has a running
time of 115 minutes, or 6900 seconds.
47. Choose a Strategy A basketball league
has 288 players and 24 teams, with an equal
number of players per team. If the number
of teams is reduced by 6 but the total number
of players stays the same, there will be ___?____ players per team.
A
6 more
B
4 more
C
4 fewer
D
6 fewer
48. Write About It A student says that the algebraic expression 5 x 7
can also be written as 5 7x. Is the student correct? Explain.
49. Challenge Can the expressions 2x and x 2 ever have the same value?
If so, what must the value of x be?
AF1.2
50. Multiple Choice What is the value of the expression 3x 4 for x 2?
A
4
B
6
C
9
D
10
51. Multiple Choice A bakery charges $7 for a dozen muffins and $2 for a
loaf of bread. If a customer bought 2 dozen muffins and 4 loaves of bread,
how much did she pay?
A
$22
B
$38
C
$80
D
$98
52. Gridded Response What is the value of 5(x y) for x 19 and y 6?
Identify the odd number(s) in each list of numbers. (Previous course)
53. 15, 18, 22, 34, 21, 61, 71, 100
54. 101, 114, 122, 411, 117, 121
55. 4, 6, 8, 16, 18, 20, 49, 81, 32
56. 9, 15, 31, 47, 65, 93, 1, 3, 43
Find each sum, difference, product, or quotient. (Previous course)
57. 200 2
58. 200 2
59. 200 2
60. 200 2
61. 200 0.2
62. 200 0.2
63. 200 0.2
64. 200 0.2
1-1 Evaluating Algebraic Expressions
9
Writing Algebraic
Expressions
1-2
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).
Who uses this? Advertisers
can write an algebraic expression
to represent the cost of airing a
commercial a given number of
times. (See Example 3.)
To use algebra to solve
real-world problems, you may
need to translate word phrases
into mathematical symbols.
Sixty-eight different commercials aired during
the 2005 Super Bowl.
Word Phrases
EXAMPLE
1
Expression
• add 5 to a number
• sum of a number and 5
• 5 more than a number
n5
• subtract 11 from a number
• difference of a number and 11
• 11 less than a number
x 11
• 3 multiplied by a number
• product of 3 and a number
3 m or 3m
• 7 divided into a number
• quotient of a number and 7
a or a 7
7
Translating Word Phrases into Math Expressions
Write an algebraic expression for each word phrase.
4 times the difference of a number n and 2
4 times the difference of n and 2
Addition is
commutative, so order
does not matter in
Example 1B. 1 12p
and 12p 1 are both
correct. Subtraction is
not commutative, so
“1 less than the
product of 12 and p”
is written as 12p 1,
not 1 12p.
4
4(n 2)
1 more than the product of 12 and p
1 more than the product of 12 and p
(12 p)
12p 1
10
(n 2)
Chapter 1 Principles of Algebra
1
EXAMPLE
2
Translating Math Expressions into Word Phrases
Write a word phrase for the algebraic expression 4 7b.
4
7
b
4 minus the product of 7 and b
4 minus the product of 7 and b
To solve a word problem, first interpret the action you need to
perform and then choose the correct operation for that action.
EXAMPLE
3
A company aired its 30-second commercial during Super
Bowl XXXIX at a cost of $2.4 million each time. Write an
algebraic expression to determine what the cost would be if the
commercial had aired n times. Then evaluate the expression for
2, 3, and 4 times.
$2.4 million n
Combine n equal amounts of $2.4 million.
When a word
problem involves
groups of equal size,
use multiplication or
division. Otherwise,
use addition or
subtraction.
EXAMPLE
Writing and Evaluating Expressions in Word Problems
2.4n
4
Cost in millions of dollars
n
2.4n
Cost
2
2.4(2)
$4.8 million
3
2.4(3)
$7.2 million
4
2.4(4)
$9.6 million
Evaluate for n 2, 3, and 4.
Writing a Word Problem from a Math Expression
Write a word problem that can be evaluated by the algebraic
expression 14,917 m. Then evaluate the expression for m 633.
At the beginning of the month, Benny’s car had 14,917 miles on the
odometer. If Benny drove m miles during the month, how many miles
were on the odometer at the end of the month?
14,917 m
14,917 633 15,550
Substitute 633 for m.
The car had 15,550 miles on the odometer at the end of the month.
Think and Discuss
1. Give two words or phrases that can be used to express each
operation: addition, subtraction, multiplication, and division.
2. Express 5 7n in words in at least two different ways.
1-2 Writing Algebraic Expressions
11
1-2
Exercises
California
Standards Practice
AF1.1, AF1.2
KEYWORD: MT8CA 1-2
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
See Example 2
Write an algebraic expression for each word phrase.
1. 5 less than the product of 3 and p
2. 77 more than the product of 2 and u
3. 16 more than the quotient of d and 7
4. 6 minus the quotient of u and 2
Write a word phrase for each algebraic expression.
5. 18 43s
See Example 3
See Example 4
22
6. r 37
y
7. 10 3
1
8. 29b 93
9. Mark is going to work for his father’s pool cleaning business during the
summer. Mark’s father will pay him $5 for each pool he helps clean. Write
an algebraic expression to determine how much Mark will earn if he
cleans n pools. Then evaluate the expression for 15, 25, 35, or 45 pools.
10. Write a word problem that can be evaluated by the algebraic expression
x 450. Then evaluate the expression for x 1325.
INDEPENDENT PRACTICE
See Example 1
Write an algebraic expression for each word phrase.
11. 1 more than the quotient of 5 and n
12. 2 minus the product of 3 and p
13. 45 less than the product of 78 and j
14. 4 plus the quotient of r and 5
15. 14 more than the product of 59 and q
See Example 2
Write a word phrase for each algebraic expression.
16. 142 19t
17. 16g 12
5
18. 14 d
w
51
19. 182
See Example 3
20. A community center is trying to raise $1680 to purchase exercise equipment.
The center is hoping to receive equal contributions from members of the
community. Write an algebraic expression to determine how much will be
needed from each person if n people contribute. Then evaluate the
expression for 10, 12, 14, or 16 people.
See Example 4
21. Write a word problem that can be evaluated by the algebraic expression
372 r. Then evaluate the expression for r 137.
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP2.
12
Write an algebraic expression for each word phrase.
22. 6 times the sum of 4 and y
23. the product of 6 and y increased by 9
1
24. 3 of the sum of 4 and p
25. 1 divided by the sum of 3 and g
26. half the sum of m and 5
27. 6 less than the product of 13 and y
28. 2 less than m divided by 8
29. twice the quotient of m and 35
3
30. 4 of the difference of p and 7
31. 8 times the sum of 3 and x
Chapter 1 Principles of Algebra
2
Translate each algebraic expression into words.
32. 4b 3
16 7
34. 8x
33. 8(m 5)
35. 17 w
36. At age 2, a cat or a dog is considered 24 “human” years old. Each year after
age 2 is equivalent to 4 “human” years. Let a represent the age of a cat or dog.
Fill in the expression [24 (a 2)] so that it represents the age of a cat
or dog in “human” years. Copy the chart and use your expression to
complete it.
Age
24 (a 2)
Age
(human years)
2
3
4
5
E
RIT
W
T
NO OK
DO N BO
I
6
37. Reasoning Write two different algebraic expressions for the word phrase
“14 the sum of x and 7.”
38. What’s the Error? A student wrote an algebraic expression for “5 less
n
5
than the quotient of n and 3” as 3 . What error did the student make?
39. Write About It Paul used addition to solve a word problem about the
weekly cost of commuting by toll road for $1.50 each day. Fran solved the
same problem by multiplying. They both got the correct answer. How is
this possible?
40. Challenge Write an expression for the sum of 1 and twice a number n. If
you let n be any number, will the result always be an odd number? Explain.
NS1.2, AF1.1, AF1.2
41. Multiple Choice Which expression means “3 times the difference of y and 4”?
A
3y4
B
3 (y 4)
C
3 (y 4)
D
3 (y 4)
42. Multiple Choice Which expression represents the product of a number n and 32?
A
n 32
B
n 32
C
n 32
D
32 n
43. Short Response A company prints n books at a cost of $9 per book.
Write an expression to represent the total cost of printing n books. What is
the total cost if 1050 books are printed?
Simplify. (Previous course)
44. 32 8 4
45. 24 2 3 6 1
46. (20 8) 2 2
Evaluate each expression for the given value of the variable. (Lesson 1-1)
47. 2(4 x) 3 for x 1
48. 3(8 x) 2 for x 2
1-2 Writing Algebraic Expressions
13
1-3
California
Standards
NS2.5 Understand the
meaning of the absolute value of
a number; interpret the absolute
value as the distance of the
number from zero on a number
line; and determine the absolute
value of real numbers.
Also covered: NS1.1
Vocabulary
integer
opposite
absolute value
Integers and Absolute Value
Why learn this? You can use integers
to represent scores in disc golf.
In disc golf, a player throws a disc
toward a target, or “hole,” in as few
throws as possible. The expected
number of throws to complete an
entire course is called “par.” A player’s
score can be written as an integer to
show how many throws he or she is
above or below par.
If you complete a course in 5 fewer throws than par your score is
5 under par, or 5. If you complete the same course in 4 more throws
than par your score is 4 over par, or 4.
Integers are the set of whole
numbers and their opposites.
Opposites are numbers that are
the same distance from 0 on a
number line, but on opposite
sides of 0.
Opposites
5 4 3 2 1
0
Negative integers
1
2
3
4
5
Positive integers
0 is neither negative nor positive.
EXAMPLE
1
Sports Application
After one round of disc golf, the scores are Fred 5, Trevor 3,
Monique 4, and Julie 2.
Numbers on a
number line increase
in value as you move
from left to right.
Use , , or to compare Trevor’s and Julie’s scores.
Trevor’s score is 3, and Julie’s score is 2.
5 4 3 2 1
0
1
2
3
4
5
2 3
Place the scores on a
number line.
2 is to the left of 3.
Julie’s score is less then Trevor’s.
List the golfers in order from the lowest score to the highest.
The scores are 5, 3, 4, and 2.
Place the scores on a
5 4 3 2 1
0
1
2
3
4
5
number line and read
them from left to right.
In order from the lowest score to the highest,
the golfers are Fred, Julie, Trevor, and Monique.
14
Chapter 1 Principles of Algebra
EXAMPLE
2
Ordering Integers
Write the integers 7, 4, and 3 in order from least to greatest.
Graph the integers on a number line. Then read them from left
to right.
⫺4
⫺3
⫺2
⫺1
0
1
2
3
4
5
6
7
The integers in order from least to greatest are 4, 3, and 7.
A number’s absolute value is its distance from 0 on a number line.
Absolute value of a nonzero number is always positive because
distance is always positive. “The absolute value of 4” is written as
⏐4⏐. Opposites have the same absolute value.
4 units
4 units
⏐4⏐ 4
EXAMPLE
3
⏐4⏐ 4
5 4 3 2 1
0
1
2
3
4
5
Simplifying Absolute-Value Expressions
Simplify each expression.
