Download Introduction to Integers and Algebraic Expressions

2
Introduction to
Integers and
Algebraic
Expressions
Real-World Application
Surface temperatures on Mars vary from 128
C
during polar night to 27
C at the equator during
midday at the closest point in orbit to the sun. Find
the difference between the highest value and the
lowest value in this temperature range.
Source : Mars Institute
Integers and the Number Line
Addition of Integers
Subtraction of Integers
Multiplication of Integers
Division of Integers and
Order of Operations
2.6 Introduction to Algebra
and Expressions
2.7 Like Terms and Perimeter
2.8 Solving Equations
ISBN 0-558-68755-5
This problem appears
as Exercise 73 in
Section 2.3.
2.1
2.2
2.3
2.4
2.5
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Objectives
State the integer that
corresponds to a realworld situation.
Form a true sentence using
or .
Find the absolute value of
any integer.
Find the opposite of any
integer.
2.1
INTEGERS AND THE NUMBER LINE
In this section, we extend the set of whole numbers to form the set of
integers. You have probably already used negative numbers. For example,
the outside temperature could drop to negative five degrees and a credit
card statement could indicate activity of negative forty-eight dollars.
To create the set of integers, we begin with the set of whole numbers, 0, 1,
2, 3, and so on. For each number 1, 2, 3, and so on, we obtain a new number
the same number of units to the left of zero on a number line.
For the number 1, there is the opposite number 1 (negative 1).
For the number 2, there is the opposite number 2 (negative 2).
For the number 3, there is the opposite number 3 (negative 3), and
so on.
The integers consist of the whole numbers and these new numbers. We
illustrate them on a number line as follows.
Integers
To the student:
In the Preface, at the front of
the text, you will find a Student
Organizer card. This pullout
card will help you keep track of
important dates and useful
contact information. You can
also use it to plan time for class,
study, work, and relaxation. By
managing your time wisely, you
will provide yourself the best
possible opportunity to be
successful in this course.
0, neither positive nor negative
Negative integers
6 5 4 3 2 1
Positive integers
0
1
2
3
4
5
6
Opposites
The integers to the left of zero on the number line are called negative integers and those to the right of zero are called positive integers. Zero is neither
positive nor negative and serves as its own opposite.
INTEGERS
The integers: . . . , 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, . . .
Study Tips
LEARN DEFINITIONS
Integers correspond to many real-world problems and situations.
The following examples will help you get ready to translate problem
situations to mathematical language.
EXAMPLE 1 Tell which integer
corresponds to this situation:
Researcher Robert Ballard discovered
the wreck of the Titanic 12,500 ft
below sea level.
Source: Office of Naval Research
12,500 ft below sea level
corresponds to the
integer 12,500.
96
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
Take time to learn the
definitions in each section.
Try to go beyond memorizing
the words of a definition to
understanding the meaning.
Asking yourself questions
about a definition can aid in
understanding. For example,
for the word integers, first
memorize the list of integers,
and then see if you can think
of some numbers that are
not integers.
Integers and the Real World
Tell which integers correspond to this situation: Elaine reversed
the disc in her DVD player 17 min and then advanced it 25 min.
EXAMPLE 2
The integers 17 and 25 correspond to the situation. The integer 17 corresponds to the reversing and 25 corresponds to the advancing.
Tell which integers correspond to
each situation.
1. The halfback gained 8 yd on
first down. The quarterback was
sacked for a 5-yd loss on
second down.
Do Exercises 1–5.
Order on the Number Line
Numbers are written in order on the number line, increasing as we move to the
right. For any two numbers on the line, the one to the left is less than the one to
the right.
Since the symbol means “is less than,” the sentence 5 9 is read “5
is less than 9.” The symbol means “is greater than,” so the sentence 4 8
is read “4 is greater than 8.”
Use either or for
EXAMPLES
9 8 7 6 5 4 3 2 1
3. 9
13
5. 19
0
1
2
3
4
5
6
7
8
9
Sources: National Climatic Data Center,
Asheville, NC, and Storm Phillips,
STORMFAX, INC.
Since 9 is to the left of 2, we have 9 2.
2
4. 7
to form a true sentence.
2. Temperature High and Low.
The highest recorded temperature in Nevada is 125
F (degrees
Fahrenheit) on June 29, 1994, in
Laughlin. The lowest recorded
temperature in Nevada is 50
F
below zero on January 8, 1937,
in San Jacinto.
Since 7 is to the right of 13, we have 7 13.
6 Since 19 is to the left of 6, we have 19 6.
3. Stock Decrease. The stock of
Wendy’s decreased from $41 per
share to $38 per share over a
recent period.
Do Exercises 6–9 on the next page.
Absolute Value
Source: The New York Stock Exchange
From the number line, we see that some integers, like 5 and 5, are the same
distance from zero.
6 5 4 3 2 1
5 units
0
1
2
3
4
5
6
5 units
How far is 5 from 0? How far is 5 from 0? Since distance is never negative (it
is “nonnegative,” that is, either positive or zero), it follows that both 5 and 5
are 5 units from 0.
4. At 10 sec (seconds) before liftoff,
ignition occurs. At 148 sec after
liftoff, the first stage is detached
from the rocket.
ABSOLUTE VALUE
The absolute value of a number is its distance from zero on a
number line. We use the symbol 兩x兩 to represent the absolute value
of a number x.
5. Jacob owes $137 to the
bookstore. Fortunately, he has
$289 in a savings account.
ISBN 0-558-68755-5
Like distance, the absolute value of a number is never negative; it is always
either positive or zero.
Answers on page A-4
97
2.1 Integers and the Number Line
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Use either or for
true sentence.
6. 13
7
8. 13
to form a
7. 12
3
3
9. 4
EXAMPLES
Find the absolute value of each number.
6. 兩3兩 The distance of 3 from 0 is 3, so 兩3兩 3.
7. 兩25兩
The distance of 25 from 0 is 25, so 兩25兩 25.
8. 兩0兩
The distance of 0 from 0 is 0, so 兩0兩 0.
20
To find a number’s absolute value:
Find the absolute value.
10. 兩18兩
1. If a number is positive or zero, use the number itself.
2. If a number is negative, make the number positive.
11. 兩9兩
Do Exercises 10–13.
12. 兩29兩
13. 兩52兩
Answers on page A-4
Opposites
Recall that the set of integers can be represented on a number line. Given a
number on one side of 0, we can get a number on the other side by reflecting
the number across zero. For example, the reflection of 2 is 2. We can read 2
as “negative 2” or “the opposite of 2.”
CALCULATOR
CORNER
Entering Negative
Numbers To enter a negative number on many calculators, we use the 兾 key.
This key gives the opposite of
whatever number is currently
displayed. Thus, to enter 27,
we press 2 7 兾 . Other
calculators have a () key. To
enter 27 on such a calculator,
we simply press () 2 7 .
Be careful not to confuse the
兾 or () key with the
key which is used for
subtraction.
Exercises: Press the
appropriate keys so that your
calculator displays each of the
following numbers.
1. 63
2. 419
3. 2004
Keystrokes will vary by calculator.
1
2
3
4
5
6
NOTATION FOR OPPOSITES
The opposite of a number x is written x (read “the opposite of x”).
EXAMPLE 9
If x is 3, find x.
To find the opposite of x when x is 3, we reflect 3 to the other side
of 0.
6 5 4 3 2 1
0
1
2
3
4
5
6
When x 3, x 共3兲. We substitute 3 for x. We have 共3兲 3. The
opposite of 3 is 3.
EXAMPLE 10
Find x when x is 0.
When we try to reflect 0 “to the other side of 0,” we go nowhere:
x 0 when
x is 0.
The opposite of 0 is 0.
In Examples 9 and 10, the variable was replaced with a number. When this
occurs, we say that we are evaluating the expression.
EXAMPLE 11
Evaluate x when x is 4.
To find the opposite of x when x is 4, we reflect 4 to the other side of 0.
6 5 4 3 2 1
98
0
0
1
2
3
4
5
6
We have 共4兲 4. The opposite of 4 is 4.
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
4. List the keystrokes
needed to compute
共7 4兲.
6 5 4 3 2 1
Do Exercises 14–16.
In each case draw a number line,
if necessary.
A negative number is sometimes said to have a negative sign. A positive
number is said to have a positive sign, even though it rarely is written in.
14. Find x when x is 1.
EXAMPLES
12. 7
Determine the sign of each number.
13. 23
Negative
Positive
15. Find x when x is 2.
Replacing a number with its opposite, or additive inverse, is sometimes
called changing the sign.
EXAMPLES
Change the sign (find the opposite, or additive inverse) of each
16. Evaluate x when x is 0.
number.
14. 6
共6兲 6
15. 10
共10兲 10
16.
共0兲 0
17.
共14兲 14
0
14
Do Exercises 17–20.
Change the sign. (Find the opposite,
or additive inverse.)
17. 4
18. 13
19. 39
20. 0
If x is 2, find 共x兲.
EXAMPLE 18
We replace x with 2:
共x兲
Read “the opposite of the opposite of x”
共2兲.
We copy the expression, replacing x with 2
The opposite of the opposite of 2 is 2, or 共2兲 2.
Evaluate 共x兲 for x 4.
EXAMPLE 19
21. If x is 7, find 共x兲.
We replace x with 4:
共x兲
共共4兲兲
Using an extra set of parentheses to avoid
notation like 4
共
Changing the sign of 4
兲
4
4.
22. If x is 1, find 共x兲.
Changing the sign of 4
Thus, 共共4兲兲 4.
23. Evaluate 共x兲 for x 6.
When we change a number’s sign twice, we return to the
original number.
24. Evaluate 共x兲 for x 2.
Do Exercises 21–24.
It is important not to confuse parentheses with absolute-value symbols.
EXAMPLE 20
Evaluate 兩x兩 for x 2.
We replace x with 2:
25. Find 兩7兩.
兩x兩
兩2兩
Replacing x with 2
2.
The absolute value of 2 is 2.
26. Find 兩39兩.
ISBN 0-558-68755-5
Thus, 兩2兩 2.
Note that 共2兲 2, whereas 兩2兩 2.
Answers on page A-4
Do Exercises 25 and 26.
99
2.1 Integers and the Number Line
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
2.1
EXERCISE SET
MathXL
For Extra Help
MyMathLab
InterAct
Math
Math Tutor
Center
Video
Lectures
on CD
Disc 1
Student’s
Solutions
Manual
Tell which integers correspond to each situation.
1. Pollution Fine. In 2003, The Colonial Pipeline
Company was fined a record $34 million for pollution.
2. Highest and Lowest Temperatures. The highest
temperature ever created on earth was 950,000,000
F.
The lowest temperature ever created was approximately
460
F below zero. 950,000,000; 460
Source: Green Consumer Guide.com
34,000,000
Source: The Guinness Book of Records, 2004
3. The recycling program for Colchester once received
$40 for a ton of office paper. More recently, they’ve
had to pay $15 to get rid of a ton of office paper.
4. The space shuttle stood ready, 3 sec before liftoff. Solid
fuel rockets were released 128 sec after liftoff.
3; 128
40; 15
5. At tax time, Janine received an $820 refund while David
owed $541.
6. Oceanography. At a depth of 2438 meters researchers
found the first hydrothermal vent ever seen by humans.
This depth is approximately 8000 ft below sea level.
820; 541
Source: Office of Naval Research
2438; 8000
8. Geography. The Dead Sea, between Jordan and Israel,
is 1286 ft below sea level; Mt. Rainier in Washington
State is 14,410 ft above sea level. 1286; 14,410
er
ra
n
Sea
ea
nS
ea
7. Geography. Death Valley, California, is 280 ft below sea
level. Mt. Whitney, the highest point in California, has
an elevation of 14,491 ft. 280; 14,491
ed
it
Dead
M
JORDAN
ISRAEL
EGYPT
Use either or for
9. 8
14. 6
19. 13
0 6 9 to form a true sentence.
10. 7
0 11. 9
0 12. 7
0 13. 8
18. 5
15. 6
4 16. 1
7 17. 8
5 20. 5
11 21. 3
4 22. 6
5 8 3 CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
100
Find the absolute value.
23. 兩57兩 57
24. 兩11兩 11
25. 兩0兩 0
26. 兩4兩 4
27. 兩24兩 24
28. 兩36兩 36
29. 兩53兩 53
30. 兩54兩 54
31. 兩8兩 8
32. 兩79兩 79
Find x when x is each of the following.
33. 7 7
34. 6 6
35. 7 7
36. 6 6
37. 0 0
38. 1 1
39. 19 19
40. 50 50
41. 42 42
42. 73 73
Change the sign. (Find the opposite, or additive inverse.)
43. 8 8
44. 7 7
45. 7 7
46. 10 10
47. 29 29
48. 14 14
49. 22 22
50. 0 0
51. 1 1
52. 53 53
Evaluate 共x兲 when x is each of the following.
53. 7 7
54. 8 8
55. 9 9
56. 3 3
57. 17 17
58. 19 19
59. 23 23
60. 0 0
61. 1 1
62. 73 73
63. 85 85
64. 37 37
ISBN 0-558-68755-5
Evaluate 兩x兩 when x is each of the following.
65. 47 47
66. 92 92
67. 345 345
68. 729 729
69. 0 0
70. 1 1
71. 8 8
72. 3 3
101
Exercise Set 2.1
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
73.
DW Does x always represent a negative number? Why
74.
or why not?
DW Explain in your own words why 共x兲 x.
SKILL MAINTENANCE
75. Add: 327 498.
76. Evaluate: 53.
[1.2b] 825
[1.9b] 125
77. Multiply: 209 34. [1.5a] 7106
78. Solve: 300 x 1200. [1.7b] 4
79. Evaluate: 9 2.
80. Multiply: 31 50. [1.5a] 1550
[1.9b] 81
81. Simplify: 5共8 6兲. [1.9c] 10
82. Simplify: 7共9 3兲. [1.9c] 42
SYNTHESIS
83.
DW If a b is true, does it follow that b a is also
true? Why or why not?
84.
DW Does 兩x兩 always represent a positive number? Why
or why not?
85.
On your calculator list the sequence of keystrokes
needed to find the opposite of the sum of 549 and 387.
86.
On your calculator list the sequence of keystrokes
needed to find the opposite of the product of 438
and 97.
5
4
9
3
8
7
兾 . Answers may vary.
4
Use either , , or for
87. 兩5兩
3
8
9
7
兾 . Answers may vary.
to write a true sentence.
兩2兩 88. 兩4兩
兩7兩 89. 兩8兩
兩8兩 Simplify.
90. 兩3兩 3
91. 兩8兩 8
92. 兩2兩 2
93. 兩7兩 7
Solve. Consider only integer replacements.
94. 兩x兩 7 7, 7
95. 兩x兩 2 1, 0, 1
96. Simplify 共x兲, 共共x兲兲, and 共共共x兲兲兲.
97. List these integers in order from least to greatest.
x, x, x
210, 5, 兩6兩, 4, 兩3兩, 100, 0, 27, 72, 10 2
102
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
100, 5, 0, 兩3兩, 4, 兩6兩, 7 2, 10 2, 2 7, 2 10
2.2
ADDITION OF INTEGERS
Objective
Add integers without using
a number line.
Addition
To explain addition of integers, we can use the number line. Once our understanding is developed, we will streamline our approach.
Add, using a number line.
1. 3 共4兲
ADDING INTEGERS
To perform the addition a b, we start at a, and then move
according to b.
a) If b is positive, we move to the right.
b) If b is negative, we move to the left.
c) If b is 0, we stay at a.
EXAMPLE 1
2. 3 共5兲
Add: 2 共5兲.
Move
5 units to the left.
3. 3 7
Start at 2.
4. 5 5
6 5 4 3 2 1
EXAMPLE 2
0
1
2
3
4
5
2 共5兲 3
6
Add: 1 共3兲.
Move
3 units to the left.
For each illustration, write a
corresponding addition sentence.
Start at 1.
5.
5 4 3 2 1
9 8 7 6 5 4 3 2 1
EXAMPLE 3
0
1
2
0
1
2
3
4
5
6 5 4 3 2 1
0
1
2
3
4
2
3
4
5
6
1 共3兲 4
3
Add: 4 9.
Start at 4.
6 5 4 3 2 1
6.
Move
9 units to the right.
0
1
2
3
4
5
6
7
8
9
4 9 5
Do Exercises 1–7.
You may have noticed a pattern in Example 2 and Margin Exercises 2 and
6. When two negative integers are added, the result is negative.
7.
4 3 2 1
0
1
ISBN 0-558-68755-5
ADDING NEGATIVE INTEGERS
To add two negative integers, add their absolute values and change
the sign (making the answer negative).
Answers on page A-4
103
2.2 Addition of Integers
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Add. Do not use a number line
except as a check.
8. 5 共6兲
9. 9 共3兲
10. 20 共14兲
EXAMPLES
4. 5 共7兲 12
Think : Add the absolute values: 5 7 12.
Make the answer negative, 12.
5. 8 共2兲 10
We can visualize the number line without actually
drawing it.
Do Exercises 8–11.
Note that the sum of two positive integers is positive, and the sum of two
negative integers is negative.
When the number 0 is added to any number, that number remains unchanged. For this reason, the number 0 is referred to as the additive identity.
EXAMPLES
11. 11 共11兲
Add.
Add.
6. 4 0 4
7. 0 共9兲 9
8. 17 0 17
Do Exercises 12–14.
Add.
12. 0 共17兲
When we add a positive integer and a negative integer, as in Examples 1
and 3, the sign of the number with the greater absolute value is the sign of
the answer.
ADDING POSITIVE AND
NEGATIVE INTEGERS
13. 49 0
14. 56 0
Add, using a number line only as
a check.
15. 4 6
To add a positive integer and a negative integer, find the difference of
their absolute values.
a) If the negative integer has the greater absolute value, the answer is
negative.
b) If the positive integer has the greater absolute value, the answer is
positive.
EXAMPLES
Add.
9. 3 共5兲 2
Think : The absolute values are 3 and 5. The difference
is 2. Since the negative number has the larger absolute
value, the answer is negative, 2.
10. 11 共8兲 3
Think : The absolute values are 11 and 8. The difference
is 3. The positive number has the larger absolute value,
so the answer is positive, 3.
11. 1 共6兲 5
12. 7 4 3
13. 7 共3兲 4
14. 6 10 4
16. 7 3
17. 5 共7兲
Do Exercises 15–18.
18. 10 共7兲
Sometimes a is referred to as the additive inverse of a. This terminology
is used because adding any number to its additive inverse always results in
the additive identity, 0.
8 8 0,
14 共14兲 0, and 0 0 0.
104
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
Answers on page A-4
ADDING OPPOSITES
Add, using a number line only as
a check.
For any integer a,
19. 5 共5兲
a 共a兲 a a 0.
(The sum of any number and its additive inverse, or opposite, is 0.)
Do Exercises 19–22.
Suppose we wish to add several numbers, positive and negative:
20. 6 6
15 共2兲 7 14 共5兲 共12兲.
Because of the commutative and associative laws for addition, we can group
the positive numbers together and the negative numbers together and add
them separately. Then we add the two results.
EXAMPLE 15
21. 10 10
Add: 15 共2兲 7 14 共5兲 共12兲.
First add the positive numbers:
Then add the negative numbers:
Finally, add the results:
15 7 14 36.
2 共5兲 共12兲 19.
36 共19兲 17.
We can also add in any other order we wish, say, from left to right:
15 共2兲 7 14 共5兲 共12兲 13 7 14 共5兲 共12兲
20 14 共5兲 共12兲
34 共5兲 共12兲
29 共12兲
22. 89 共89兲
17.
Add.
23. 共15兲 共37兲 25 42 共59兲 共14兲
Do Exercises 23–25.
Study Tips
Make the most of
tutoring sessions
by doing what you
can ahead of time
and knowing the
topics with which
you need help.
HELP SESSIONS
Often students find that a tutoring session would be
helpful. The following comments may help you to
make the most of such sessions.
䊏
䊏
Work on the topics before you go to the help or
tutoring session. Do not regard yourself as an
24. 42 共81兲 共28兲 24 18 共31兲
empty cup that the tutor will fill with knowledge.
The primary source of your ability to learn is
within you. When students go to help or tutoring
sessions unprepared, they waste time and, in
many cases, money. Go to class, study the
textbook, work exercises, and mark trouble spots.
Then use the help and tutoring sessions to work
on the trouble spots.
25. 35 17 14 共27兲 31 共12兲
Do not be afraid to ask questions in these
sessions! The more you talk to your tutor, the
ISBN 0-558-68755-5
more the tutor can help you.
䊏
Try being a “tutor” yourself. Explaining a topic
to someone else—a classmate, your instructor—
is often the best way to master it.
Answers on page A-4
105
2.2 Addition of Integers
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
2.2
EXERCISE SET
For Extra Help
MathXL
MyMathLab
InterAct
Math
Math Tutor
Center
Video
Lectures
on CD
Disc 1
Student’s
Solutions
Manual
Add, using a number line.
1. 7 2 5
2. 1 共5兲 4
3. 9 5 4
4. 8 共3兲 5
6. 9 共9兲 0
7. 7 7 0
8. 8 共5兲 13
9. 3 共1兲 4
11. 4 共9兲 5
12. 4 13 9
13. 7 12 5
5. 3 9 6
10. 2 共9兲 11
14. 3 2 1
Add. Use a number line only as a check.
15. 3 共9兲 12
16. 3 共7兲 10
17. 6 共5兲 11
18. 10 共14兲 24
19. 5 共5兲 0
20. 10 共10兲 0
21. 2 2 0
22. 3 3 0
23. 0 6 6
24. 7 0 7
25. 13 共13兲 0
26. 17 17 0
27. 25 0 25
28. 43 0 43
29. 0 共27兲 27
30. 0 共19兲 19
31. 31 31 0
32. 12 共12兲 0
33. 8 0 8
34. 11 0 11
35. 9 共4兲 5
36. 7 8 1
37. 4 共5兲 9
38. 0 共3兲 3
39. 0 共5兲 5
40. 10 共12兲 2
41. 14 共5兲 9
42. 3 14 11
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
106
ISBN 0-558-68755-5
43. 11 8 3
44. 0 共34兲 34
45. 19 19 0
46. 10 3 7
47. 16 6 10
48. 15 5 10
49. 17 共7兲 24
50. 15 共5兲 20
51. 11 共16兲 5
52. 8 14 6
53. 15 共6兲 21
54. 8 8 0
55. 11 共9兲 2
56. 14 共19兲 33
57. 11 17 6
58. 19 共19兲 0
59. 15 共7兲 1 21
60. 23 共5兲 4 22
61. 30 共10兲 5 25
62. 40 共8兲 5 37
63. 23 共9兲 15 17
64. 25 25 共9兲 9
65. 40 共40兲 6 6
66. 63 共18兲 12 57
67. 12 共65兲 共12兲 65
68. 35 共63兲 35 63
69. 24 共37兲 共19兲 共45兲 共35兲 160
70. 75 共14兲 共17兲 共5兲 39
71. 28 共44兲 17 31 共94兲 62
72. 27 共54兲 共32兲 65 46 52
73. 19 73 共23兲 19 共73兲 23
74. 35 共51兲 29 51 共35兲 29
107
Exercise Set 2.2
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
75.
DW Explain in your own words why the sum of two
76.
negative numbers is always negative.
DW A student states “45 is bigger than 21.” What
mistake do you think the student is making?
SKILL MAINTENANCE
Subtract.
77.
[1.3d]
5 4 3
2 1 9
78.
324
6 3 1 4
2 6 8 9
79.
3625
81. Write in expanded notation: 39,417. [1.1b]
2 8 9 1
1 4 0 7
80.
1484
4 3,2 1 3
1 9,8 7 6
23,337
82. Round to the nearest hundred: 746.
[1.4a] 700
3 ten thousands 9 thousands 4 hundreds 1 ten 7 ones
83. Round to the nearest thousand: 32,831. [1.4a] 33,000
84. Multiply: 42 56. [1.5a] 2352
85. Divide: 288 9.
86. Round to the nearest ten: 3496. [1.4a] 3500
[1.6c] 32
SYNTHESIS
87.