⏐6⏐
6 units
⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
4
6 is 6 units from 0, so⏐6⏐ 6.
⏐20 20⏐
⏐20 20⏐⏐0⏐
0
⏐9⏐⏐7⏐
⏐9⏐⏐7⏐ 9 7
16
Subtract first: 20 20 0.
Then find the absolute value: 0 is 0
units from 0.
Find the absolute values first: 9 is 9 units
from 0. 7 is 7 units from 0. Then add.
⏐9 3⏐⏐2 3⏐
⏐9 3⏐⏐2 3⏐⏐6⏐⏐5⏐ 9 3 6; 2 3 5
65
6 is 6 units from 0. 5 is 5 units from 0.
1
Think and Discuss
1. Explain the steps you would take to simplify ⏐5⏐⏐2⏐.
2. Explain whether an absolute value is ever negative.
1-3 Integers and Absolute Value
15
1-3
California
Standards Practice
NS1.1,
NS2.5
Exercises
KEYWORD: MT8CA 1-3
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
See Example 2
1. After the first round of the 2005 Masters golf tournament, scores relative to
par were Tiger Woods 2, Vijay Singh 4, Phil Mickelson 2, and Justin
Leonard 5. Use , , or to compare Vijay Singh’s and Phil Mickelson’s
scores. Then list the golfers in order from the lowest score to the highest.
Write the integers in order from least to greatest.
2. 5, 2, 3
See Example 3
3. 17, 6, 8
4. 9, 21, 14
5. 3, 7, 0
8. ⏐3⏐ ⏐11⏐
9. ⏐10⏐⏐4 1⏐
Simplify each expression.
6. ⏐9⏐
7. ⏐22 7⏐
10. ⏐0⏐
11. ⏐12⏐⏐9⏐
12. ⏐28 18⏐
13. ⏐8 7⏐⏐6 5⏐
INDEPENDENT PRACTICE
See Example 1
14. During a very cold week, the temperature in Philadelphia was 7°F on
Monday, 4°F on Tuesday, 2°F on Wednesday, and 3°F on Thursday.
Use , , or to compare the temperatures on Wednesday and Thursday.
Then list the days in order from the coldest to the warmest.
See Example 2
Write the integers in order from least to greatest.
15. 6, 5, 2
See Example 3
16. 8, 11, 5
17. 25, 30, 27 18. 4, 2, 1
Simplify each expression.
19. ⏐6 3⏐
20. ⏐3⏐
21. ⏐6⏐⏐8 3⏐
22. ⏐7 2⏐⏐9 8⏐
23. ⏐5⏐
24. ⏐7⏐⏐14⏐
25. ⏐8 8⏐
26. ⏐19⏐⏐13⏐
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP2.
Compare. Write , , or .
27. 9
15
31. ⏐7⏐
⏐6⏐
17
28. 13
32. ⏐3⏐
⏐3⏐
29. 23
23
33. ⏐13⏐
⏐2⏐
30. 14
34. ⏐20⏐
0
⏐21⏐
Write the integers in order from least to greatest.
35. 24, 16, 12
36. 46, 31, 52
37. 45, 35, 25
Simplify each expression.
38. ⏐17⏐⏐24⏐
39. ⏐22⏐⏐28⏐
40. ⏐53 37⏐
41. ⏐21 20⏐
42. ⏐7⏐⏐9⏐
43. ⏐6⏐⏐12⏐
44. ⏐72⏐⏐8⏐
45. ⏐3⏐ ⏐3⏐
46. Reasoning Two integers A and B are graphed on a number line. If A is less
than B, is ⏐A⏐ always less than ⏐B⏐? Explain your answer.
16
Chapter 1 Principles of Algebra
Simplify each expression.
Science
47. ⏐51 23⏐⏐8 11⏐
48. ⏐61⏐⏐28 9⏐
49. ⏐37 14⏐⏐25 10⏐
50. ⏐24⏐⏐13 5⏐
51. ⏐35 11⏐⏐15 7⏐
52. ⏐17 26⏐⏐29⏐
53. Science The table shows the lowest
recorded temperatures for each
continent. Write the continents in order
from the lowest recorded temperature to
the highest recorded temperature.
The Ice Hotel in
Jukkasjärvi,
Sweden, is rebuilt
every winter from
30,000 tons of
snow and 4000
tons of ice.
54. Chemistry The boiling point of nitrogen
is 196°C. The boiling point of oxygen is
183°C. Which element has the greater
boiling point? Explain your answer.
55. Reasoning Write rules for using
absolute value to compare two integers.
Be sure to take all of the possible
combinations into account.
Lowest Recorded Temperatures
Continent
Temperature
Africa
11°F
Antarctica
129°F
Asia
90°F
Australia
9°F
Europe
67°F
North America
81°F
South America
27°F
56. Write About It Explain why there is no number that
can replace n to make the equation ⏐n⏐ 1 true.
57. Challenge List the integers that can replace n to make
the statement ⏐8⏐ n ⏐5⏐ true.
NS1.1,
NS2.5, AF1.1, AF1.2
58. Multiple Choice Which set of integers is in order from greatest to least?
A
10, 8, 5
B
8, 5, 10
C
5, 8, 10
D
10, 5, 8
D
5
D
⏐21⏐
59. Multiple Choice Which integer is between 4 and 2?
A
0
B
3
C
4
60. Multiple Choice Which expression has the greatest value?
A
⏐12⏐
B
⏐15⏐
C
⏐18⏐
Evaluate each expression for a 3, b 2.5, and c 24. (Lesson 1-1)
61. c 15
62. 9a 8
63. 8(a 2b)
64. bc a
Write an algebraic expression for each word phrase. (Lesson 1-2)
65. 8 more than the product of 7 and a number t
66. A pizzeria delivered p pizzas on Thursday. On Friday, it delivered 3 more
than twice the number of pizzas delivered on Thursday. Write an
expression to show the number of pizzas delivered on Friday.
1-3 Integers and Absolute Value
17
1-4
California
Standards
NS1.2 Add, subtract, multiply,
and divide rational numbers
(integers, fractions, and terminating
decimals) and take positive rational
numbers to whole-number powers.
AF1.3 Simplify numerical
expressions by applying properties of
rational numbers (e.g., identity, inverse,
distributive, associative, commutative)
and justify the process used.
Vocabulary
Adding Integers
Why learn this? You can
track your daily calorie count
by adding integers. (See
Example 4.)
Suppose you eat a piece of grapefruit
that contains 35 calories. Then you
walk 2 laps around a track and burn
35 calories. Using opposites, you can
represent this situation as 35 (35) 0. When
you add two opposites, or additive inverses , the sum is zero.
You can model integer addition on a number line.
additive inverse
EXAMPLE
1
Using a Number Line to Add Integers
Use a number line to find each sum.
3 (7)
To add a positive
number, move to
the right. To add a
negative number,
move to the left.
7
3
5 4 3 2 1
0
1
2
3
4
5
Start at 0. Move right
3 units. From 3, move left
7 units.
You finish at 4, so 3 (7) 4.
2 (5)
5
2
7 6 5 4 3 2 1
0
1
2
3
Start at 0. Move left
2 units. From 2, move
left 5 units.
You finish at 7, so 2 (5) 7.
Another way to add integers is to use absolute value.
ADDING TWO INTEGERS
If the signs are the same. . .
. . . find the sum of the absolute
values. Use the same sign as
the integers.
18
Chapter 1 Principles of Algebra
If the signs are different. . .
. . . find the difference of the
absolute values. Use the sign
of the integer with the greater
absolute value.
EXAMPLE
2
Using Absolute Value to Add Integers
Add.
EXAMPLE
3
4 (6)
4 (6)
10
Think: Find the sum of ⏐4⏐ and ⏐6⏐.
8 (9)
8 (9)
1
Think: Find the difference of ⏐8⏐ and ⏐9⏐.
5 11
5 11
6
Think: Find the difference of ⏐5⏐ and ⏐11⏐.
Same sign; use the sign of the integers.
9 8; use the sign of 9.
11 5; use the sign of 11.
Evaluating Expressions with Integers
Evaluate b 11 for b 6.
b 11 6 11
Replace b with 6.
Think: Find the difference of ⏐11⏐ and ⏐6⏐.
6 11 5
11 6; use the sign of 11.
When finding the sum of three or more integers, you may want to
group opposites, or additive inverses. You may also want to group
positive integers together and negative integers together.
EXAMPLE
ing
Monday Morn
Calories
Oatmeal
Toast w /jam
8 fl oz juice
145
62
111
Calories burned
110
Walked six laps 40
ps
la
Swam six
4
Health Application
Melanie wants to check her calorie count after breakfast and
exercise. Use information from the journal entry to find her total.
Use positive for calories eaten and negative for calories burned.
145 62 110 (110) (40) (65)
145 62 (110 110) (40) (65) Group opposites.
145 62 0 (40) (65)
(145 62) (40 65)
Group integers with same signs.
207 (105)
Add integers within each group.
102
207 105; use the sign of 207.
Melanie’s calorie count after breakfast and exercise is 102 calories.
Think and Discuss
1. Compare the sums of 10 (22) and 10 22.
2. Describe how to add the following expressions on a number line:
9 (13) and 13 9. Then compare the sums.
1-4 Adding Integers
19
1-4
California
Standards Practice
NS1.2,
AF1.3
Exercises
KEYWORD: MT8CA 1-4
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
Use a number line to find each sum.
1. 5 1
See Example 2
3. 7 9
4. 4 (2)
6. 7 (3)
7. 11 17
8. 6 (8)
Add.
5. 12 5
See Example 3
2. 6 (4)
Evaluate each expression for the given value of the variable.
9. t 16 for t 5
See Example 4
10. m 7 for m 5
12. Lee opens a checking account. In the first
month, he makes two deposits and writes
three checks, as shown at right. Find what his
balance is at the end of the month. (Hint:
Checks count as negative amounts.)
11. p (5) for p 5
Checks
Deposits
$134
$600
$56
$225
$302
INDEPENDENT PRACTICE
See Example 1
Use a number line to find each sum.
13. 5 (7)
See Example 2
See Example 3
14. 7 7
15. 4 (9)
16. 4 7
17. 8 14
18. 6 (7)
19. 8 (8)
20. 19 (5)
21. 22 (15)
22. 17 9
23. 20 (12)
24. 18 7
Add.
Evaluate each expression for the given value of the variable.
25. q 13 for q 10
See Example 4
26. x 21 for x 7
28. On Monday morning, a
mechanic has no cars in her
shop. The table at right
shows the number of cars
dropped off and picked up
each day. Find the total
number of cars left in her
shop on Friday.
27. z (7) for z 16
Cars Dropped
Off
Cars Picked
Up
Monday
8
4
Tuesday
11
6
9
12
14
9
7
6
Wednesday
Thursday
Friday
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP2.
Write an addition equation for each number line diagram.
29.
8 7 6 5 4 3 2 1
30.