DW Without using the words “absolute value,” explain
how to find the sum of a positive number and a
negative number.
88.
DW Why is it important to understand the associative
and commutative laws when adding more than two
integers at a time?
Add.
89. 兩27兩 共兩13兩兲 40
90. 兩32兩 共兩15兩兲 17
91.
3496 共2987兲 6483
92.
497 共3028兲 2531
93.
7846 5978 1868
94.
7623 4839 2784
95. For what numbers x is x positive? All negative
96. For what numbers x is x negative? All positive
Tell whether each sum is positive, negative, or zero.
97. If n is positive and m is negative, then n m is
negative
.
99. If n is negative and m is less than n, then n m is
negative
.
98. If n m and n is negative, then n 共m兲 is
positive
.
100. If n is positive and m is greater than n, then n m is
positive
.
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
108
2.3
SUBTRACTION OF INTEGERS
Objectives
Subtract integers and
simplify combinations of
additions and subtractions.
Subtraction
We now consider subtraction of integers. To find the difference a b, we look
for a number to add to b that gives us a.
Solve applied problems
involving addition and
subtraction of integers.
THE DIFFERENCE a ⴚ b
The difference a b is the number that when added to b gives a.
Subtract.
1. 6 4
For example, 45 17 28 because 28 17 45. Let’s consider an
example in which the answer is a negative number.
EXAMPLE 1
Think: What number can be
added to 4 to get 6?
Subtract: 5 8.
Think: 5 8 is the number that when added to 8 gives 5. What number
can we add to 8 to get 5? The number must be negative. The number is 3:
2. 7 共10兲
Think: What number can be
added to 10 to get 7?
5 8 3.
That is, 5 8 3 because 8 共3兲 5.
Do Exercises 1–3.
3. 7 共2兲
The definition of a b above does not always provide the most efficient
way to subtract. To understand a faster way to subtract, consider finding 5 8
using a number line. We start at 5. Then we move 8 units to the left to do the
subtracting. Note that this is the same as adding the opposite of 8, or 8, to 5.
Move 8 units
to the left.
Think: What number can be
added to 2 to get 7?
Complete the addition and compare
with the subtraction.
Start at 5.
4. 4 6 2;
4 共6兲 6 5 4 3 2 1
0
1
2
3
4
5
6
5 8 3
Look for a pattern in the following table.
SUBTRACTIONS
ADDING AN OPPOSITE
5 8 3
6 4 10
7 共10兲 3
7 共2兲 5
5 共8兲 3
6 共4兲 10
7 10 3
7 2 5
ISBN 0-558-68755-5
Do Exercises 4–7.
5. 3 8 11;
3 共8兲 6. 5 共9兲 4;
5 9 7. 5 共3兲 2;
5 3 Perhaps you have noticed that we can subtract by adding the opposite of
the number being subtracted. This can always be done.
Answers on page A-4
109
2.3 Subtraction of Integers
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Equate each subtraction with a
corresponding addition. Then write
the equation in words.
8. 3 10
SUBTRACTING BY ADDING
THE OPPOSITE
To subtract, add the opposite, or additive inverse, of the number
being subtracted:
a b a 共b兲.
9. 13 5
This is the method generally used for quick subtraction of integers.
Equate each subtraction with a corresponding addition. Then
write the equation in words.
EXAMPLES
10. 12 共9兲
2. 12 30;
12 30 12 共30兲
Adding the opposite of 30
Negative twelve minus thirty is negative twelve plus negative thirty.
11. 12 10
3. 20 共17兲;
20 共17兲 20 17
Adding the opposite of 17
Negative twenty minus negative seventeen is negative twenty plus
seventeen.
12. 14 共14兲
Do Exercises 8–12.
Subtract.
13. 7 11
Once the subtraction has been rewritten as addition, we add as in
Section 2.2.
EXAMPLES
Subtract.
4. 2 6 2 共6兲
14. 6 10
The opposite of 6 is 6. We change the
subtraction to addition and add the opposite.
Instead of subtracting 6, we add 6.
4
5. 4 共9兲 4 9
15. 13 8
The opposite of 9 is 9. We change the
subtraction to addition and add the opposite.
Instead of subtracting 9, we add 9.
13
6. 4 8 4 共8兲
16. 7 共9兲
We change the subtraction to addition and
add the opposite.
Instead of subtracting 8, we add 8.
12
7. 10 7 10 共7兲
17. 8 共2兲
We change the subtraction to addition and
add the opposite.
Instead of subtracting 7, we add 7.
3
8. 4 共9兲 4 9
18. 5 共8兲
5
9. 7 共3兲 7 3
Answers on page A-4
To check, note that 5 共9兲 4.
Instead of subtracting 3, we add 3.
Check: 4 共3兲 7.
Do Exercises 13–18.
110
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
4
Instead of subtracting 9, we add 9.
When several additions and subtractions occur together, we can make
them all additions. The commutative law for addition can then be used.
EXAMPLE 10
Simplify.
19. 6 共2兲 共4兲 12 3
Simplify: 3 共5兲 9 4 共6兲.
3 共5兲 9 4 共6兲 3 5 共9兲 4 6
Adding opposites
3 共9兲 5 4 6
Using a
commutative law
20. 9 共6兲 7 9 8 共20兲
12 15
3
Do Exercises 19 and 20.
Applications and Problem Solving
We need addition and subtraction of integers to solve a variety of applied
problems.
21. E-Z Pass. (See Example 11.)
Suppose the Ramones had a
balance of $11 in their account,
accumulated $25 in tolls, and
had to pay $85 in fines and
administrative fees. By how
much would the Ramones
be in debt?
EXAMPLE 11 Toll Roads. The E-Z Pass program allows drivers in the
Northeast to travel certain toll roads without having to stop to pay. Instead,
a transponder attached to the vehicle is scanned as the vehicle rolls through
a toll booth. Recently the Ramones began a trip to New York City with a balance of $12 in their E-Z Pass account. Their trip accumulated $15 in tolls,
and because they overspent their balance, the Ramones had to pay $80 in
fines and administrative fees. By how much were the Ramones in debt as a
result of their travel on the toll roads?
Source: State of New Jersey
We solve by first subtracting the cost of the tolls from the original balance in the account. Then we subtract the cost of the fees and fines from
the new balance in the account:
12 15 12 共15兲
Adding the opposite of 15
3,
and
3 80 3 共80兲
Adding the opposite of 80
83.
The Ramones were $83 in debt as a result of their travel on toll roads.
Do Exercises 21 and 22.
Study Tips
ISBN 0-558-68755-5
It is easy to become
overdependent on
the answer section.
USING THE ANSWER SECTION
When using the answers listed in the back of this
book, try not to “work backward” from the answer.
If you find that you frequently require two or more
attempts to answer an exercise correctly, you
probably need to work more carefully and/or reread
the section preceding the exercise set. Remember that
on quizzes and tests you have only one attempt to
answer each question.
22. Temperature Extremes. In
Churchill, Manitoba, Canada,
the average daily low
temperature in January is
31
C (degrees Celsius). The
average daily low temperature
in Key West, Florida, is 50
warmer. What is the average
daily low temperature in
Key West, Florida?
Answers on page A-4
111
2.3 Subtraction of Integers
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
2.3
EXERCISE SET
For Extra Help
MathXL
MyMathLab
InterAct
Math
Math Tutor
Center
Video
Lectures
on CD
Disc 1
Student’s
Solutions
Manual
Subtract.
1. 2 7 5
2. 3 8 5
3. 0 8 8
4. 0 9 9
5. 7 共4兲 3
6. 6 共8兲 2
7. 11 共11兲 0
8. 6 共6兲 0
9. 13 17 4
10. 14 19 5
11. 20 27 7
12. 30 4 26
13. 9 共4兲 5
14. 7 共9兲 2
15. 40 共40兲 0
16. 9 共9兲 0
17. 7 7 0
18. 9 9 0
19. 7 共7兲 14
20. 4 共4兲 8
21. 8 共3兲 11
22. 7 4 11
23. 6 8 14
24. 6 共10兲 16
25. 3 共9兲 6
26. 14 2 16
27. 1 9 8
28. 2 8 6
29. 6 共5兲 1
30. 4 共3兲 1
31. 8 共10兲 18
32. 5 共6兲 11
33. 0 10 10
34. 0 23 23
35. 5 共2兲 3
36. 3 共1兲 2
37. 7 14 21
38. 9 16 25
39. 0 共5兲 5
40. 0 共1兲 1
41. 8 0 8
42. 9 0 9
43. 7 共5兲 12
44. 7 共4兲 11
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
112
45. 6 25 19
46. 18 63 45
47. 42 26 68
48. 18 63 81
49. 72 9 81
50. 49 3 52
51. 24 共92兲 116
52. 48 共73兲 121
53. 50 共50兲 0
54. 70 共70兲 0
55. 30 共85兲 55
56. 25 共15兲 10
Simplify.
57. 7 共5兲 4 共3兲 19
58. 5 共8兲 3 共7兲 13
59. 31 共28兲 共14兲 17 62
60. 43 共19兲 共21兲 25 22
61. 34 28 共33兲 44 139
62. 39 共88兲 29 共83兲 5
63. 93 共84兲 41 共56兲
64. 84 共99兲 44 共18兲 43
65. 5 共30兲 30 40 共12兲
6
4
66. 14 共50兲 20 共32兲
116
107
67. 132 共21兲 45 共21兲
219
68. 81 共20兲 14 共50兲 53
190
ISBN 0-558-68755-5
Solve.
69. Reading. Before falling asleep, Alicia read from the
top of page 37 to the top of page 62 of her book. How
many pages did she read? 25 pages
70. Writing. During a weekend retreat, James wrote from
the bottom of page 29 to the bottom of page 37 of his
memoirs. How many pages did he write? 8 pages
71. Through exercise, Rod went from 8 lb above his “ideal”
body weight to 9 lb below it. How many pounds did
Rod lose? 17 lb
72. Laura has a charge of $476.89 on her credit card, but
she then returns a sweater that cost $128.95. How
much does she now owe on her credit card? $347.94
113
Exercise Set 2.3
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
73. Surface Temperatures on Mars. Surface temperatures
on Mars vary from 128
C during polar night to 27
C
at the equator during midday at the closest point in
orbit to the sun. Find the difference between the
highest value and the lowest value in this
temperature range. 155
C
74. Carla is completing the production work on a track
that is to appear on her band’s upcoming CD. In
doing so, she resets the digital recorder to 0, advances
the recording 16 sec, and then reverses the recording
25 sec. What reading will the recorder then display?
9
Source: Mars Institute
75. While recording a 60-minute television show, the
reading on Kate’s VCR changes from 21 min to
29 min. How many minutes have been recorded?
Has she recorded the entire show? 50 min; No
76. As a result of coaching, Cedric’s average golf score
improved from 3 over par to 2 under. By how many
strokes did his score change? 5 strokes
77. Temperature Changes. One day the temperature in
Lawrence, Kansas, is 32
at 6:00 A.M. It rises 15
by
noon, but falls 50
by midnight when a cold front
moves in. What is the final temperature? 3
78. Midway through a movie, Lisa resets the counter on
her DVD player to 0. She then reverses the disc 8 min,
and then advances the movie 11 min. What does the
counter now read? 3
79. Profit. Teapots and Treasures lost $5000 in 2004. In
2005, the store made a profit of $8000. How much
more did the store make in 2005 than in 2004?
80. Tallest Mountain. The tallest mountain in the world,
when measured from base to peak, is Mauna Kea
(White Mountain) in Hawaii. From its base 19,684 ft
below sea level in the Hawaiian Trough, it rises
33,480 ft. What is the elevation of the peak?
Source: The Guinness Book of Records
$13,000
114
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
13,796 ft above sea level
81. Offshore Oil. In 1998, the elevation of the world’s
deepwater drilling record was 7718 ft. In 2005, the
deepwater drilling record was 2293 ft deeper. What was
the elevation of the deepwater drilling record in 2005?
Source: www.deepwater.com/FactsandFirsts.cfm
1998
82. Oceanography. The deepest point in the Pacific
Ocean is the Marianas Trench, with a depth of
11,033 m. The deepest point in the Atlantic Ocean
is the Puerto Rico Trench, with a depth of 8648 m.
What is the difference in the elevation of the
two trenches? 2385 m
2005
Pacific
Ocean
ic
Atlant
– 7718 ft
,033
– 11
?
Ocean
48 m
– 86
m
Marianas
Trench
Puerto Rico
Trench
10,011 ft
83. Toll Roads. The Murrays began a trip with $13 in their
E-Z Pass account (see Example 11). They accumulated
$20 in tolls and had to pay $80 in fines and administrative fees. By how much were the Murrays in debt as a
result of their travel on toll roads?
84. Toll Roads. Suppose the Murrays (see Exercise 83)
incurred $25 in tolls and $85 in fines and administrative
fees. By how much would the Murrays be in debt?
97, or $97 in debt
87, or $87 in debt
85.
DW Write a subtraction problem for a classmate to
solve. Design the problem so that the solution is
“Clara ends up $15 in debt.”
86.
DW If a negative number is subtracted from a positive
number, will the result always be positive? Why or
why not?
SKILL MAINTENANCE
Evaluate.
87. 43
[1.9b] 64
88. 68 72 [1.5a] 4896
91. How many 12-oz cans of soda can be filled with 96 oz
of soda? [1.8a] 8 cans
89. 17 [1.9b] 1
90. 143 29
[1.5a] 4147
92. A case of soda contains 24 bottles. If each bottle
contains 12 oz, how many ounces of soda are in
the case? [1.8a] 288 oz
ISBN 0-558-68755-5
Simplify.
93. 5 42 2 7 [1.9c] 35
94. 45 共22 11兲 [1.9c] 3
95. 共9 7兲 共9 7兲 [1.9c] 32
96. 共13 2兲 共13 2兲 [1.9c] 165
115
Exercise Set 2.3
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
SYNTHESIS
97.
DW Explain why the commutative law was used in
Example 10.
98.
DW Is subtraction of integers associative? Why or
why not?
Subtract.
99.
123,907 433,789 309,882
100.
23,011 共60,432兲 83,443
For Exercises 101–106, tell whether each statement is true or false for all integers a and b. If false, show why.
101. a 0 0 a False; 3 0 0 3
102. 0 a a False; 0 3 3
103. If a b, then a b 0. True
104. If a b, then a b 0. True
105. If a b 0, then a and b are opposites. True
106. If a b 0, then a b.
False; 3 3 0 , but 3 3 .
107. If a 54 is 37, find the value of a. 17
108. If x 48 is 15, find the value of x. 33
109. Doreen is a stockbroker. She kept track of the weekly changes in the stock market over a period of 5 weeks. By how many
points (pts) had the market risen or fallen over this time? Up 15 points
WEEK 1
WEEK 2
WEEK 3
WEEK 4
WEEK 5
Down 13 pts
Down 16 pts
Up 36 pts
Down 11 pts
Up 19 pts
110. Blackjack Counting System. The casino game of
blackjack makes use of many card-counting systems
to give players an advantage if the count becomes
negative. One such system is called High –Low, first
developed by Harvey Dubner in 1963. Each card
counts as 1, 0, or 1 as follows:
2, 3, 4, 5, 6
count as 1;
7, 8, 9
count as 0;
10, J, Q, K, A
count as 1.
Source: Patterson, Jerry L. Casino Gambling. New York:
Perigee, 1982
a) Find the total count on the sequence of cards 2
K, A, 2, 4, 5, 10, J, 8, Q, K, 5.
b) Does the player have a winning edge? Yes
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
116
2.4
MULTIPLICATION OF INTEGERS
Objectives
Multiply integers.
Multiplication
Find products of three or
more integers and simplify
powers of integers.
Multiplication of integers is like multiplication of whole numbers. The difference is that we must determine whether the answer is positive or negative.
MULTIPLICATION OF A POSITIVE INTEGER AND A NEGATIVE INTEGER
To see how to multiply a positive integer and a negative integer, consider the
following pattern.
This number decreases
by 1 each time.
4 5 20
3 5 15
2 5 10
5
15
0
05
1 5 5
2 5 10
3 5 15
1. Complete, as in the example.
4 10 40
3 10 30
2 10 1 10 0 10 1 10 2 10 3 10 This number decreases
by 5 each time.
Do Exercise 1.
According to this pattern, it looks as though the product of a negative integer and a positive integer is negative. To confirm this, use repeated addition:
Multiply.
2. 3 6
1 5 5 共1兲 1 共1兲 共1兲 共1兲 共1兲 5
2 5 5 共2兲 2 共2兲 共2兲 共2兲 共2兲 10
3 5 5 共3兲 3 共3兲 共3兲 共3兲 共3兲 15
3. 20 共5兲
MULTIPLYING A POSITIVE AND A NEGATIVE INTEGER
To multiply a positive integer and a negative integer, multiply their
absolute values and make the answer negative.
4. 9共1兲
EXAMPLES
Multiply.
1. 8共5兲 40
2. 50共1兲 50
3. 7 6 42
5. Complete, as in the example.
Do Exercises 2–4.
MULTIPLICATION OF TWO NEGATIVE INTEGERS
How do we multiply two negative integers? Again we look for a pattern.
ISBN 0-558-68755-5
This number decreases
by 1 each time.
Do Exercise 5.
4 (5) 3 (5) 2 (5) 1 (5) 0 (5) 1 (5) 2 (5) 3 (5) 20
15
10
5
0
5
10
15
This number increases
by 5 each time.
3 (10) 30
2 (10) 20
1 (10) 0 (10) 1 (10) 2 (10) 3 (10) Answers on page A-5
117
2.4 Multiplication of Integers
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Multiply.
6. 共3兲 共4兲
According to the pattern, the product of two negative integers is positive.
This leads to the second part of the rule for multiplying integers.
MULTIPLYING TWO
NEGATIVE INTEGERS
7. 9共5兲
To multiply two negative integers, multiply their absolute values. The
answer is positive.
EXAMPLES
Multiply.
4. 共2兲 共4兲 8
8. 共1兲 共6兲
5. 共10兲 共7兲 70
6. 共9兲 共1兲 9
Do Exercises 6–8.
The following is another way to state the rules for multiplication.
Multiply.
9. 0 共5兲
To multiply two integers:
a) Multiply the absolute values.
b) If the signs are the same, the answer is positive.
c) If the signs are different, the answer is negative.
10. 23 0
MULTIPLICATION BY ZERO
No matter how many times 0 is added to itself, the answer is 0. This leads to
the following result.
Answers on page A-5
For any integer a,
a 0 0.
(The product of 0 and any integer is 0.)
Study Tips
SLEEP WELL
Multiply.
7. 19 0 0
8. 0共7兲 0
Do Exercises 9 and 10.
Multiplication of More Than Two Integers
Because of the commutative and the associative laws, to multiply three or
more integers, we can group as we please.
EXAMPLES
Multiply.
9. a) 8 2共3兲 16共3兲
48
b) 8 2共3兲 24 2
118
48
Multiplying the first two numbers
Multiplying the results
Multiplying the negatives
The result is the same as above.
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
Being well rested, alert, and
focused is very important
when studying math. Often,
problems that may seem
confusing to a sleepy person
are easily understood after a
good night’s sleep. Using your
time efficiently is always
important, so you should be
aware that an alert, wideawake student can often
accomplish more in
10 minutes than a sleepy
student can accomplish in
30 minutes.
EXAMPLES
10. 7共1兲 共4兲 共2兲 共7兲8
Multiplying the first two numbers and the
last two numbers
56
11. a) 5 共2兲 共3兲 共6兲 10 18
Multiply.
11. 2 共5兲 共4兲 共3兲
Each pair of negatives gives a
positive product.
180
b) 5 共2兲 共3兲 共6兲 共1兲 10 18 共1兲
Making use of
Example 11(a)
180
12. 共4兲 共5兲 共2兲 共3兲 共1兲
We can see the following pattern in the results of Examples 9–11.
The product of an even number of negative integers is positive.
The product of an odd number of negative integers is negative.
13. 共1兲 共1兲 共2兲 共3兲 共1兲 共1兲
Do Exercises 11–13.
POWERS OF INTEGERS
A positive number raised to any power is positive. When a negative number is
raised to a power, the sign of the result depends upon whether the exponent
is even or odd.
Simplify.
EXAMPLES
Simplify.
2
12. 共7兲 共7兲 共7兲 49
14. 共2兲3
The result is positive.
3
13. 共4兲 共4兲 共4兲 共4兲
16共4兲
64
The result is negative.
4
14. 共3兲 共3兲 共3兲 共3兲 共3兲
15. 共9兲2
99
81
The result is positive.
5
15. 共2兲 共2兲 共2兲 共2兲 共2兲 共2兲
4 4 共2兲
16共2兲
32
The result is negative.
16. 共1兲9
Perhaps you noted the following.
When a negative number is raised to an even exponent, the result is
positive.
When a negative number is raised to an odd exponent, the result is
negative.
17. 25
ISBN 0-558-68755-5
Do Exercises 14–17.
When an integer is multiplied by 1, the result is the opposite of that
integer.
For any integer a,
1 a a
Answers on page A-5
119
2.4 Multiplication of Integers
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
18. Simplify: 52.
EXAMPLE 16
Simplify: 72.
2
Since 7 lacks parentheses, the base is 7, not 7. Thus we regard 72
as 1 72:
72 1 72
1 7 7
The rules for order of operations tell us to
square first.
1 49
49.
Compare Examples 12 and 16 and note that 共7兲2 72. In fact, the
expressions 共7兲2 and 72 are not even read the same way: 共7兲2 is read
“negative seven squared,” whereas 72 is read “the opposite of seven
squared.”
Do Exercises 18–20.
19. Simplify: 共5兲2.
CALCULATOR CORNER
Exponential Notation When using a calculator to calculate
expressions like 共39兲4 or 394, it is important to use the correct
sequence of keystrokes.
Calculators with 兾 key: On some calculators, a 兾 key must be
pressed after a number is entered to make the number negative. For these
calculators, appropriate keystrokes for 共39兲4 are
3
兾
9
xy
4
.
To calculate 394, we must first raise 39 to the power 4. Then the sign of
the result must be changed. This can be done with the keystrokes
3
xy
9
4
兾
or by multiplying 39 by 1:
4
3
2
2
20. Write 共8兲 and 8 in words.
9
xy
4
1
兾
.
Calculators with () key: On some calculators, the () key is pressed
before a number to indicate that the number is negative. This is similar to
the way the expression is written on paper. For these calculators, 共39兲4 is
found by pressing
(
()
3
9
)
⵩
9
⵩
4
4
ENTER and 394 is found by pressing
()
3
ENTER .
You can either experiment or consult a user’s manual if you are unsure of
the proper keystrokes for your calculator.
Exercises: Use a calculator to determine each of the following.
5. 9 6 531,441
2. 共17兲5 1,419,857
6. 76 117,649
3. 共104兲3 1,124,864
7. 65 7776
10
4. 共4兲
1,048,576
8. 39 19,683
Answers on page A-5
120
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
1. 共23兲6 148,035,889
2.4
EXERCISE SET
For Extra Help
MathXL
MyMathLab
InterAct
Math
Math Tutor
Center
Video
Lectures
on CD
Disc 1
Student’s
Solutions
Manual
Multiply.
1. 2 8 16
2. 7 3 21
3. 9 2 18
4. 7 7 49
5. 8 共6兲 48
6. 8 共3兲 24
7. 10 3 30
8. 9 8 72
9. 3共5兲 15
10. 8 共2兲 16
11. 9 共2兲 18
12. 共8兲 共9兲
13. 共6兲 共7兲 42
14. 8 共3兲 24
15. 10共2兲 20
16. 9共8兲
17. 12共10兲 120
18. 15共8兲 120
19. 6共50兲 300
20. 25共8兲
21. 共72兲 共1兲 72
22. 41共3兲 123
23. 共20兲17 340
24. 共1兲43 43
25. 47 0 0
26. 17 0 0
27. 0共14兲 0
28. 0共38兲 0
72
72
200
ISBN 0-558-68755-5
Multiply.