0
1
2
5 4 3 2 1
20
Chapter 1 Principles of Algebra
0
1
2
3
4
5
Find each sum.
Economics
31. 9 (3)
32. 16 (22)
33. 34 17
34. 44 39
35. 45 (67)
36. 14 85
37. 52 (9)
38. 31 (31)
Evaluate each expression for the given value of the variable.
The number one
category of
imported goods in
the United States
is industrial
supplies, including
petroleum and
petroleum
products. In 2004,
this category
accounted for
over $412 million
of imports.
39. c 17 for c 9
40. k (12) for k 4
41. b (6) for b 24
42. 13 r for r 19
43. 9 w for w 6
44. 3 n (8) for n 5
45. Economics Refer to the
Exports
Imports
data at right about U.S.
Goods
$807,584,000,000 $1,473,768,000,000
international trade for
Services $338,553,000,000 $290,095,000,000
the year 2004. Consider
Source: U.S. Census Bureau
values of exports as
positive quantities and values of imports as negative quantities.
a. What was the total of U.S. exports in 2004?
b. What was the total of U.S. imports in 2004?
c. The sum of exports and imports is called the balance of trade.
Approximate the 2004 U.S. balance of trade to the nearest billion
dollars.
46. What’s the Error? A student evaluated 4 d for d 6 and gave an
answer of 2. What might the student have done wrong? Give the correct
answer.
47. Write About It Explain the different ways it is possible to add two
integers and get a negative answer.
48. Challenge What is the sum of 3 (3) 3 (3) . . . when there
are 10 terms? 19 terms? 24 terms? 25 terms? Explain any patterns that
you find.
NS1.2,
NS2.5, AF1.2
49. Multiple Choice Which of the following is the value of 7 3h when h 5?
A
22
B
8
C
8
22
D
50. Gridded Response Evaluate the expression 35 y for y 8.
Evaluate each expression for the given values of the variables. (Lesson 1-1)
51. 2x 3y for x 8 and y 4
52. 6s t for s 7 and t 12
Simplify each expression. (Lesson 1-3)
53. ⏐3⏐ ⏐9⏐
54. ⏐22 9⏐
55. ⏐18⏐ ⏐5⏐
56. ⏐27⏐ ⏐5⏐
1-4 Adding Integers
21
1-5
California
Standards
NS1.2 Add, subtract,
multiply, and divide rational
numbers (integers, fractions, and
terminating decimals) and take
positive rational numbers to wholenumber powers.
NS2.5 Understand the
meaning of the absolute value of
a number; interpret the absolute
value as the distance of the number
from zero on a number line; and
determine the absolute value of
real numbers.
Subtracting Integers
Why learn this? You can represent the differences
between the drops and climbs of a roller coaster by
subtracting integers. (See Example 3.)
In Lesson 1-4 you modeled integer addition on a number line. You
can also model integer subtraction on a number line. The model
below shows how to find 8 10.
10
8
3 2 1
0
1
2
3
4
5
6
7
8
9
Start at 0. Move right
8 units. From 8, move
left 10 units.
When you subtract a positive integer, the difference is less than the
original number. Therefore, you move to the left on the number line.
To subtract a negative integer, you move to the right.
You can also subtract an integer by adding its opposite. You can then
use the rules for addition of integers.
SUBTRACTING TWO INTEGERS
Words
To subtract an integer,
add its opposite.
EXAMPLE
1
Numbers
Algebra
3 7 3 (7)
a b a (b)
5 (8) 5 8
a (b) a b
Subtracting Integers
Subtract.
7 7
7 7 7 (7)
14
Add the opposite of 7.
2 (4)
2 (4) 2 4
6
Add the opposite of 4.
Same sign; add; use the sign of the integers.
Same sign; add; use the sign of the integers.
13 (5)
13 (5) 13 5 Add the opposite of 5.
8
Unlike signs; subtract; 13 5; use the sign of 13.
22
Chapter 1 Principles of Algebra
EXAMPLE
2
Evaluating Expressions with Integers
Evaluate each expression for the given value of the variable.
4 s for s 9
4 s
4 (9)
Substitute 9 for s.
4 9
Add the opposite of 9.
5
9 4; use the sign of 9.
3 x for x 5
3 x
3 5
Substitute 5 for x.
3 (5)
Add the opposite of 5.
8
Same sign; use the sign of the integers.
⏐6 t⏐ 5 for t 4
⏐6 t⏐ 5
⏐6 (4)⏐ 5
Substitute 4 for t.
⏐6 4⏐ 5
Add the opposite of 4.
⏐10⏐ 5
6 4 10
10 5
The absolute value of 10 is 10.
15
Add.
EXAMPLE
3
Architecture Application
The roller coaster Desperado has a
maximum height of 209 feet and a
maximum drop of 225 feet. How
far underground does the roller
coaster go?
209 (225)
Subtract the drop
Desperado Roller Coaster
209 ft
225 ft
from the height.
209 (225)
Add the opposite of
225.
16
225 209; use the
sign of 225.
Ground level, 0 ft
? ft
Desperado goes 16 feet underground.
Think and Discuss
1. Explain why 10 (10) does not equal 10 10.
2. Describe the answer that you get when you subtract a greater
number from a lesser number. Then give an example.
1-5 Subtracting Integers
23
1-5
California
Standards Practice
NS1.2,
NS2.5
Exercises
KEYWORD: MT8CA 1-5
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
Subtract.
1. 5 9
See Example 2
2. 8 (6)
4. 11 (6)
Evaluate each expression for the given value of the variable.
5. 9 h for h 8
See Example 3
3. 8 (4)
6. 7 m for m 5
7. ⏐3 k⏐ ⏐3⏐ for k 12
8. The temperature rose from 4°F to 45°F in Spearfish, South Dakota, on
January 22, 1943, in only 2 minutes! By how many degrees did the
temperature change?
INDEPENDENT PRACTICE
See Example 1
Subtract.
9. 3 7
See Example 2
10. 14 (9)
12. 8 (2)
Evaluate each expression for the given value of the variable.
13. 14 b for b 3
See Example 3
11. 11 (6)
14. 7 q for q 15
15. ⏐5 f⏐ 7 for f 12
16. A submarine cruising at 27 m below sea level, or 27 m, descends 14 m.
What is its new depth?
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP3.
Write a subtraction equation for each number line diagram.
17.
18.
8 6 4 2
0
2
4
6 5 4 3 2 1
0
1
2
3
4
5
Perform the given operations.
19. 8 (11)
20. 24 (27)
21. 43 13
22. 26 26
23. 13 7 (6)
24. 11 (4) (9)
Evaluate each expression for the given value of the variable.
25. x 16 for x 4
26. 8 t for t 5
27. 16 y for y 8
28. s (22) for s 18
29. ⏐r 11⏐ ⏐9⏐for r 7
30. 4 ⏐2 p⏐for p 15
31. Estimation A roller coaster starts with a 160-foot climb and then
plunges 228 feet down a canyon wall. It then climbs a gradual 72 feet
before a steep climb of 189 feet. Approximately how far is the coaster
above or below its starting point?
24
Chapter 1 Principles of Algebra
6
Social Studies
Use the timeline to answer the questions. Use negative numbers for
years B.C.E. Assume that there was a year 0 (there wasn’t) and that
there have been no major changes to the calendar (there have been).
32. How long was the Greco-Roman era, when Greece and Rome
ruled Egypt?
33. Which was a longer period of time: from the Great Pyramid to
Cleopatra, or from Cleopatra to the present? By how many years?
34. Queen Neferteri ruled Egypt about 2900 years before the Turks
ruled. In what year did she rule?
35. There are 1846 years between which two events on this timeline?
36.
Write About It What is it about years B.C.E. that make
negative numbers a good choice for representing them?
37.
Challenge How would your calculations differ if you
took into account the fact that there was no year 0?
KEYWORD: MT8CA Egypt
NS1.2 ,
NS2.5, AF1.1
38. Multiple Choice Which of the following is equivalent to ⏐7 (3)⏐?
A
⏐7⏐ ⏐3⏐
B
⏐7⏐ ⏐3⏐
C
10
D
4
39. Gridded Response Subtract: 4 (12).
Write an algebraic expression to evaluate the word problem. (Lesson 1-2)
40. Tate bought a compact disc for $17.99. The sales tax on the disc was t dollars.
What was the total cost including sales tax?
Evaluate each expression for m 3. (Lesson 1-4)
41. m 6
42. m (5)
43. 9 m
44. m 3
1-5 Subtracting Integers
25
1-6
California
Standards
NS1.2 Add, subtract,
multiply, and divide rational
numbers (integers, fractions, and
terminating decimals) and take
positive rational numbers to wholenumber powers.
Multiplying and Dividing
Integers
Who uses this? Football
players can multiply integers
to represent the total change
in yards during several plays.
(See Example 3.)
If a football team loses 10 yards
in each of 3 plays, the total
change in yards can be
represented by 3(10). You can
write this product as repeated addition.
3(10) 30
3(10) 10 (10) (10) 30
2(10) 20
Notice that a positive integer multiplied by a
negative integer gives a negative product.
1(10) 10
You already know that the product of two
positive integers is a positive integer. The
pattern shown at right can help you
understand that the product of two negative
integers is also a positive integer.
0(10) 0
1(10) 10
2(10) 20
3(10) 30
10
10
10
10
10
10
MULTIPLYING AND DIVIDING TWO INTEGERS
If the signs are the same, the sign of the answer is positive.
If the signs are different, the sign of the answer is negative.
EXAMPLE
1
Multiplying and Dividing Integers
Multiply or divide.
The sum of two
negative integers is
always negative, but
the product or
quotient of two
negative integers is
always positive.
26
5(8)
Signs are different.
45
9
Signs are the same.
40
Answer is negative.
5
Answer is positive.
4(2)(3)
4(2)(3)
8(3)
8(3)
24
Multiply two integers.
32
8
Signs are different.
4
Answer is negative.
Chapter 1 Principles of Algebra
Signs are different.
Answer is negative.
Signs are the same.
Answer is positive.
EXAMPLE
2
Using the Order of Operations with Integers
Simplify.
3(2 8)
3(2 8)
3(2 8)
3(6)
18
Subtract inside the parentheses.
Add the opposite of 8 inside the parentheses.
Think: The signs are the same.
The answer is positive.
4(7) 2
4(7) 2
28 2
30
Multiply first.
Think: The signs are different; answer is negative.
Same sign; use the sign of the integers.
2(14 6)
2(14 6)
2(20)
40
EXAMPLE
3
Add inside the parentheses.
Think: The signs are different.
The answer is negative.
Sports Application
A football team runs 10 plays. On 6 plays, it has a gain of
4 yards each. On 4 plays, it has a loss of 5 yards each. Each
gain in yards can be represented by a positive integer, and
each loss can be represented by a negative integer. Find the
total net change in yards.
6(4) 4(5)
24 (20)
4
Add the losses to the gains.
Multiply.
Add.
The team gained 4 yards.