29. 3 共8兲 共1兲 24
30. 共7兲 共4兲 共1兲 28
31. 7共4兲 共3兲5 420
32. 9共2兲 共6兲7 756
33. 2共5兲 共7兲 70
34. 共2兲 共5兲 共3兲 共5兲 150
35. 共5兲 共2兲 共3兲 共1兲 30
36. 6共5兲 共9兲 270
37. 共15兲 共29兲0 8 0
38. 19共7兲 共8兲0 6 0
39. 共7兲 共1兲 共7兲 共6兲 294
40. 共5兲6共4兲5 600
121
Exercise Set 2.4
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Simplify.
41. 共6兲2 36
42. 共8兲2 64
43. 共5兲3 125
44. 共2兲4 16
45. 共10兲4 10,000
46. 共1兲5 1
47. 24 16
48. 共2兲6 64
49. 共3兲5 243
50. 10 4 10,000
51. 共1兲12 1
52. 共1兲13 1
53. 36 729
54. 26 64
55. 43 64
56. 25 32
59. 共9兲10
60. 54
Write each of the following expressions in words.
57. 84
58. 共6兲8
The opposite of eight to
the fourth power
Negative six to the
eighth power
DW Explain in your own words why 共9兲10 is positive.
61.
Negative nine to the
tenth power
62.
The opposite of five to
the fourth power
DW Explain in your own words why 9 10 is negative.
SKILL MAINTENANCE
63. Round 532,451 to the nearest hundred.
[1.4a]
532,500
64. Write standard notation for sixty million. [1.1c]
60,000,000
65. Divide: 2880 36. [1.6c] 80
66. Multiply: 75 34. [1.5a] 2550
67. Simplify: 10 23 6 2. [1.9c] 5
68. Simplify: 2 52 3 23 共3 32 兲.
69. A rectangular rug measures 5 ft by 8 ft. What is the area
of the rug? [1.5c], [1.8a] 40 sq ft
70. How many 12-egg cartons can be filled with 2880 eggs?
[1.8a] 240 cartons
71. A ferry can accommodate 12 cars and 53 cars are
waiting. How many trips will be required to ferry
them all? [1.8a] 5 trips
72. An elevator can hold 16 people and 50 people are
waiting to go up. How many trips will be required to
transport all of them? [1.8a] 4 trips
[1.9c] 48
SYNTHESIS
DW Which number is larger, 共3兲79 or 共5兲79? Why?
73.
74.
DW Describe all conditions for which a x is negative.
Simplify.
75. 共3兲5共1兲379 243
76. 共2兲3 关共1兲29 兴 46 8
79. 兩共2兲5 32兩 共3 7兲2 7
77. 9 4 共9兲4 0
78. 52共1兲29 25
80. 兩12共3兲2 53 62 共5兲2兩 294
81.
472 2209
82.
532 2809
83.
共19兲4 130,321
84.
共23兲4 279,841
85.
共73 86兲3 2197
86.
共49 34兲3 3375
87.
935共238 243兲3
88.
共17兲4共129 133兲5
116,875
85,525,504
90. After diving 95 m below the surface, a diver rises at a
rate of 7 meters per minute for 9 min. What is the diver’s
new elevation? 32 m
91. What must be true of m and n if 关共5兲m 兴 n is to be
(a) negative? (b) positive?
92. What must be true of m and n if mn is to be
(a) positive? (b) zero? (c) negative?
(a) Both m and n must be odd. (b) At least one of m and n
must be even.
122
(a) m and n must have different signs. (b) At least one of m
and n must be zero. (c) m and n must have the same sign.
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
89. Jo wrote seven checks for $13 each. If she had a balance
of $68 in her account, what was her balance after
writing the checks? $23
2.5
DIVISION OF INTEGERS AND
ORDER OF OPERATIONS
Objectives
Divide integers.
We now consider division of integers. Because of the way in which division is defined, its rules are similar to those for multiplication.
Use the rules for order of
operations with integers.
Division of Integers
Divide.
THE QUOTIENT a兾b
1. 6 共2兲
Think : What number
multiplied by 2 gives 6?
a
(or a b, or a兾b) is the number, if there is one, that
b
when multiplied by b gives a.
The quotient
Let’s use the definition to divide integers.
EXAMPLES
Divide, if possible. Check each answer.
1. 14 共7兲 2
2.
32
8
4
3. 21 7 3
2.
15
3
Think : What number
multiplied by 3 gives 15?
Think: What number multiplied by 7 gives 14?
The number is 2. Check: 共2兲 共7兲 14.
Think: What number multiplied by 4 gives 32?
The number is 8. Check: 8共4兲 32.
Think: What number multiplied by 7 gives 21?
The number is 3. Check: 共3兲 7 21.
4.
0
0
5
Think: What number multiplied by 5 gives 0?
The number is 0. Check: 0共5兲 0.
5.
5
is not defined.
0
Think : What number multiplied by 0 gives 5?
There is no such number because the product of 0
and any number is 0.
The rules for determining the sign of a quotient are the same as for
determining the sign of a product. We state them together.
3. 24 8
Think : What number
multiplied by 8 gives 24?
4.
0
4
5.
30
5
6.
45
9
To multiply or divide two integers:
a) Multiply or divide the absolute values.
b) If the signs are the same, the answer is positive.
c) If the signs are different, the answer is negative.
Do Exercises 1–6.
ISBN 0-558-68755-5
Recall that, in general, a b and b a are different numbers. In Example 4, we divided into 0. In Example 5, we attempted to divide by 0. Since any
number times 0 gives 0, not 5, we say that 5 0 is not defined or is undefined. Also, since any number times 0 gives 0, 0 0 is also not defined.
Answers on page A-5
123
2.5 Division of Integers and
Order of Operations
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Divide, if possible.
7. 34 0
EXCLUDING DIVISION BY 0
Division by 0 is not defined:
a 0, or
8. 0 共4兲
a
, is undefined for all real numbers a.
0
DIVIDING 0 BY OTHER NUMBERS
Note that 0 8 0 because 0 0 8.
DIVIDENDS OF 0
9. 52 0
Zero divided by any nonzero real number is 0:
0
0,
a
Answers on page A-5
EXAMPLES
a 0.
Divide.
6. 0 共6兲 0
Study Tips
ORGANIZATION
124
0
0
12
8.
3
is undefined.
0
Do Exercises 7–9.
Rules for Order of Operations
When more than one operation appears in a calculation or problem, we apply
the same rules that were used in Section 1.9. We repeat them here for review,
now including absolute-value symbols.
RULES FOR ORDER OF OPERATIONS
1. Do all calculations within parentheses, brackets, braces, absolutevalue symbols, numerators, or denominators.
2. Evaluate all exponential expressions.
3. Do all multiplications and divisions in order from left to right.
4. Do all additions and subtractions in order from left to right.
EXAMPLES
Simplify.
9. 17 10 2 4
With no grouping symbols or exponents, we begin with the third rule.
17 10 2 4 17 5 4 ⎫
17 20
⎬
⎭
Carrying out all multiplications and
divisions in order from left to right
3
10. 兩共2兲3 4兩 5共2兲
We first simplify within the absolute-value symbols.
兩共2兲3 4兩 5共2兲 兩8 4兩 5共2兲
共2兲 共2兲 共2兲 8
兩2兩 5共2兲
Dividing
2 5共2兲
兩2兩 2
2 共10兲
Multiplying
12
Subtracting
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
When doing homework,
consider using a spiral or
three-ring binder. You want to
be able to go over your
homework when studying for
a test. Therefore, you need to
be able to easily access any
problem. Write legibly, label
each section and each exercise
clearly, and show all steps.
Writing clearly will also be
appreciated by your instructor
if homework is collected. Most
tutors and instructors can be
more helpful if they can see
and understand all the steps
in your work.
When you are finished
with your homework, check
the answers to the oddnumbered exercises at the
back of the book or in the
Student’s Solutions Manual
and make corrections. If you
do not understand why an
answer is wrong, draw a star
by it so you can ask questions
in class or during your
instructor’s office hours.
7.
EXAMPLE 11
Simplify: 24 51 4 共37 23 2兲.
Simplify.
10. 5 共7兲 共3兲2
24 51 4 共37 23 2兲
24 51 4 共37 46兲
Carrying out all operations inside
parentheses first, multiplying 23 by 2,
following the rules for order of
operations within the parentheses
24 51 4 83
Completing the addition inside
parentheses
16 51 4 83
Evaluating exponential expressions
16 204 83
Doing all multiplications
220 83
Doing all additions and subtractions
in order from left to right
137
11. 共2兲 兩3 22兩 5
Always regard a fraction bar as a grouping symbol. It separates any calculations in the numerator from those in the denominator.
EXAMPLE 12
Simplify:
5 共3兲2
.
2
5 共3兲2 5 9 ⎫
2
2 ⎪
4
2
2
⎬
⎪
⎭
Calculating within the numerator:
共3兲2 共3兲 共3兲 9 and 5 9 4
Dividing
12. 52 5 53 共42 48 4兲
Do Exercises 10–13.
CALCULATOR CORNER
Grouping Symbols Most calculators now provide grouping symbols.
Such keys may appear as ( and ) or [(... and ...)] . Grouping symbols
can be useful when we are simplifying expressions written in fraction
form. For example, to simplify
38 142
,
2 47
13.
we press ( 3 8 1 4 2 ) ( 2 4 7 ) .
Failure to include grouping symbols in the above keystrokes would mean
that we are simplifying a different expression:
ISBN 0-558-68755-5
38 共5兲 共9兲
122
142
47.
2
Exercises: Use a calculator with grouping symbols to simplify each of
the following.
38 178
311 172
285 54
1.
4
2.
2
3. 785 5 30
2 13
17 3 51
787
Answers on page A-5
125
2.5 Division of Integers and
Order of Operations
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
2.5
EXERCISE SET
For Extra Help
MathXL
MyMathLab
InterAct
Math
Math Tutor
Center
Video
Lectures
on CD
Disc 1
Student’s
Solutions
Manual
Divide, if possible, and check each answer by multiplying. If an answer is undefined, state so.
1. 28 共4兲 7
5.
18
9
2
9.
72
9
8
2.
35
5
7
6. 22 共2兲 11
3.
28
14
2
4. 26 共13兲 2
7.
48
4
12
8. 63 共9兲 7
10.
50
2
25
11. 100 共50兲 2
13. 344 8 43
14.
128
16
8
15.
200
8
25
16. 651 共31兲 21
56
Undefined
0
18.
0
0
5
19.
88
8
11
20.
145
29
5
23.
0
0
2
24.
13
Undefined
0
17.
21. 25.
276
23
12
19
19
1
22. 26.
217
31
7
17
17
1
12.
400
50
8
27. 41 1 41
28. 23 共1兲 23
Simplify, if possible. If an answer is undefined, state so.
29. 5 2 3 6 7
30. 5 共2 3 7兲 6
31. 9 2共3 8兲 19
32. 共8 2兲 共3 9兲 36
33. 16 共24兲 50 334
34. 10 20 15 24 160
35. 40 32 23 23
36. 24 22 10 10
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
126
37. 4 共6 8兲兾共4 3兲
38. 43 10 20 82 23
8
41.
92 1
10
1 32
45. 20 5共3兲 3 9
39. 4 5 2 6 4
12
305
42.
100 62
4
共5兲2 32
46. 14 2共6兲 7 35
40. 53 4 9 共8 9 3兲
126
43. 8共7兲 6共5兲 86
44. 10共5兲 1共1兲 50
47. 18 0共32 52 7 4兲
48. 9 0 5 4 0
18
49. 4 52 10 10
50. 共2 5兲2 共9兲 1
51. 共3 8兲2 共1兲 25
52. 3 32 6
53. 17 10 3 983
54. 30 共5兲3 95
55. 2 10 2 5 22 82
56. 5 62 3 22 53
57. 12 20 3 7988
58. 20 43 共8兲 12
59. 2 10 3 5000 3000
60. 7共34 兲 18 549
61. 6关9 共3 4兲兴 60
62. 8关共6 13兲 11兴 144
63. 1000 共100兲 10
64. 256 共32兲 共4兲
1
65. 8 兩7 9兩 3 2
69.