Think and Discuss
1. List all possible multiplication and division statements for
the integers with absolute values of 5, 6, and 30.
For example, 5 6 30.
2. Compare the sign of the product of two negative integers with the
sign of the sum of two negative integers.
3. Suppose the product of two integers is positive. What do you
know about the signs of the integers?
1-6 Multiplying and Dividing Integers
27
1-6
California
Standards Practice
NS1.2, AF1.2
Exercises
KEYWORD: MT8CA 1-6
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
Multiply or divide.
1. 8(4)
See Example 2
3. 7(4)(3)
32
4. 8
6. 4(3) 9
7. 6(5 9)
8. 3 11(7 3)
Simplify.
5. 7(5 12)
See Example 3
54
2. 9
Business An investor buys shares of stock A and stock B. Stock A loses $8 per
share, and stock B gains $5 per share. Given the number of shares, how much
does the investor lose or gain?
9. stock A: 20 shares, stock B: 35 shares
10. stock A: 30 shares, stock B: 20 shares
INDEPENDENT PRACTICE
See Example 1
See Example 2
Multiply or divide.
11. 3(7)
72
12. 6
13. 12(7)
42
14. 6
3(4)(2)
15. 8
16. 13(9)
17. 5(7)(4)
63
18. 3(21)
20. 13(2) 8
21. 13(8 11)
22. 10 4(5 8)
Simplify.
19. 12(9 14)
See Example 3
Banking A student puts $50 in the bank each time he makes a deposit.
He takes $20 each time he makes a withdrawal. Given the number of
transactions, what is the net change in the student’s account?
23. deposits: 4, withdrawals: 5
24. deposits: 3, withdrawals: 8
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP3.
25. Science Ocean tides are the result of the
gravitational force between the sun, the
moon, and the earth. When ocean tides
occur, the earth’s crust also moves. This
is called an earth tide. The formula for
the height of an earth tide is y 3x, where
x is the height of the ocean tide. Fill in
the table.
Ocean Tide
Height (x)
x
3
Earth Tide
Height (y)
15
6
9
15
Perform the given operations.
28
26. 7(6)
144
27. 1
2
28. 7(7)
160
29. 40
30. 2(3)(5)
96
31. 12
32. 12(3)(2)
18(6)
33. 3
Chapter 1 Principles of Algebra
Evaluate the expressions for the given value of the variable.
Science
34. 4t 5 for t 3
35. x 2 for x 9
36. 6(s 9) for s 1
r
37. 8 for r 64
42
38. t for t 6
for y 35
39. 4
40. Science The ocean floor is
extremely uneven. It includes
underwater mountains, ridges, and
extremely deep areas called
trenches. To the nearest foot, find
the average depth of the trenches
shown.
Depths of Ocean Trenches
(Sea level) 0
–20,000
Depth (ft)
Anoplogaster
cornuta, often
called a fangtooth
or ogrefish, is a
predatory fish that
reaches a maximum
length of 15 cm.
It can be found
in tropical and
temperate waters
at 16,000 ft.
y 11
–25,000
Yap
–27,976
–30,000
Bonin
–32,788
–35,000
– 40,000
Kuril
–31,988
Mariana
–35,840
41. Reasoning A football team runs
11 plays. There are 3 plays that result
in a loss of 2 yards each and 8 plays that result in a gain of
4 yards each. To find the total yards gained, Art simplifies the expression
3(2) 8(4). Bella first finds the total yards lost, 6, and the total yards
gained, 32. Then she subtracts 6 from 32. Compare these two methods.
42. Choose a Strategy P is the set of positive factors of 30, and Q is the set
of negative factors of 18. If x is a member of set P and y is a member of set
Q, what is the greatest possible value of x y?
A
540
B
180
90
C
D
1
43. Write About It If you know that the product of two integers is negative,
what can you say about the two integers? Give examples.
44. Challenge A visiting player is positioned on his own 30-yard line. The
player loses 3 yards after a gain of 12 yards. What is the player’s position if
4 yards are lost on the next two plays?
NS1.2, AF1.1
45. Multiple Choice What is the product of 7 and 10?
A
70
B
17
C
3
D
70
46. Short Response Brenda donates part of her salary to the local children’s
hospital each month by having $15 taken out of her monthly paycheck.
Write an integer to represent the amount taken out of each paycheck.
Find an integer to represent the change in the amount of money in
Brenda’s paychecks after 1.5 years.
Write an algebraic expression for each word phrase. (Lesson 1-2)
47. j decreased by 18
48. twice b less 12
49. 22 less than y
Find each sum or difference. (Lessons 1-4 and 1-5)
50. 7 3
51. 5 (4)
52. 3 (6)
53. 513 (259)
1-6 Multiplying and Dividing Integers
29
Quiz for Lessons 1-1 Through 1-6
1-1
Evaluating Algebraic Expressions
Evaluate each expression for the given values of the variables.
1. 5x 6y for x 8 and y 4
1-2
2. 6(r 7t) for r 80 and t 8
Writing Algebraic Expressions
Write an algebraic expression for each word phrase.
3. one-sixth the sum of r and 7
4. 10 plus the product of 16 and m
Write a word phrase for each algebraic expression.
5. 7y 46
1-3
x 10
7. 3
6. 2(18 t)
p
8. 15 3
2
Integers and Absolute Value
Write the integers in order from least to greatest.
9. 17, 25, 18, 2
10. 0, 8, 9, 1
Simplify each expression.
11. ⏐14 7⏐
1-4
12. ⏐15⏐ ⏐12⏐
13. ⏐26⏐ ⏐14⏐
Adding Integers
Evaluate each expression for the given value of the variable.
14. p 14 for p 8
15. w (9) for w 4
16. In Loma, Montana, on January 15, 1972, the temperature increased 103
degrees in a 24-hour period. If the lowest temperature on that day was
54°F, what was the highest temperature?
1-5
Subtracting Integers
Subtract.
17. 12 (8)
18. 7 (5)
19. 5 (16)
20. 22 5
21. The point of highest elevation in the United States is on Mount McKinley,
Alaska, at 20,320 feet. The point of lowest elevation is in Death Valley,
California, at 282 feet. What is the difference in the elevations?
1-6
Multiplying and Dividing Integers
Multiply or divide.
22. (8)(6)
30
28
23. 7
Chapter 1 Principles of Algebra
39
24. 3
25. (2)(5)(6)
California
Standards
MR1.1
Analyze problems by
identifying relationships,
distinguishing relevant from irrelevant
information, identifying missing
information, sequencing and prioritizing
information, and observing patterns.
Also covered:
NS1.2
Solve
• Choose an operation: Addition or Subtraction
To decide whether to add or subtract to solve a word problem, you
need to determine what action is taking place in the problem. If
you are combining numbers or putting numbers together, you
need to add. If you are taking away or finding out how far apart
two numbers are, you need to subtract.
Action
Combining or
putting together
Operation
Illustration
Add
Removing or
taking away
Subtract
Finding the difference
Subtract
Jan has 10 red marbles. Joe gives her 3 more. How many marbles
does Jan have now? The action is combining marbles. Add 10 and 3.
Determine the action in each problem. Use the actions to restate the
problem. Then give the operation that must be used to solve the problem.
1 Lake Superior is the largest of the Great Lakes
and contains approximately 3000 mi3 of water.
Lake Michigan is the second largest Great
Lake by volume and contains approximately
1180 mi3 of water. Approximately how much
more water is in Lake Superior?
Superior
Lake
N
W
Upper Peninsula
Lower
Peninsula
t
L. On
Michigan
Lak
E
3 Einar has $18 to spend on his friend’s
birthday presents. He buys one present that
costs $12. How much does he have left to
spend?
S
uron
eH
ak
Lake Michig
an
L
2 The average temperature in Homer,
Alaska, is approximately 53°F in July and
approximately 24°F in December. How much
hotter is the average temperature in Homer
in July than in December?
eE
a r io
4 Dinah got 87 points on her first test and 93
points on her second test. What is her point
total for the first two tests?
ri e
Focus on Problem Solving
31
1-7
California
Standards
Preparation for
AF4.0
Students solve simple linear
equations and inequalities over the
rational numbers.
AF1.4 Use algebraic terminology
(e.g., variable, equation, term,
coefficient, inequality, expression,
constant) correctly.
Solving Equations by
Adding or Subtracting
Why learn this? You can use
an equation to represent the
forces acting on an object. (See
Example 3.)
equation
inverse operations
EXAMPLE
11
An equation is a mathematical statement that uses an equal sign
to show that two expressions have the same value. All of these
are equations.
3 8 11
Vocabulary
38
r 6 14
24 x 7
100
50
2
To solve an equation that contains a variable, find the value of the
variable that makes the equation true. This value of the variable is
called the solution of the equation.
1
Determining Whether a Number Is a Solution of an Equation
Determine which value of x is a solution of the equation.
The phrase
“subtraction ‘undoes’
addition” can be
understood with
this example:
If you start with 3
and add 4, you can
get back to 3 by
subtracting 4.
34
4
3
x 7 13; x 12 or 20
Substitute each value for x in the equation.
x 7 13
x 7 13
?
?
20 7 13
Substitute 20 for x.
12 7 13
Substitute 12 for x.
?
?
13 13 ✔
5 13 ✘
So 20 is a solution.
So 12 is not a solution.
Adding and subtracting by the same number are inverse operations .
Inverse operations “undo” each other. To solve an equation, use
inverse operations to isolate the variable. In other words, get the
variable alone on one side of the equal sign.
The properties of equality allow you to perform inverse operations.
These properties show that you can perform the same operation on
both sides of an equation.
You can use the properties of equality along with the Identity
Property of Addition to solve addition and subtraction equations.
IDENTITY PROPERTY OF ADDITION
Words
The sum of any number
and zero is that number.
32
Chapter 1 Principles of Algebra
Numbers
505
2 0 2
Algebra
a0a
EXAMPLE
2
Solving Equations Using Addition and Subtraction Properties
Solve.
6 t 28
6t
6
0t
t
28
6
22
22
Check
6 t 28
?
6 22 28
?
28 28 ✔
m 8 14
m 8 14
8
8
m 0 6
m
6
Check
m 8 14
?
6 8 14
?
14 14 ✔
Since 6 is added to t, subtract 6 from both sides
to undo the addition.
Identity Property of Addition: 0 t t
To check your solution, substitute 22 for t in the
original equation.
Since 8 is subtracted from m, add 8 to both
sides to undo the subtraction.
Identity Property of Addition: m 0 m
To check your solution, substitute 6 for m in
the original equation.
15 w (14)
15 w (14)
15 (14) w (14) (14)
29 w 0
29 w
w 29
Since 14 is added to w, subtract
14 from both sides to undo the
addition.
Identity Property of Addition
Definition of Equality
PROPERTIES OF EQUALITY
Words
Addition Property of Equality
You can add the same
number to both sides of an
equation, and the statement
will still be true.
Numbers
Algebra
23 5
4 4
27 9
If x y, then
x z y z.
4 7 11
3 3
44 8
If x y, then
x z y z.
Subtraction Property of Equality
You can subtract the same
number from both sides
of an equation, and the
statement will still be true.
1-7 Solving Equations by Adding or Subtracting
33
EXAMPLE
3
PROBLEM SOLVING APPLICATION
Net force is the sum of all forces acting on an object. Expressed in
newtons (N), it tells you in which direction and how quickly the
object will move. If two dogs are playing tug-of-war, and the dog
on the right pulls with a force of 12 N, what force is the dog on
the left exerting on the rope if the net force is 2 N?
1 Understand the Problem
1.
Force is measured in
newtons (N). The
number of newtons
tells the size of the
force and the sign
tells its direction.
Positive is to the
right and up, and
negative is to the
left and down.
The answer is the force that the
left dog exerts on the rope.
List the important information:
• The dog on the right pulls with a force of 12 N.
• The net force is 2 N.
Show the relationship of the information:
net force left dog’s force right dog’s force
2.2 Make a Plan
Write an equation and solve it. Let f represent the left dog’s force
on the rope, and use the equation model.
2 Math
Builders
For more on writing
equations from words,
see the Graph and
Equation Builder on
page MB2.
f
12
3.3 Solve
2 f 12
12
12
10 f
Subtract 12 from both sides.
The left dog is exerting a force of 10 newtons on the rope.
4.4 Look Back
The problem states that the net force is 2 N, which means that the
dog on the right must be pulling with more force. The absolute
value of the left dog’s force is less than the absolute value of the
right dog’s force, 10 12, so the answer is reasonable.
Think and Discuss
1. Explain what the result would be in the tug-of-war match in
Example 3 if the dog on the left pulled with a force of 7 N and
the dog on the right pulled with a force of 6 N.
2. Describe the steps to solve y 5 16.
34
Chapter 1 Principles of Algebra
1-7
California
Standards Practice
NS1.2, AF1.4, Preparation for
AF4.0
Exercises
KEYWORD: MT8CA 1-7
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
Determine which value of x is a solution of each equation.
1. x 6 18; x 10, 12, or 25
See Example 2
See Example 3
2. x 7 14; x 2, 7, or 21
Solve.
3. m 9 23
4. 8 t 13
5. p (13) 10
6. q (25) 81
7. 26 t 13
8. 52 p (41)
9. A team of mountain climbers descended 3600 feet to a camp that was at
an altitude of 12,035 feet. At what altitude did they start?
INDEPENDENT PRACTICE
See Example 1
Determine which value of x is a solution for each equation.
10. x 14 8; x 6, 22, or 32
See Example 2
See Example 3
11. x 23 55; x 15, 28, or 32
Solve.
12. 9 w (8)
13. m 11 33
14. 4 t 16
15. z (22) 96
16. 102 p (130)
17. 27 h (8)
18. Olivia owns 43 CDs. This is 15 more CDs than Angela owns. How many
CDs does Angela own?
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP3.
Solve. Check your answer.
19. 7 t 12
20. h 21 52
21. 15 m (9)
22. m 5 10
23. h 8 11
24. 6 t 14
25. 1785 t (836)
26. m 35 172
27. x 29 81
28. p 8 23
29. n (14) 31
30. 20 8 w
31. 8 t 130
32. 57 c 28
33. 987 w 797
34. Social Studies In 1990, the population of Cheyenne, Wyoming, was
73,142. By 2000, the population had increased to 81,607. Write and solve
an equation to find n, the increase in Cheyenne’s population from 1990
to 2000.
35. Astronomy Mercury’s surface temperature has a range of 600°C. This
range is the broadest of any planet in the solar system. Given that the
lowest temperature on Mercury’s surface is 173°C, write and solve an
equation to find the highest temperature.
1-7 Solving Equations by Adding or Subtracting
35
Determine which value of the variable is a solution of the equation.
36. d 4 24; d 6, 20, or 28
37. k (13) 27; k 40, 45, or 50
38. d 17 36; d 19, 17, or 19
39. k 3 4; k 1, 7, or 17
40. 12 14 s; s 20, 26, or 32
41. 32 27 g; g 58, 25, 59
42. Science An ion is a charged particle. Each proton in an
ion has a charge of 1 and each electron has a charge
of 1. The ion charge is the electron charge plus the
proton charge. Write and solve an equation to find the
electron charge for each ion.
Hydrogen sulfate ion (HSO4)
Name of Ion
Proton Charge
Electron Charge
Ion Charge
Aluminum ion (Al3)
13
3
Hydroxide ion (OH)
9
1
Oxide ion (O2)
8
2
Sodium ion (Na)
11
1
43. What’s the Error? A student evaluated the expression 7 (3) and
came up with the answer 10. What did the student do wrong?
44. Write About It Write a set of rules to use when solving addition and
subtraction equations.
45. Challenge Explain how you could solve for h in the equation 14 h 8
using algebra. Then find the value of h.
NS1.2, Prep for
AF4.0
46. Multiple Choice Which value of x is the solution of the equation x (5) 8?
A
x3
B
x 11
C
x 13
D
x 15
47. Multiple Choice Len bought a pair of $12 flip-flops and a shirt. He paid $30
in all. Which equation can you use to find the price p he paid for the shirt?
A
12 p 30
B
12 p 30
C
30 p 12
D
p 12 30
48. Gridded Response What value of x is the solution of the equation
x 23 19?
Add. (Lesson 1-4)
49. 5 (9)
50. 16 (22)
51. 64 51
52. 82 (75)
38
55. 19
56. 8(13)
Multiply or divide. (Lesson 1-6)
53. 7(8)
36
54. 63 (7)
Chapter 1 Principles of Algebra
1-8
California
Standards
Preparation for
AF4.0
Students solve simple linear
equations and inequalities over the
rational numbers.
Also covered: AF1.3
Solving Equations by
Multiplying or Dividing
Why learn this? You can write a multiplication
equation to solve problems about person-hours.
(See Example 3.)
Just as with addition and subtraction equations, you can use an
identity property to solve multiplication and division equations.
IDENTITY PROPERTY OF MULTIPLICATION
Vocabulary
Words
Identity Property of
Multiplication
Division Property of
Equality
Numbers
The product of any number
and one is that number.
515
Algebra
2 1 2
a1a
You can use the properties of equality along with the Identity Property
of Multiplication to solve multiplication and division equations.
DIVISION PROPERTY OF EQUALITY
EXAMPLE
1
Words
Numbers
You can divide both sides
of an equation by the same
nonzero number, and the
statement will still be true.
4 3 12
4 3
12
2
2
12
6
2
Algebra
If x y and z 0,
x
y
then z z .
Solving Equations Using Division
Solve.
8x 32
8x 32
Since x is multiplied by 8, divide both sides
by 8 to undo the multiplication.
8x
32
8
8
1x 4
x4
Identity Property of Multiplication: 1 x x
7y 91
7y 91
7y
91
7
7
1y 13
y 13
Since y is multiplied by 7, divide both sides by
7 to undo the multiplication.
Identity Property of Multiplication: 1
yy
1-8 Solving Equations by Multiplying or Dividing
37
MULTIPLICATION PROPERTY OF EQUALITY
EXAMPLE
2
Words
Numbers
Algebra
You can multiply both sides of an
equation by the same number,
and the statement will still be true.
236
42346
8 3 24
If x y, then
zx zy.
Solving Equations Using Multiplication
h
6.
Solve 3
h
6
3
h
3
3 3
6
Since h is divided by 3, multiply both sides
by 3 to undo the division.
1h 18
h 18
EXAMPLE
A person-hour is a
unit of work. It
represents the
amount of work
that can be
completed by one
person in one hour.
3
Identity Property of Multiplication: 1
hh
Sports Application
It takes 96 person-hours to prepare a baseball stadium for a
playoff game. The stadium’s manager assigns 6 people to the job.
Each person works the same number of hours. How many hours
does each person work?
Let x represent the number of hours each person works.
number of
people
6
number of hours
total number of
each person works person-hours
x
96
6x 96
Write the equation.
6x
96
6
6
Since x is multiplied by 6, divide both sides by 6
to undo the multiplication.
1x 16
Identity Property of Multiplication: 1 x x
x 16
Each person works for 16 hours.
Think and Discuss
1. Explain the steps you would use to solve 2k.5 6.
2. Tell how the Multiplication and Division Properties of Equality are
similar to the Addition and Subtraction Properties of Equality.
38
Chapter 1 Principles of Algebra
1-8
California
Standards Practice
AF1.1, Preparation for
AF4.0
Exercises
KEYWORD: MT8CA 1-8
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
See Example 2
See Example 3
Solve.
1. 4x 28
2. 3y 42
3. 12q 24
4. 25m 125
l
4
5. 15
h
6. 1
3
9
m
1
7. 6
t
8. 1
52
3
9. Business It takes 270 person-days to prepare one issue of a magazine
for publication. Each week, it takes 5 days to prepare an issue of the
magazine. Assuming each person works the same number of days, how
many people work on the magazine?
INDEPENDENT PRACTICE
See Example 1
See Example 2
Solve.
10. 3d 57
11. 7x 105
12. 4g 40
14. 8p 88
15. 17n 34
16. 212b 424 17. 41u 164
n
18. 9 63
h
2
19. 27
a
20. 6 102
21. 8 12
d
23. 7 23
t
24. 5 60
3
25. 84
y
11
22. 9
See Example 3
13. 16y 112
j
p
26. A kilowatt-hour is one kilowatt of power that is supplied for one hour.
The lighting for a 3-hour concert uses a total of 210 kilowatt-hours of
electricity. How many kilowatts of electricity do the lights require?
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP3.
Solve.
27. 2x 14
28. 4y 80
29. 6y 12
30. 9m 9
k
31. 8 7
x
32. 5 121
b
33. 6 12
n
34. 1
1
5
35. 3x 51
36. 15g 75
37. r 92 115
38. 23 x 36
b
12
39. 4
m
40. 2
24
4
41. a 31 16
42. y 25 5
43. Nutrition One serving of milk contains 8 grams of protein, and one
serving of steak contains 32 grams of protein. Write and solve an equation
to find the number of servings of milk n needed to get the same amount
of protein as there is in one serving of steak.
x
11 be greater than 11 or less than
44. Reasoning Will the solution of 5
11? Explain how you know.
45. Multi-Step Joy earns $8 per hour at an after-school job. Each month
she earns $128. How many hours does she work each month? After six
months, she gets a $2 per hour raise. How much money does she earn
per month now?
1-8 Solving Equations by Multiplying or Dividing
39
Translate each sentence into an equation. Then solve the equation.
46. The quotient of a number d and 4 is 3.
47. The sum of 7 and a number n is 15.
48. The product of a number b and 5 is 250.
49. Twelve is the difference of a number q and 8.
50. Zero is the product of 38 and a number t.
51. Recreation While on vacation, Milo drove his car a total of 370 miles.
This was 5 times as many miles as he drives in a normal week. How many
miles does Milo drive in a normal week?