63 7 34 25 9
共1 23 兲3 73
66. 兩8 7 9兩 2 1 17
70.
Undefined
73.
共5兲3 17
10共2 6兲 2共5 2兲
2
77.
71.
2
74.
共3 5兲2 共7 13兲
共2 5兲3 2 4
2 32 共32 共2 1兲兲
52 62 22共3兲
68. 9 兩5 7兩3 1
72.
3
75.
10
DW Explain in your own words why 17 0 is
undefined.
ISBN 0-558-68755-5
4 2 42 3 2
共7 4兲3 2 5 4
67. 9 兩7 32兩 7
264
2 43 4 32
19 3 174
0
78.
5 62 共22 5兲 72
32 42 共2兲3 2
40
76.
16 28 22
5 25 53
Undefined
DW Without performing any calculations, Stefan
2
2
2
2
reports that 共19 17 兲兾共16 18 兲 is negative.
How do you think he reached this conclusion?
127
Exercise Set 2.5
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
SKILL MAINTENANCE
79. Fabrikant Fine Diamonds ran a 4-in. by 7-in.
advertisement in The New York Times. Find the
area of the ad. [1.5c], [1.8a] 28 sq in.
80. A classroom contains 7 rows of chairs with 6 chairs
in each row. How many chairs are there in the
classroom? [1.8a] 42 chairs
81. Cindi’s Ford Focus gets 32 mpg (miles per gallon). How
many gallons will it take to travel 384 mi? [1.8a]
82. Craig’s Chevy Blazer gets 14 mpg. How many gallons
will it take to travel 378 mi? [1.8a]
12 gal
27 gal
83. A 7-oz bag of tortilla chips contains 1050 calories. How
many calories are in a 1-oz serving? [1.8a]
84. A 7-oz bag of tortilla chips contains 8 g (grams) of fat
per ounce. How many grams of fat are in a carton
containing 12 bags of chips? [1.8a] 672 g
150 cal
85. There are 18 sticks in a large pack of Trident gum. If
4 people share the pack equally, how many whole
pieces will each person receive? How many extra
pieces will remain? [1.8a] 4 pieces; 2 pieces
86. A bag of Ricola throat lozenges contains 24 cough
drops. If 5 people share the bag equally, how many
lozenges will each person receive? How many extra
lozenges will remain? [1.8a] 4 lozenges; 4 lozenges
SYNTHESIS
87.
DW Ty claims that 8 32 1 is 2. What mistake do
88.
DW Bryn contends that 13 10兾2 5 is 1. What
mistake do you think she is making?
90.
73 9 62 8 43 6
Undefined
52 25
you think he is making?
Simplify, if possible.
89.
9 32
0
2 4 52 9 82 7
91.
共25 42 兲3
共共6兲2 62 兲
172 162
2
92.
0
共7 8兲37
共98 72 2兲
72 82
95.
282 362兾42 172 992
97.
Write down the keystrokes needed to calculate
98.
1
5
x2
5
yx
3
(
3
x2
Write down the keystrokes needed to calculate
(
1
6
x2
2
4
2
3
)
(
3
x2
4
)
162 24 23
.
3 4 52
4
100. Evaluate the expression for which the keystrokes are 4
x2
5
1
6
Determine the sign of each expression if m is negative and n is positive.
102.
Positive
n
m
Negative
冉 冊
103. n
m
Negative
)
1
(
0
2
2
6
) .
冉 冊
104. n
m
Negative
6 .
5
2
冉 冊
105. n
m
Positive
128
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
n
m
159
152 53
.
32 42
99. Evaluate the expression for which the keystrokes are as follows: 4
101.
195 共15兲3
195 7 52
9 3 363兾122 9 2 486
96.
)
94.
2
0
(
19 172
132 34
93.
2.6
INTRODUCTION TO ALGEBRA
AND EXPRESSIONS
In this section, we will learn to write equivalent expressions by making use
of the distributive law, both of which are very important concepts.
Objectives
Evaluate an algebraic
expression by substitution.
Use the distributive law to
find equivalent expressions.
Algebraic Expressions
In arithmetic, we work with expressions such as
37 86,
7 8,
19 7, and
3
.
8
1. Evaluate a b for a 38 and
b 26.
In algebra, we use both numbers and letters and work with algebraic
expressions such as
x 86,
7 t,
19 y, and
a
.
b
We have already worked with expressions like these.
When a letter can stand for various numbers, we call the letter a variable.
A number or a letter that stands for just one number is called a constant. Let
c the speed of light. Then c is a constant. Let a the speed of an Amtrak
metroliner. Then a is a variable since the value of a can vary.
An algebraic expression consists of variables, constants, numerals, and
operation signs. When we replace a variable with a number, we say that we are
substituting for the variable. This process is called evaluating the expression.
EXAMPLE 1
Evaluate x y for x 37 and y 29.
2. Evaluate x y for x 57 and
y 29.
3. Evaluate 5t for t 14.
We substitute 37 for x and 29 for y and carry out the addition:
x y 37 29 66.
The number 66 is called the value of the expression.
Algebraic expressions involving multiplication, like “8 times a,” can be
written as 8 a, 8 a, 8共a兲, or simply 8a. Two letters written together without
an operation symbol, such as ab, also indicate multiplication.
EXAMPLE 2
Answers on page A-5
Evaluate 3y for y 14.
3y 3共14兲 42
Parentheses are required here.
Study Tips
Do Exercises 1–3.
READING A MATH TEXT
Algebraic expressions involving division can also be written several ways.
8
For example, “8 divided by t ” can be written as 8 t, 8兾t, or .
t
a
a
EXAMPLE 3 Evaluate
and
for a 35 and b 7.
b
b
ISBN 0-558-68755-5
We substitute 35 for a and 7 for b:
35
a
5;
b
7
a 35
5.
b
7
Do not expect a math text to
read like a magazine or novel.
On one hand, most assigned
readings in a math text consist
of only a few pages. On the
other hand, every sentence
and word is important and
should make sense. If they
don’t, seek help as soon as
possible.
129
2.6 Introduction to Algebra
and Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
For each number, find two equivalent
expressions with negative signs in
different places.
6
4.
x
m
5. n
6.
r
4
EXAMPLE 4
Evaluate a
a a
,
, and
for a 15 and b 3.
b b
b
We substitute 15 for a and 3 for b:
15
a
5;
b
3
a 15
5;
b
3
a
15
5.
b 3
Examples 3 and 4 illustrate the following.
a
a
and represent the same number.
b
b
a a
a
all represent the same number.
,
, and
b b
b
Do Exercises 4–7.
a a
a
,
, and for
b b
b
a 28 and b 4.
7. Evaluate
8. Find the Fahrenheit temperature
that corresponds to 10 degrees
Celsius (see Example 5).
EXAMPLE 5
9C
32 for C 20.
5
This expression can be used to find the Fahrenheit temperature that corresponds to 20 degrees Celsius:
9 20
180
9C
32 32 32 36 32 68.
5
5
5
Do Exercise 8.
EXAMPLE 6
9. Evaluate 3x 2 for x 4 and
x 4.
Evaluate
Evaluate 5x 2 for x 3 and x 3.
The rules for order of operations specify that the replacement for x be
squared. That result is then multiplied by 5:
5x 2 5共3兲2 5共9兲 45;
5x 2 5共3兲2 5共9兲 45.
10. Evaluate a 4 for a 3 and
a 3.
Example 6 illustrates that when opposites are raised to an even power, the
results are the same.
Do Exercises 9 and 10.
11. Evaluate 共x兲2 and x 2 for
x 3.
EXAMPLE 7
Evaluate 共x兲2 and x 2 for x 7.
When we evaluate 共x兲2 for x 7, we have
共x兲2 共7兲2 共7兲 共7兲 49.
12. Evaluate 共x兲2 and x 2 for
x 2.
13. Evaluate x 5 for x 2 and
x 2.
To evaluate x 2, we again substitute 7 for x. We must recall that taking the opposite of a number is the same as multiplying that number by 1.
x 2 1 x 2
The opposite of a number is the same as
multiplying by 1.
72 1 72
Substituting 7 for x
1 49 49.
Using the rules for order of operations;
calculating the power before multiplying
Example 7 shows that 共x兲2 x 2.
Do Exercises 11–13.
130
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
Answers on page A-5
Substitute 7 for x. Then evaluate
the power.
Equivalent Expressions and the Distributive Law
It is useful to know when two algebraic expressions will represent the same
number. In many situations, this will help with problem solving.
EXAMPLE 8
Complete each table by evaluating
each expression for the given values.
14.
3x ⴙ 2x
x4
Evaluate x x and 2x for x 3 and x 5 and compare the
x 2
results.
x x 3 3 6;
20
20
10
10
0
0
x0
We substitute 3 for x in x x and again in 2x:
5x
2x 2 3 6.
Next we repeat the procedure, substituting 5 for x :
x x 5 共5兲 10;
2x 2共5兲 10.
The results can be shown in a table. It
appears that x x and 2x represent the
same number.
15.
xⴙx
x3
x 5
4x ⴚ x
2x
6
6
10
10
x2
x 2
3x
6
6
6
6
0
0
x0
Do Exercises 14 and 15.
Example 8 suggests that x x and 2x represent the same number for any
replacement of x. When this is known to be the case, we can say that x x and
2x are equivalent expressions.
Answers on page A-5
CALCULATOR
CORNER
EQUIVALENT EXPRESSIONS
Two expressions that have the same value for all allowable
replacements are called equivalent.
a
a
and are equivalent
b
b
but that the expressions 共x兲2 and x 2 are not equivalent.
An important concept, known as the distributive law, is useful for finding
equivalent algebraic expressions. The distributive law involves two operations: multiplication and either addition or subtraction.
To review how the distributive law works, consider the following:
In Examples 3 and 7 we saw that the expressions
1
4
兾
xy
3
兾
4 5
7
3 5
2 8 0
This is 7 5.
This is 7 40.
3 1 5
This is the sum 7 40 7 5.
To carry out the multiplication, we actually added two products. That is,
7 45 7共40 5兲 7 40 7 5.
The distributive law says that if we want to multiply a sum of several
numbers by a number, we can either add within the grouping symbols and
then multiply, or multiply each of the terms separately and then add.
ISBN 0-558-68755-5
Evaluating Powers To
evaluate an expression like
x 3 for x 14 with a
calculator, it is imperative that
we keep in mind the rules for
order of operations. On some
calculators, this expression is
evaluated by pressing
. Other calculators use
the keystrokes () ( () 1
4 ) ⵩ 3 ENTER .
Consult your owner’s manual,
an instructor, or simply
experiment if your calculator
behaves differently.
Exercises: Evaluate.
1. a 5 for a 3 243
2. x 5 for x 4 1024
3. x 5 for x 2 32
4. x 5 for x 5 3125
131
2.6 Introduction to Algebra
and Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Use the distributive law to write an
equivalent expression.
THE DISTRIBUTIVE LAW
16. 5共a b兲
For any numbers a, b, and c,
a共b c兲 ab ac.
EXAMPLE 9
Evaluate a共b c兲 and ab ac for a 3, b 4, and c 2.
We have
a共b c兲 3共4 2兲 3 6 18
17. 6共x y z兲
and
ab ac 3 4 3 2 12 6 18.
It is impossible to overemphasize the importance of the parentheses in
the statement of the distributive law. Were we to omit the parentheses, we
would have ab c. To see that a共b c兲 ab c, note that 3共4 2兲 18,
but 3 4 2 14.
EXAMPLE 10
Use the distributive law to write an
equivalent expression.
18. 4共x y兲
Use the distributive law to write an expression equivalent to
2共l w兲.
2共l w兲 2 l 2 w
Note that the sign between l and w now
appears between 2 l and 2 w.
2l 2w.
Try to go directly to this step.