52. Reasoning In the equation ax 12, a is an integer. Is it possible to choose a
value of a so that the solution of the equation is x 0? Why or why not?
53. Music Jason wants to buy the
trumpet advertised in the classified
ads. He has saved $156. Using the
information from the ad, write and
solve an equation to find how much
more money he needs to buy the
trumpet.
54. Write a Problem Write a problem
that can be solved by using the
equation 15x 75. What is the
solution of the problem?
m
55. Write About It Explain how to estimate the solution of 97.
8
56. Challenge During a winter storm, the temperature dropped 3°F every
hour. The overall change in the temperature was 18°F. Write and solve an
equation to determine how long the storm lasted.
NS1.2, Prep for
AF4.0
57. Multiple Choice Solve the equation 7x 42.
A
x 49
B
x 35
C
x 6
D
x6
58. Gridded Response On a game show, Paul missed q questions, each
worth 100 points. Paul received a total of 900 points. How many
questions did he miss?
Subtract. (Lesson 1-5)
59. 8 8
60. 3 (7)
61. 10 2
62 11 (9)
65. 17 x 9
66. 19 x 11
Solve each equation. (Lesson 1-7)
63. 4 x 13
40
64. x 4 9
Chapter 1 Principles of Algebra
Model Two-Step Equations
1-9
Use with Lesson 1-9
KEYWORD: MT8CA Lab1
KEY
REMEMBER
= 1
= 1
= variable
=0
California
Standards
AF4.1 Solve two-step
• You can perform the same operation with
the same numbers on both sides of an
equation without changing the value of
the equation.
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.
You can use algebra tiles to model and solve two-step equations.
To solve a two-step equation, you use two different operations.
Activity
1 Use algebra tiles to model
and solve 3s 4 10.
3s
4
10
Two steps are needed to solve this equation.
Step 1: Remove 4 yellow tiles from each side.
3s
6
Step 2: Divide each side into 3 equal groups.
s
2
Substitute to check:
3s 4 10
?
3(2) 4 10
?
6 4 10
?
10 10 ✔
Lab continued on next page.
1-9 Hands-On Lab
41
2 Use algebra tiles to model
and solve 2r 4 6.
2r
Step 1: Since 4 is being added to 2r,
add 4 red tiles to both sides and
remove the zero pairs on the
left side.
4
6
Step 2: Divide each side into
2 equal groups.
Add 4 to both sides.
r
2r
5
10
Substitute to check:
2r 4 6
?
2(5) 4 6
?
10 4 6
?
6 6 ✔
Think and Discuss
1. Why can you add zero pairs to one side of an equation without
having to add them to the other side as well?
2. Show how you could have modeled to check your solution for
each equation.
Try This
Use algebra tiles to model and solve each of the following equations.
1. 2x 3 5
2. 4p 3 9
3. 5r 6 11
4. 3n 5 4
5. 6b 8 2
6. 2a 2 6
7. 4m 4 4
8. 7h 8 41
9. Gerry walked dogs five days a week and got paid the same amount
each day. One week his boss added on a $15 bonus. That week
Gerry earned $90. What was his daily salary?
42
Chapter 1 Principles of Algebra
1-9
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.
Also covered: AF1.1
Solving Two-Step Equations
Who uses this? Two-step equations can help pet
owners decide how much pet food they should purchase.
(See Example 4.)
Two-step equations contain two operations. For example, the
equation 6x 2 10 contains multiplication and subtraction.
Multiplication
6x 2 10
Subtraction
EXAMPLE
1
Translating Sentences into Two-Step Equations
Translate each sentence into an equation.
3 more than the product of 5 and a number y is 17.
3 more than the product of 5 and a number y is 17.
5 y 3 17
5y 3 17
The quotient of a number n and 4, minus 8, is 42.
The quotient of a number n and 4, minus 8, is 42.
[n (4)] 8 42
n
8 42
4
Math
Builders
For more on solving
two-step equations, see
the Step-by-Step
Solution Builder on
page MB4.
Equations that contain two operations require two steps to solve.
Identify the operations and the order in which they are applied to the
variable. Then use inverse operations to undo them in the reverse order.
Operations in the Equation
To Solve
1 First x is multiplied by 6.
1 Add 2 to both sides of the equation.
2 Then 2 is subtracted.
2 Then divide both sides by 6.
1-9 Solving Two-Step Equations
43
EXAMPLE
2
Solving Two-Step Equations Using Division
Solve.
2x 1 7
Step 1: 2x 1 7
11
2x
8
Reverse the order of
operations when
solving equations
that have more than
one operation.
Note that x is multiplied by 2. Then 1 is
added. Work backward: Since 1 is
added to 2x, subtract 1 from both sides.
2x
8
2
2
Step 2:
Since x is multiplied by 2, divide both
sides by 2 to undo the multiplication.
x 4
Check
2x 1 7
?
2(4) 1 7
8 1 7
?
7 7 ✔
To check your solution, substitute 4
for x in the original equation.
4 is a solution.
18 7n 3
18 7n 3
3
3
21 7n
Since 3 is subtracted from 7n, add 3
to both sides to undo the subtraction.
21
7n
7
7
Since n is multiplied by 7, divide both
sides by 7 to undo the multiplication.
3 n
EXAMPLE
3
Solving Two-Step Equations Using Multiplication
Solve.
k
6
5 13
k
6
5 13
Step 1:
5
5
k
6
k
6
8
(6) (6)8
Step 2:
k 48
Note that k is divided by 6. Then 5 is added.
k
Work backward: Since 5 is added to ,
6
subtract 5 from both sides.
Since k is divided by 6, multiply both
sides by 6 to undo the division.
v
1 9 10
v
1 9 10
10 v
11 9
10
v
(9)11 (9)
9
99 v
44
Chapter 1 Principles of Algebra
v
Since 10 is subtracted from , add 10
9
to both sides to undo the subtraction.
Since v is divided by 9, multiply both
sides by 9 to undo the division.
EXAMPLE 4
Consumer Math Application
Tyrell is ordering cat food online. Shipping is $12 no matter how
many cases he orders. Each case of cat food costs $16. Tyrell has
$130 to spend. How many cases of cat food can he order?
Let c represent the number of cases that Tyrell can order. That means
Tyrell can spend $16c plus the cost of shipping.
Reasoning
cost of shipping cost of cat food total cost
$12
16c
12 16c 130
12
12
16c 118
$130
Subtract 12 from both sides.
16c
118
16
16
Divide both sides by 16.
c 7.375
Since you cannot order part of a case, Tyrell can order 7 cases.
Think and Discuss
1. Explain how you decide which operation to use first when solving
a two-step equation.
2. Describe the steps you would follow to solve 5 7x 16.
1-9
Exercises
California
Standards Practice
AF1.1,
AF4.1
KEYWORD: MT8CA 1-9
KEYWORD: MT8CA Parent
GUIDED PRACTICE
See Example 1
Translate each sentence into an equation.
1. 6 more than 9 times a number x is 12.
2. The quotient of a number t and 7, plus 2, is 4.
See Example 2
See Example 3
Solve.
3. 4x 7 15
4. 8 5s 53
5. 3n 18 6
6. 12 5p 2
7. 6 2y 4
8. 6w 8 20
Solve.
m
3
9. 4 12
x
12. 13 11
3
x
7
10. 7 9
w
11. 2 5
r
13. 6 5 8
14. 8 15 6
z
2
1-9 Solving Two-Step Equations
45
See Example 4
15. Consumer Math A Web site that rents DVDs charges a one-time fee of $7 to
become a member of the site. It costs $3 to rent each DVD. Diana joined the
site and spent a total of $61. How many DVDs did she rent?
INDEPENDENT PRACTICE
See Example 1
Translate each sentence into an equation.
16. 11 less than the quotient of a number b and 12 is 5.
17. 18 more than the product of 4 and a number z is 54.
18. The difference of twice a number n and 9 is 17.
See Example 2
See Example 3
Solve.
19. 5y 16 21
20. 14 2x 16
21. 6 5t 24
22. 10w 3 53
23. 7q 8 27
24. 12 5m 13
25. 9 6g 3
26. 16y 41 7
27. 3z 11 38
Solve.
w
5
x
31. 17 12
8
t
34. 8 5
7
28. 2 9
See Example 4
r
2
y
29. 10 8
30. 4 9 4
w
32. 5 16
33. 3 8
4
g
82
35. 3
n
5
x
36. 2 3
8
37. Bob’s Auto Shop charges $375 to replace the timing belt in a car. The belt costs
$50 and the labor to install the belt costs $65 per hour. How many hours does
it take to install the timing belt?
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP3.
Translate each equation into words. Then solve the equation.
x
38. 5 12
6
w
2
39. 4y 9 3
40. 7 4
42. 6 4p 2
43. 7j 12 9
45. 5 2z 5
r
46. 3 8
Solve and check.
m
3
x
44. 4 1
3
41. 11 4
4
Translate each sentence into an equation. Then solve the equation.
47. Three less than the quotient of a number x and 9 is 8.
48. Thirteen plus the product of 15 and a number a is 58.
49. If 5 is decreased by 3 times a number m, the result is 4.
50. Hobbies Mike has 60 baseball cards in his collection. Each week he buys a
pack of cards to add to the collection. Each pack contains the same number of
cards. After 12 weeks, Mike has 156 cards. Write and solve an equation to find
the number of cards in a pack.
46
Chapter 1 Principles of Algebra
Social Studies
In 1956, during President Eisenhower’s term,
construction began on the United States interstate
highway system. The original plan was for 42,000
miles of highways to be completed within 16 years.
It actually took 37 years to complete. The last part,
Interstate 105 in Los Angeles, was completed
in 1993.
51. Write and solve an equation to show how many
miles m needed to be completed per year for
42,000 miles of highways to be built in 16 years.
52. Interstate 35 runs north and south from Laredo,
Texas, to Duluth, Minnesota, covering 1568 miles.
There are 505 miles of I-35 in Texas and 262 miles in Minnesota.
Write and solve an equation to find m, the number of miles
of I-35 that are not in either state.
53. A portion of I-476 in Pennsylvania, known as the Blue Route,
is about 22 miles long. The length of the Blue Route is about
one-sixth the total length of I-476. Write and solve an equation
to calculate the length of I-476 in miles m.
54.
Challenge Interstate 80 extends from California to New
Jersey. At right are the number of miles of Interstate 80 in each
state the highway passes through.
a. __?__ has 134 more miles than __?__.
b. __?__ has 174 fewer miles than __?__.
Number of I-80 Miles
State
Miles
California
Nevada
Utah
Wyoming
Nebraska
Iowa
Illinois
Indiana
Ohio
Pennsylvania
New Jersey
195
410
197
401
455
301
163
167
236
314
68
AF4.1
t
5
55. Multiple Choice Solve the equation 4 8.
A
t 60
t 40
B
C
t 20
D
t 10
D
12 15
56. Multiple Choice For which equation is m 3 a solution?
A
5m 6 21
m
1 1
3
B
C
2 3m 11
m
3
Multiply or divide. (Lesson 1-6)
57. 4(11)
12
58. 59. 6(5)
42
60. 62. 16m 48
p
63. 21
64. 12g 108
3
7
Solve. (Lesson 1-8)
y
61. 9
8
7
1-9 Solving Two-Step Equations
47
Quiz for Lessons 1-7 Through 1-9
1-7
Solving Equations by Adding or Subtracting
Solve.
1. p 12 5
2. w (9) 14
3. t (14) 8
4. 23 k 5
5. 52 p 17
6. y (6) 74
7. The approximate surface temperature of Pluto, the coldest planet, is
391°F. This is approximately 1255 degrees cooler than the approximate
surface temperature of Venus, the hottest planet. What is the approximate
surface temperature of Venus?
1-8
Solving Equations by Multiplying or Dividing
Solve.
x
8. 6 48
9. 3x 21
10. 14y 84
y
11. 1
72
2
r
d
3
14. 1
10
15. 8y 96
13. 7
2
16. Ahmed’s baseball card collection consists of 228 cards. This is 4 times as
many cards as Ming has. How many baseball cards are in Ming’s collection?
12. 5p 75
17. It takes 72 person-hours to clean an office building each night. A crew of
12 people cleans the building and each person works the same number of
hours. How many hours does each person work?
1-9
Solving Two-Step Equations
Solve.
18. 3x 1 8
k
4
21. 6 14
24. 41 2 13n
y
19. 2 5 8
20. 7n 4 24
22. 13 9 24
23. 27 5x 2
y
x
25. 3 3
7
k
6
26. 15 5 27. Richard is collecting bottle caps for a school fundraiser. He has collected
335 caps toward his goal of 500 caps. If he collects 15 bottle caps per week,
how many more weeks will it take before he reaches his goal?
28. One week Ella worked 36 hours as a prep chef. Her earnings for the week
totaled $210, which included $30 in tips that she made as a waitress. How
much money does Ella make per hour?
48
Chapter 1 Principles of Algebra
Have a Ball
A physical education class is
playing a variation of basketball. When a team
makes a basket from inside the three-point line, the
team gets a “Climb” (C), or 2 points. When a team
makes a basket from outside the three-point line,
the other team gets a “Slide” (S), or 1 point.
1. During the first 20 minutes of the game, a team gets
the following: C, C, S, S, C, S, C, C, and S. Simplify the
expression 2 2 (1) (1) 2 (1) 2 2 (1) to determine the team’s score.
Game Results
Team 1
Team 2
C
C
2. The points scored by two teams during a game are
shown in the table. Which team won the game? What
was the difference in the teams’ scores?
S
C
S
C
S
S
3. Diego’s team gets 3 Climbs and 2 Slides, but not
necessarily in that order. Find his team’s score by
substituting S 1 and C 2 in the expression
3C 2S.
S
S
C
S
C
S
C
S
4. After four consecutive baskets are made, Leann’s team’s
score is 8. After the next basket is made, the team’s score is
6. Write and solve an equation for the last made basket.
5. Daryl’s team finishes the game with a score of 12. If his team
scored 9 times, how many Climbs did the team get?
6. Is it possible to finish with a score of 2 after five
baskets are made? Explain your reasoning.
Concept Connection
49
Math Magic
You can guess what your friends are thinking by learning to
“operate” your way into their minds! For example, try this
math magic trick.
Think of a number. Multiply the number by 8, divide by
2, add 5, and then subtract 4 times the original number.
No matter what number you choose, the answer will always
be 5. Try another number and see. You can use what you
know about variables to prove it. Here’s how:
What you say:
What the person thinks:
What the math is:
Step 1:
Pick any number.
6 (for example)
n
Step 2:
Multiply by 8.
8(6) 48
8n
Step 3:
Divide by 2.
48 2 24
8n 2 4n
Step 4:
Add 5.
24 5 29
4n 5
Step 5:
Subtract 4 times the
original number.
29 4(6) 29 24 5
4n 5 4n 5
Invent your own math magic trick that has at least five steps.
Show an example using numbers and variables. Try it on a friend!
Crazy
Crzzy Cubes
This game, called The Great Tantalizer around 1900, was
reintroduced in the 1960s as “Instant Insanity™.” Make four
cubes with paper and tape, numbering each side as shown.
1
4
4
2
4
2
3
2
1
3
4
2
2
3
3
1
2
2
4
3
4
3
1
1
The goal is to line up the cubes so that 1, 2, 3, and 4 can be
seen along the top, bottom, front, and back of the row of cubes.
They can be in any order, and the numbers do not have to be
right-side up.
50
Chapter 1 Principles of Algebra
Materials
• sheet of
decorative paper
(812 by 11 in.)
• ruler
• pencil
• scissors
• glue
• markers
PROJECT
Note-Taking
Taking Shape
A
3 _58 in.
3 _58 in.
Make this notebook to help you organize examples of
algebraic expressions.
Directions
1 Hold the sheet of paper horizontally. Make two vertical
lines 358 inches from each end of the sheet. Figure A
2 Fold the sheet in half lengthwise. Then cut it in half by
cutting along the fold. Figure A
B
A
3 On one half of the sheet, cut out rectangles A and B. On
the other half, cut out rectangles C and D. Figure B
Rectangle A: 34 in. by 358 in.
Rectangle B: 112 in. by 358 in.
Rectangle C: 214 in. by 358 in.
Rectangle D: 3 in. by 358 in.
4 Place the piece with the taller rectangular panels on
top of the piece with the shorter rectangular panels.
Glue the middle sections of the two pieces together.
Figure C
5 Fold the four panels into the center, starting with the
tallest panel and working your way down to the shortest.
Taking Note of the Math
Write “Addition,” “Subtraction,” “Multiplication,”
and “Division” on the tabs at the top of each
panel. Use the space below the name of each
operation to list examples of verbal, numerical,
and algebraic expressions.
B
C
C
D
Vocabulary
absolute value . . . . . . . . . . 15
Addition Property
of Equality . . . . . . . . . . . . . 33
equation . . . . . . . . . . . . . . . 32
evaluate . . . . . . . . . . . . . . . . . 6
integer . . . . . . . . . . . . . . . . . 14
inverse operations . . . . . . 32
expression . . . . . . . . . . . . . . . 6
additive inverse . . . . . . . . . 8
algebraic expression . . . . . 6
Identity Property
of Addition . . . . . . . . . . . . . 32
numerical expression . . . . 6
opposite . . . . . . . . . . . . . . . . 14
Division Property
of Equality . . . . . . . . . . . . . 37
Identity Property
of Multiplication . . . . . . . 37
Subtraction Property
of Equality . . . . . . . . . . . . . 33
variable . . . . . . . . . . . . . . . . . 6
Complete the sentences below with vocabulary words from the list above.
1. A(n) ___?___ is a statement that two expressions have the same value.
2. ___?___ is another word for “additive inverse.”
3. The ___?___ of 3 is 3.
1-1 Evaluating Algebraic Expressions
EXAMPLE
■
EXERCISES
Evaluate 4x 9y for x 2 and y 5.
4x 9y
4(2) 9(5)
8 45
53
Evaluate each expression.
4. 9a 7b for a 7 and b 12
5. 17m 3n for m 10 and n 6
Substitute 2 for x and 5 for y.
Multiply.
Add.
6. 1.5r 19s for r 8 and s 14
1-2 Writing Algebraic Expressions
Write an algebraic expression for the
word phrase “2 less than a number n.”
n2
■
Write as subtraction.
Write a word phrase for 25 13t.
25 plus the product of 13 and t
Write an algebraic expression for each phrase.
7. twice the sum of k and 4
8. 5 more than the product of 4 and t
Write a word phrase for each algebraic
expression.
9. 5b 10
10
11. r 12
52
AF1.1
(pp. 10–13)
EXERCISES
EXAMPLE
■
AF1.2, AF1.4
(pp. 6–9)
Chapter 1 Principles of Algebra
10. 32 23s
y
12. 16 8
■
1-3 Integers and Absolute Value
(pp. 14–17)
EXAMPLE
EXERCISES
Simplify each expression.
⏐9⏐ ⏐3⏐
93
6
13. ⏐7 6⏐
15. ⏐15⏐ ⏐19⏐
14. ⏐8⏐ ⏐7⏐
16. ⏐14 7⏐
17. ⏐16 2⏐
18. ⏐7⏐ ⏐8⏐
⏐9⏐ 9 and ⏐3⏐ 3
Subtract.
EXERCISES
Add.
Add.
8 2
6
Find the difference of⏐8⏐ and⏐2⏐.
8 2; use the sign of the 8.
1-5 Subtracting Integers
20. 3 (9)
21. 4 (7)
22. 4 (3)
Evaluate.
24. k 11 for k 3
25. 6 m for m 2
NS1.2,
(pp. 22–25)
EXAMPLE
■
19. 6 4
23. 11 (5) (8)
Evaluate.
4 a for a 7
Substitute.
4 (7)
11
Same sign
■
NS1.2, AF1.3
(pp. 18–21)
EXAMPLE
■
NS2.5
EXERCISES
Subtract.
Subtract.
3 (5)
3 5
2
26. 7 9
27. 8 (9)
28. 2 (5)
29. 13 (2)
30. 5 17
31. 16 20
Add the opposite of 5.
5 3; use the sign of the 5.
Evaluate.
9 d for d 2
Substitute.
9 2
9 (2)
Add the opposite of 2.
11
Same sign
Evaluate.
32. 9 h for h 7
33. 12 z for z 17
1-6 Multiplying and Dividing Integers
EXAMPLE
Multiply or divide.
4(9)
36
The signs are different.
The answer is negative.
■ 33
11
The signs are the same.
3
The answer is positive.
■
NS2.5
Simplify the expression |9| |3|.
1-4 Adding Integers
■
NS1.1,
NS1.2
(pp. 26–29)
EXERCISES
Multiply or divide.
34. 7(–5)
72
35. 4
36. 4(13)
100
37. 4
38. 8(3)(5)
10(5)
39. 25
Study Guide: Review
53
1-7 Solving Equations by Adding or Subtracting
EXAMPLE
EXERCISES
Solve.
■
■
(pp. 32–36)
x 7 12
7 7
x0 5
x 5
y 3 1.5
3 3
y 0 4.5
y 4.5
40. z 9 14
41. t 3 11
Subtract 7 from both sides.
42. 6 k 21
43. x 2 13
Identity Property of Zero
Write an equation and solve.
Add 3 to both sides.
Identity Property of Zero
EXAMPLE
44. A polar bear weighs 715 lb, which is
585 lb less than a sea cow. How much
does the sea cow weigh?
45. The Mojave Desert, at 15,000 mi2, is
11,700 mi2 larger than Death Valley.
What is the area of Death Valley?
(pp. 37–40)
Prep for
EXERCISES
Solve.
AF4.0
Solve and check.