Do Exercises 16 and 17.
Since subtraction can be regarded as addition of the opposite, it follows
that the distributive law holds in cases involving subtraction.
19. 3共a b c兲
EXAMPLE 11 Use the distributive law to write an expression equivalent to
each of the following:
a) 7共a b兲;
b) 9共x 5兲;
c) 共a 7兲b;
d) 4共x 2y 3z兲
a) 7共a b兲 7 a 7 b
7a 7b
Try to go directly to this step.
20. 共m 4兲6
b) 9共x 5兲 9x 9共5兲
9x 45
c) 共a 7兲b b共a 7兲
21. 8共2a b 3c兲
Again, try to go directly to this step.
Using a commutative law
bab7
Using the distributive law
ab 7b
Using a commutative law to write ba
alphabetically and b 7 with the constant first
d) 4共x 2y 3z兲 4 x 共4兲 共2y兲 共4兲 共3z兲
4x 共4 2兲y 共4 3兲z
Using the
distributive law
Using an associative
law (twice)
4x 共8y兲 共12z兲
Answers on page A-5
Try to go directly to this step.
Do Exercises 18–21.
132
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
4x 8y 12z
2.6
EXERCISE SET
For Extra Help
MathXL
MyMathLab
InterAct
Math
Math Tutor
Center
Video
Lectures
on CD
Disc 1
Student’s
Solutions
Manual
Evaluate.
1. 12n, for n 2
2. 39n, for n 2
(The cost, in cents, of sending 2 text messages)
(The cost, in cents, of sending 2 letters)
24¢
78¢
3.
x
, for x 6 and y 3 2
y
4.
m
, for m 18 and n 2 9
n
5.
2q
, for p 6 and q 3 1
p
6.
5y
, for y 15 and z 25 3
z
7.
72
, for r 4
r
8.
72
, for i 2
i
(The approximate doubling time, in years, for an
investment earning 4% interest per year)
(The approximate doubling time, in years, for an
investment earning 2% interest per year)
18 yr
36 yr
9. 3 5 x, for x 2 13
11. 2l 2w, for l 3 and w 4
10. 9 2 x, for x 3 3
12. 3共a b兲, for a 2 and b 4
(The perimeter, in feet, of a 3-ft by 4-ft rectangle)
18
14 ft
13. 2共l w兲, for l 3 and w 4
14. 3a 3b, for a 2 and b 4
(The perimeter, in feet, of a 3-ft by 4-ft rectangle)
18
14 ft
15. 7a 7b, for a 5 and b 2 21
16. 4x 4y, for x 6 and y 1 20
17. 7共a b兲, for a 5 and b 2 21
18. 4共x y兲, for x 6 and y 1 20
19. 16t 2, for t 5
20.
ISBN 0-558-68755-5
(The distance, in feet, that an object falls in 5 sec)
400 ft
49t 2
, for t 10
10
(The distance, in meters, that an object falls in 10 sec)
490 m
133
Exercise Set 2.6
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
21. a 共b a兲2, for a 6 and b 4 10
22. 共x y兲2, for x 2 and y 3 25
23. 9a 9b, for a 13 and b 13 0
24. 8x 8y, for x 17 and y 17 0
25.
n2 n
, for n 9
2
26.
(For determining the number of handshakes possible
among 9 people) 36
27. m 3 m 2, for m 5 100
5共F 32兲
, for F 50
9
(For converting 50 degrees Fahrenheit to degrees
Celsius) 10
28. a 6 a, for a 2 66
For each expression, write two equivalent expressions with negative signs in different places.
5 5
;
t t
30.
7 7
7
;
x
x x
31.
n n
n
;
b
b b
32. 9 9
9
;
p
p p
34.
u u
u
;
5
5 5
35.
14 14
14
;
w
w
w
36.
29. 33.
5
t
Evaluate
3
r
3 3
;
r r
23 23
23
;
m
m m
a a
a
,
, and for the given values.
b b
b
37. a 45, b 9 5; 5; 5
38. a 40, b 5 8; 8; 8
39. a 81, b 3 27; 27; 27
40. a 56, b 7 8; 8; 8
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
134
Evaluate.
41. 共3x兲2 and 3x 2, for x 2 36; 12
42. 共2x兲2 and 2x 2, for x 3 36; 18
43. 5x 2, for x 3 and x 3 45; 45
44. 2x 2, for x 5 and x 5 50; 50
45. x 3, for x 6 and x 6 216; 216
46. x 6, for x 2 and x 2 64; 64
47. x 8, for x 1 and x 1 1; 1
48. x 5, for x 3 and x 3 243; 243
49. a 5, for a 2 and a 2 32; 32
50. a 7, for a 1 and a 1 1; 1
Use the distributive law to write an equivalent expression.
51. 5共a b兲 5a 5b
52. 7共x y兲 7x 7y
53. 4共x 1兲 4x 4
54. 6共a 1兲 6a 6
55. 2共b 5兲 2b 10
56. 3共x 6兲 3x 18
57. 7共1 t兲 7 7t
58. 4共1 y兲 4 4y
59. 6共5x 2兲 30x 12
60. 9共6m 7兲 54m 63
61. 8共x 7 6y兲 8x 56 48y
62. 4共5x 8 3p兲 20x 32 12p
63. 7共 y 2兲 7y 14
64. 9共 y 7兲 9y 63
65. 共x 2兲3 3x 6
66. 共x 4兲2 2x 8
67. 4共x 3y 2z兲 4x 12y 8z
68. 8共2x 5y 8z兲 16x 40y 64z
69. 8共a 3b c兲 8a 24b 8c
70. 6共a 2b c兲 6a 12b 6c
71. 4共x 3y 7z兲 4x 12y 28z
72. 5共9x y 8z兲
73. 共4a 5b c 2d兲5
74. 共9a 4b 3c d兲7
45x 5y 40z
75.
20a 25b 5c 10d
DW Does x always represent a negative number?
y
76.
63a 28b 21c 7d
DW Is x 2 always negative? Why or why not?
ISBN 0-558-68755-5
Why or why not?
135
Exercise Set 2.6
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
SKILL MAINTENANCE
77. Write a word name for 23,043,921.
[1.1c]
Twenty-three million, forty-three thousand, nine hundred
twenty-one
79. Estimate by rounding to the nearest ten. Show your
work. [1.4b]
78. Multiply: 17 53. [1.5a]
901
80. Divide: 2982 3.
[1.6c]
994
5 2 8 3
2 4 7 5
5280 2480 2800
81. On January 6, it snowed 9 in., and on January 7, it
snowed 8 in. How much did it snow altogether? [1.8a]
82. On March 9, it snowed 12 in., but on March 10, the sun
melted 7 in. How much snow remained? [1.8a]
17 in.
5 in.
83. For Brett’s party, his wife ordered two cheese pizzas at
$11 apiece and two pepperoni pizzas for $13 apiece.
How much did she pay for the pizza? [1.8a]
$48
84. For Tania’s graduation party, her husband ordered
three buckets of chicken wings at $12 apiece and
3 trays of nachos at $9 a tray. How much did he pay for
the wings and nachos? [1.8a]
$63
SYNTHESIS
85.
DW Under what condition(s) will the expression ax 2
86.
be nonnegative? Explain.
DW Ted evaluates a a 2 for a 5 and gets 100 as the
result. What mistake did he probably make?
87. A car’s catalytic converter works most efficiently after it is heated to about 370°C. To what Fahrenheit temperature does
this correspond? (Hint : see Example 5.) 698
F
Evaluate.
88.
x 2 xy 2 2 y, for x 24 and y 6 2016
89.
a b 3 17a, for a 19 and b 16 4438
90.
x 2 23y y 3, for x 18 and y 21 8454
91.
r 3 r 2t rt 2, for r 9 and t 7 279
92.
a 3b a 2b 2 ab 3, for a 8 and b 6 2496
93. a 1996 a 1997, for a 1 2
94. x 1492 x 1493, for x 1 2
95. 共m 3 mn兲m, for m 4 and n 6 2,560,000
96. 5a 3a4, for a 2 20
Replace the blanks with , , , or to make each statement true.
97.
32
共88
29兲 1888
32 共88 29兲 1888
98.
59
17
59
8 1475
59 17 59 8 1475
Classify each statement as true or false. If false, write an example showing why.
99. For any choice of x, x 2 共x兲2.
True
101. For any choice of x, x 6 x 4 共x兲6 共x兲4.
True
103.
100. For any choice of x, x 3 x 3.
False; 2 3 8; but 23 8
102. For any choice of x, 共3x兲2 9x 2.
True
(Hint: see Example 5.)
136
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
DW If the Fahrenheit temperature is doubled, does it follow that the corresponding Celsius temperature is also doubled?
2.7
LIKE TERMS AND PERIMETER
Objectives
Combine like terms.
One common way in which equivalent expressions are formed is by combining like terms. In this section we learn how this is accomplished and apply
the concept to geometry.
Determine the perimeter of a
polygon.
Combining Like Terms
A term is a number, a variable, a product of numbers and/or variables, or a
quotient of numbers and/or variables. Terms are separated by addition signs.
If there are subtraction signs, we can find an equivalent expression that uses
addition signs.
EXAMPLE 1
What are the terms of 3xy 4y 3xy 4y 2
2
3xy 共4y兲 z
z
The terms are 3xy, 4y, and
What are the terms of each
expression?
1. 5x 4y 3
2
?
z
Separating parts with signs
2
.
z
2. 4y 2x Do Exercises 1 and 2.
x
y
Terms in which the variable factors are exactly the same, such as 9x and
4x, are called like, or similar, terms. For example, 3y 2 and 7y 2 are like terms,
whereas 5x and 6x 2 are not. Constants, like 7 and 3, are also like terms.
EXAMPLES
Identify the like terms.
2
2. 7x 5x 2x 8 5x 3 1
7x and 2x are like terms; 8 and 1 are like terms.
3. 5ab a 3 a 2b 2ab 7a 3
5ab and 2ab are like terms; a 3 and 7a 3 are like terms.
Identify the like terms.
3. 9a 3 4ab a 3 3ab 7
Do Exercises 3 and 4.
When an algebraic expression contains like terms, an equivalent expression can be formed by combining, or collecting, like terms. To combine like terms, we rely on the distributive law.
EXAMPLE 4
Combine like terms to form an equivalent expression.
a) 4x 3x
b) 6mn 7mn
c) 7y 2 3y 5
d) 2a 5 9ab 3 a 5 7 4ab
a) 4x 3x 共4 3兲x
7x
Using the distributive law (in “reverse”)
We usually go directly to this step.
b) 6mn 7mn 共6 7兲mn
ISBN 0-558-68755-5
4. 3xy 5x 2 y 2 4xy y
Try to do this mentally.
1mn, or simply mn
c) 7y 2 3y 5 7y 共2兲 共3y兲 5
Rewriting as addition
7y 共3y兲 共2兲 5
Using a commutative law
4y 3
Try to go directly to this step.
Answers on page A-6
137
2.7 Like Terms and Perimeter
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
d) 2a 5 9ab 3 a 5 7 4ab
Combine like terms to form an
equivalent expression.
2a 5 9ab 3 a 5 共7兲 共4ab兲
5. 2a 7a
2a 5 a 5 9ab 共4ab兲 3 共7兲
5
3a 5ab 共4兲
5
Rearranging terms
5
Think of a as 1a ; 2a 5 a 5 3a 5
3a 5 5ab 4
6. 5x 2 9 2x 2 3
Do Exercises 5–7.
Perimeter
PERIMETER OF A POLYGON
7. 4m 2n2 5 n2 m 9
A polygon is a closed geometric figure with three or more sides. The
perimeter of a polygon is the distance around it, or the sum of the
lengths of its sides.
Find the perimeter of each polygon.
EXAMPLE 5
8.
11 cm
5m
4m
4 cm
2 cm
2 cm
Find the perimeter of this polygon.
6m
A polygon with five sides
is called a pentagon.
5m
7 cm
9m
We add the lengths of all sides. Since all the units are the same, we are
effectively combining like terms.
9.