4h 24
4h
24
4
4
1h 6
h6
t
16
4
t
4 4 4 16
46. 7g 56
47. 108 12k
Divide both sides by 4.
48. 0.1p 8
441
1hh
50. 20 2
49. 4 12
z
51. 2
8
4
■ 1t 64
t 64
Multiply both sides by 4.
441
1tt
y
53. Luz will pay a total of $9360 on her car
loan. Her monthly payment is $390. For
how many months is the loan?
(pp. 43–47)
EXAMPLE
EXERCISES
k
3
k
6 3
5
6
6
k
1
3
k
(3) (3)(1)
3
k 3
w
52. The Lewis family drove 235 mi toward
their destination. This was 13 of the total
distance. What was the total distance?
1-9 Solving Two-Step Equations
Solve 6 5.
54
AF4.0
Solve and check.
1-8 Solving Equations by Multiplying or Dividing
■
AF1.4, Prep for
AF4.1
Solve.
Subtract 6 from
both sides.
Multiply both sides
by 3.
Chapter 1 Principles of Algebra
y
6
9 25
54. 15 3 3x
55.
n
56. 1 8
7
n
58. 17 29
2
57. 8x 10 42
59. 5y 12 67
60. A home computer repair service charges
$68 for the first hour and $48 for each
additional hour of service. If a repair bill
is $212, how many additional hours did
the repair take?
Evaluate each expression for the given value of the variable.
1. 16 p for p 12
2. t 7 for t 14
3. 13 x (2) for x 4
4. 8y 27 for y 9
Write an algebraic expression for each word phrase.
5. 15 more than the product of 33 and y
6. 18 less than the quotient of x and 7
7. 4 times the sum of 7 and h
8. 18 divided by the difference of t and 9
Write each set of integers in order from least to greatest.
9. 7, 7, 2, 3, 0, 1
10. 12, 45, 13, 100, 20
11. 120, 7, 54, 41, 7
13. ⏐21⏐ ⏐9⏐
14. ⏐16⏐ ⏐8 3⏐
Simplify each expression.
12. ⏐12 5⏐ ⏐7⏐
Perform the given operations.
15. 9 (12)
16. 11 17
17. 6(22)
18. (20) (4)
19. 42 (5)
20. 18 3
21. 9 (13)
22. 12 (6) (5)
23. 2(21 17)
24. (15 3) (4) 25. (54 6) (1)
26. (16 4) 20
27. The temperature on a winter day increased 37°F. If the beginning
temperature was 9°F, what was the temperature after the increase?
Solve.
28. y 19 9
t
32. 7 12
29. 4z 32
30. 52 p 3
33. 9p 27
q
18
34. 5
w
31. 9
3
g
35. 4 11
36. Acme Sporting Products manufactures 3216 tennis balls a day. Each
container holds 3 balls. How many tennis ball containers are needed daily?
37. It takes 105 person-hours to set up new computers in an office. A crew
sets up the computers and each person works the same number of hours.
It takes 7 hours to complete the job. How many people are on the crew?
Solve.
38. 9x 11 110
y
41. 2 6 13
n
3
39. 8 6x 62
40. 14 4
42. 35 27 4n
43. 40 53 5
y
44. A round-trip ticket to a soccer camp costs $250. If the total cost for the ticket
and 5 days at camp is $715, what is the cost to attend the camp per day?
45. Group rates at Putt-Putt Pfun include charges of $4 per person and entry
to the game room at $10 per group. How many people are in a group that
pays $74 for putt-putt and games?
Chapter 1 Test
55
Multiple Choice: Eliminate Answer Choices
With some multiple-choice test items, you can use logical reasoning or
estimation to eliminate some of the answer choices. Test writers often create
the incorrect choices, called distracters, using common student errors.
Which expression represents “4 times the sum of x and 8”?
A
4 (x 8)
C
4x8
B
4 (x 8)
D
4 (x 8)
Read the question. Then try to eliminate some of the answer choices.
Use logical reasoning.
Times means “to multiply,” and sum means “to add.” You can eliminate
any option without a multiplication symbol and an addition symbol.
You can eliminate B and D.
The sum of x and 8 is being multiplied by 4, so you need to add before
you multiply. Because multiplication comes before addition in the order
of operations, x 8 should be in parentheses. The correct answer is A.
Which value for k is a solution of the equation k 3.5 12?
A
k 8.5
C
k 42
B
k 15.5
D
k 47
Read the question. Then try to eliminate some of the answer choices.
Use estimation.
You can eliminate C and D immediately because they are too large.
Estimate by rounding 3.5 to 4. If x 47, then 47 4 43. This is not
even close to 12. Similarly, if x 42, then 42 4 38, which is also
too large to be correct.
Choice A is called a distracter because it was created using a
common student error, subtracting 3.5 from 12 instead of adding
3.5 to 12. Therefore, A is also incorrect. The correct answer is B.
56
Chapter 1 Principles of Algebra
Even if the answer you calculated is an
answer choice, it may not be the correct
answer. It could be a distracter. Always
check your answers!
Read each test item and answer the
questions that follow.
Item A
The table shows average high
temperatures for Nome, Alaska. Which
answer choice lists the months in order
from coolest to warmest?
Month
Temperature (°C)
Jan
11
Feb
10
Mar
8
Apr
3
May
6
Jun
12
Jul
15
Aug
13
Sep
9
Oct
1
Nov
5
Dec
9
Item B
Which value for p is a solution of the
equation p 5.2 15?
A
p 30.2
C
p 20.2
B
p 9.8
D
p 78
3. Which choices can you eliminate by
using estimation? Explain your
reasoning.
4. What common error does choice C
represent?
Item C
Solve x 20 8 for x.
4
A
x 112
C
x 12
B
x 112
D
x 48
5. Which two choices can you eliminate
by using logic?
6. What common error does choice C
represent?
Item D
Which word phrase can be translated
into the algebraic expression 2x 6?
Jul, Aug, Jun, Sep, May, Oct, Apr, Nov,
Mar, Dec, Feb, Jan
A
six more than twice a number
B
the sum of twice a number and six
B
Jul, Jun, Aug, Jan, Feb, Sep, Dec, Mar,
May, Nov, Apr, Oct
C
twice the difference of a number and
six
C
Jan, Apr, Jun, Jul, Sep, Nov, Feb, Mar,
May, Jul, Sep, Nov
D
six less than twice a number
D
Jan, Feb, Dec, Mar, Nov, Apr, Oct,
May, Sep, Jun, Aug, Jul
A
1. Which two choices can you eliminate
by using logic? Explain your reasoning.
2. What common error does choice A
represent?
7. Can you eliminate any of the choices
immediately by using logic? Explain
your reasoning.
8. Describe how you can determine
the correct answer from the remaining
choices.
Strategies for Success
57
KEYWORD: MT8CA Practice
Cumulative Assessment, Chapter 1
Multiple Choice
1. Which expression has a value of 12
when x 2, y 3, and z 1?
A
3xyz
C
3xz 2y
A
⏐5⏐ 5
B
2x 3y z
D
4xyz 2
B
8 0 8
C
6 3 3 (6)
D
4 (1 7) (4 1) 7
2. The word phrase “10 less than 4 times
a number” can be represented by
which expression?
A
10 4x
C
10 4x
B
4x 10
D
10x 4
3. A copy center prints c copies at a cost
of $0.10 per copy. What is the total cost
of the copies?
A
0.10c
B
0.10 c
0.10
c
c
D 0.10
C
4. If both x and y are negative, which
expression is always negative?
A
xy
C
xy
B
xy
D
xy
5. What is the solution of s (12) 16?
A
s4
C
s 28
B
s8
D
s 192
6. Carlos owes his mother money. His
paycheck is $105. If he pays his mother
the money he owes her, he will have
$63 left. Which equation represents
this situation?
A
x 63 105
B
x 63 105
C
105 x 63
D
58
7. Which equation uses the Identity
Property of Addition?
x 105 63
Chapter 1 Principles of Algebra
8. Which addition equation represents
the number line diagram below?
5 4 3 2 1
0
1
A
4 (2) 2
B
4 (6) 2
C
4 6 10
D
4 (6) 10
2
3
4
5
6
9. Which equation has the solution
x 16?
A
x 16 4
C
2x 32
B
x
32
2
D
x 2 16
10. Solve 6x 6 12 for x.
A
x1
C
x 3
B
x3
D
x 1
11. A scuba diver swimming at a depth of
35 ft below sea level, or 35 ft, dives
another 15 ft deeper to get a closer
look at a fish. Which integer represents
the diver’s new depth?
A
50 ft
C
20 ft
B
20 ft
D
50 ft
The incorrect answer choices in a
multiple-choice test item are called
distracters. They are the results of
common mistakes. Be sure to check
your work!
12. Which set of numbers is in order from
least to greatest?
A
15, 13, 10
C
10, 15, 13
B
13, 10, 15
D
15, 10, 13
13. Which expression is equivalent to
⏐9 (5)⏐?
A
⏐9⏐ ⏐5⏐
C 14
B
⏐9⏐ ⏐5⏐
D
4
14. Dennis spent a total of $340 on flowers
and bushes for his yard. He spent $116
on flowers, and the bushes cost $28
each. To find how many bushes b
Dennis bought, solve the equation
28b 116 340.
A
3 bushes
C
10 bushes
B
8 bushes
D
16 bushes
Gridded Response
15. What is the value of the expression
2xy y when x 3 and y 5?
16. What is the solution of the equation
x 27 16?
17. Evaluate m 11 (3) for m 5.
18. Nora collects 15 magazines every week
for 6 weeks to use for an art project.
After 6 weeks, she still does not have
enough magazines to complete the
project. If Nora needs 20 more
magazines, how many total magazines
does she need?
19. Patricia works twice as many days as
Laura works each month. Laura works
3 more days than Jaime. If Jaime works
10 days each month, how many days
does Patricia work?
Short Response
20. The Hun family plans to visit the Sea
Center. Admission tickets cost $7 each.
a. Write an expression to represent the
cost of t tickets.
b. How much will it cost the Hun
family if they buy 6 tickets? Explain
your answer.
c. Mrs. Hun pays with three $20 bills.
How much change should she get
back? Explain your answer.
21. It costs $0.15 per word to place an
advertisement in the school newspaper.
Let w represent the number of words
in an advertisement and C represent
the cost of the advertisement.
a. Write an equation that relates the
number of words to the cost of the
advertisement.
b. If Bernard has $12.00, how
many words can he use in his
advertisement? Explain your answer.
Extended Response
22. Student Council members at a school
are either elected members or work-on
members. The number of elected
student council members at a school is
ten times the number of officers. The
55-member student council includes
5 work-on members.
a. Write an equation to represent the
number of student council officers x.
Explain the meaning of each term in
the equation.
b. Solve the equation in part a. Check
your answer.
c. At the beginning of the school
year, the school’s enrollment is
1106 students. By the end of the
year, 6 students have transferred
out of the school. Write and solve
an equation to find the number of
students each student council
member represented at the end of
the year.
Cumulative Assessment, Chapter 1
59