Perimeter 6 m 5 m 4 m 5 m 9 m
7 mm
共6 5 4 5 9兲 m
7 mm
29 m
7 mm
7 mm
Using the distributive law
Try to go directly to this step.
Caution!
When units of measurement are given in the statement of a problem, as in
Example 5, the solution should also contain units of measurement.
7 mm
Do Exercises 8 and 9.
10. Find the perimeter of a
rectangle that is 2 cm by 4 cm.
A rectangle is a polygon with four sides and four 90
angles. Opposite
sides of a rectangle have the same measure. The symbol CR or RC indicates
a 90
angle. A 90
angle is often referred to as a right angle.
EXAMPLE 6
Find the perimeter of a rectangle that is 3 cm by 4 cm.
2 cm
4 cm
4 cm
90
Perimeter 3 cm 3 cm 4 cm 4 cm
共3 3 4 4兲 cm
14 cm
Answers on page A-6
Do Exercise 10.
138
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
3 cm
The perimeter of the rectangle in Example 6 is 2 3 cm 2 4 cm, or
equivalently 2共3 cm 4 cm兲. This can be generalized, as follows.
11. Find the perimeter of a 4-ft by
8-ft sheet of plywood.
PERIMETER OF A RECTANGLE
The perimeter P of a rectangle of length l and width w is given by
l
P 2l 2w, or P 2 共l w兲.
w
w
l
12. Find the perimeter of a square
with sides of length 10 km.
EXAMPLE 7
A common door size is 3 ft by 7 ft. Find the perimeter of such
a door.
P 2l 2w
We could also use P 2共l w兲.
10 km
2 7 ft 2 3 ft
共2 7兲 ft 共2 3兲 ft
Try to do this mentally.
14 ft 6 ft
10 km
20 ft
Combining like terms
The perimeter of the door is 20 ft.
Do Exercise 11.
A square is a rectangle in which all sides have the same length.
EXAMPLE 8
Find the perimeter of a square with sides of length 9 mm.
9 mm
13. Find the perimeter of a square
sandbox with sides of length 6 ft.
P 9 mm 9 mm 9 mm 9 mm
共9 9 9 9兲 mm
36 mm
Note that
9 9 9 9 4 9.
9 mm
Do Exercise 12.
Answers on page A-6
PERIMETER OF A SQUARE
The perimeter P of a square is four times s, the length of a side:
Pssss
4s.
Study Tips
s
s
s
s
EXAMPLE 9
Find the perimeter of a square garden with sides of length
12 ft.
P 4s
ISBN 0-558-68755-5
4 12 ft
48 ft
The perimeter of the garden is 48 ft.
Do Exercise 13.
UNDERSTAND YOUR
MISTAKES
When you receive a graded
quiz, test, or assignment back
from your instructor, it is
important to review and
understand what your
mistakes were. Too often
students simply file away old
papers without first making an
effort to learn from their
mistakes.
139
2.7 Like Terms and Perimeter
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
1. Yarn Cost. It costs $8 for the
yarn for each scarf that Annette
knits. If she used $120 worth of
yarn, how many scarves did
she knit?
2. Elevation. Genine started
hiking a 10-mi trail at an
elevation that was 150 ft below
sea level. At the end of the trail,
she was 75 ft above sea level.
How many feet higher was she
at the end of the trail than at
the beginning?
Translating
for Success
The goal of these matching
questions is to practice step (2),
Translate, of the five-step problemsolving process. Translate each
word problem to an equation and
select a correct translation from
equations A–O.
A. 75 x 120
B. 15 750 x
6. Account Balance. Lorenzo had
$75 in his checking account. He
then wrote a check for $150.
What was the balance in his
account?
7. Laptop Computers. Great
Graphics purchased a laptop
computer for each of its
15 employees. If each laptop
costs $750, how much did the
computers cost?
C. 10 15 x
D. 8 120 x
3. Community Service. In order
to fulfill the requirements for a
sociology class, Glen must log
120 hr of community service.
So far, he has spent 75 hr
volunteering at a youth center.
How many more hours must
he serve?
E. 150 75 x
F.
15 750 x
G. 75 共150兲 x
8. Basketball. A basketball team
scored 75 points in one game. In
the next game, the team scored
a record 120 points. How many
points did the team score in the
two games?
H. 2 150 2 75 x
4. Disaster Relief. Each package
that is prepared for a disaster
relief effort contains 15 meal
bars. How many packages can
be filled from a donation of
750 bars?
I.
10 共15兲 x
J.
8 x 120
9. Pizza Sales. A youth club sold
120 pizzas for a fundraiser. If
each pizza sold for $8, how
much money was taken in?
K. 750 15 x
L. 15 共10兲 x
M. 75 120 x
5. Perimeter. A rectangular
building lot is 75 ft wide and
150 ft long. What is the
perimeter of the lot?
N. 75 150 x
10. Temperature. The temperature
in Fairbanks was 10
at 6:00 P.M.
and fell another 15
by midnight.
What was the temperature at
midnight?
O. 75 120 x
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
Answers on page A-6
2.7
EXERCISE SET
For Extra Help
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Student’s
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List the terms of each expression.
1. 2a 5b 7c 2a, 5b, 7c
2. 4x 6y 7z 4x, 6y, 7z
3. 9mn 6n 8 9mn, 6n, 8
4. 7rs 4s 5 7rs, 4s, 5
5. 3x 2y 4y 2 2z 3 3x 2y , 4y 2, 2z 3
6. 4a3b ab2 9b3 4a 3b, ab 2, 9b 3
Combine like terms to form an equivalent expression.
7. 5x 9x 14x
8. 9a 7a 16a
9. 10a 13a 3a
11. 2x 6z 9x 11x 6z
12. 3a 5b 7a 10a 5b
13. 27a 70 40a 8 13a 62
14. 42x 6 4x 2 38x 4
15. 9 5t 7y t y 13 4 4t 6y
16. 8 4a 9b 7a 3b 15 7 3a 6b
17. a 3b 5a 2 b 6a 4b 2
18. x 7y 5 2y 3x 4x 5y 5
19. 8 11a 5b 6a 7b 7
1 17a 12b
22. 8y 3 3z 4y 3
12y 3 3z
ISBN 0-558-68755-5
10. 16x x 15x
20. 8x 5x 6 3y y 4
3x 2 2y
23. 11x 4 2y 3 4x 4 y 3
7x 4 y 3
21. 8x 2 3y x 2
7x 2 3y
24. 13a5 9b4 2a5 4b4
11a 5 5b 4
141
Exercise Set 2.7
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
25. 9a2 4a a 3a2
26. 3a2 7a3 a2 5 a3
2
2
6a 3a
27. x 3 5x 2 2x 3 3x 2 4
3
3x 3 8x 2 4
2a 8a 5
28. 9xy 4y 2 2xy 2y 2 1
29. 7a3 4ab 5 7ab 8
7xy 6y 2 1
30. 8a2b 3ab2 4a2b 2ab
7a 3 3ab 3
31. 9x 3y 4xy 3 6xy 3 3xy
4a 2b 3ab 2 2ab
32. 3x 4 2y 4 8x 4y 4 7x 4 8y 4
9x 3y 2xy 3 3xy
33. 3a6 9b4 2a6b4 7a6 2b4
4x 4 6y 4 8x 4y 4
4a 6 11b 4 2a 6b 4
34. 9x 6 5y 5 3x 6y 8x 6 4y 5 x 6 y 5 3x 6y
Find the perimeter of each polygon.
35.
36.
37.
2 ft
Each side
7 km
5 in.
3 ft
10 ft
5 in.
42 km
20 in.
38.
39.
4 mm
6 mm
40.
3m
1m
1m
4m
4m
3m
7 mm
8m
5m
4m
17 mm
1m
18 m
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
142
Tennis Court. A tennis court contains many rectangles.
Use the diagram of a regulation tennis court to calculate the
perimeters in Exercises 41–44.
41. The perimeter of a singles court 210 ft
42. The perimeter of a doubles court 228 ft
Baseline
Service line
18 ft
Center service line
Service line
Doubles sideline
Singles sideline
42 ft
27 ft
Singles
43. The perimeter of the rectangle formed by the service
lines and the singles sidelines 138 ft
36 ft
Doubles
Singles sideline
Doubles sideline
44. The perimeter of the rectangle formed by a service line,
a baseline, and the singles sidelines 90 ft
78 ft
45. Find the perimeter of a rectangular 8-ft by
10-ft bedroom. 36 ft
46. Find the perimeter of a rectangular 3-ft by
4-ft doghouse. 14 ft
47. Find the perimeter of a checkerboard that is 14 in. on
each side. 56 in.
48. Find the perimeter of a square skylight that is 2 m on
each side. 8 m
49. Find the perimeter of a square frame that is 65 cm on
each side. 260 cm
50. Find the perimeter of a square garden that is 12 yd on
each side. 48 yd
51. Find the perimeter of a 12-ft by 20-ft rectangular deck.
52. Find the perimeter of a 40-ft by 35-ft rectangular
backyard. 150 ft
64 ft
ISBN 0-558-68755-5
53.
DW Explain in your own words what it means for two
algebraic expressions to be equivalent.
54.
DW Can the formula for the perimeter of a rectangle be
used to find the perimeter of a square? Why or why not?
143
Exercise Set 2.7
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
SKILL MAINTENANCE
55. A box of Shaw’s Corn Flakes contains 510 grams (g) of
corn flakes. A serving of corn flakes weighs 30 g. How
many servings are in one box? [1.8a] 17 servings
Simplify.
56. Estimate the difference by rounding to the nearest ten.
[1.4b] 210
7 0 4
4 8 6
[1.9c]
57. 5 3 23 29
58. 共9 7兲4 32 7
59. 12 3 2 8
60. 27 3共2 1兲 27
61. 15 3 2 7 16
62. 30 42 8 2 26
Solve.
[1.7b]
63. 25 t 9 16
64. 19 x 6 13
65. 45 3x 15
66. 50 2t
25
SYNTHESIS
67.
DW Does doubling the length of a square’s side double
the perimeter of the original square? Why or why not?
68.
DW Why was it necessary to introduce the distributive
law before discussing how to combine like terms?
Simplify. (Multiply and then combine like terms.)
69. 5共x 3兲 2共x 7兲 7x 1
70. 3共a 7兲 7共a 4兲 10a 7
71. 2共3 4a兲 5共a 7兲 29 3a
72. 7共2 5x兲 3共x 8兲 10 32x
73. 5共2 3x 4y兲 7共2x y兲
74. 3共4 2x兲 5共9x 3y 1兲
10 x 27y
17 39x 15y
75. In order to save energy, Andrea plans to run a bead of
caulk sealant around 3 exterior doors and 13 windows.
Each window measures 3 ft by 4 ft, each door measures
3 ft by 7 ft, and there is no need to caulk the bottom of
each door. If each cartridge of caulk seals 56 ft and costs
$5.95, how much will it cost Andrea to seal the windows
and doors? $29.75
76. Eric is attaching lace trim to small tablecloths that are
5 ft by 5 ft, and to large tablecloths that are 7 ft by 7 ft.
If the lace costs $1.95 per yard, how much will the trim
cost for 6 small tablecloths and 6 large tablecloths?
77.
78. A rectangular box is used to store six Christmas
ornaments. Find the perimeter of such a box if each
ornament has a diameter of 72 mm. 720 mm
A square wooden rack is used to store the
15 numbered balls as well as the cue ball in pool.
If a pool ball has a diameter of 57 mm, find the
inside perimeter of the storage rack. 912 mm
$187.20
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
144
2.8
SOLVING EQUATIONS
Objectives
In Section 1.7, we learned to solve certain equations by writing a “related
equation.” We now extend this approach to include negative integers, as well
as equations that involve both addition and multiplication.
Use the addition principle to
solve equations.
Use the division principle to
solve equations.
Decide which principle
should be used to solve
an equation.
The Addition Principle
In Section 1.7, we learned to solve an equation like x 12 27 by writing
the related subtraction, x 27 12, or x 15. Note that x 15 is an equation, not a solution. Of course, the solution of the equation x 15 is obviously 15. The solution of x 12 27 is also 15. Because their solutions are
identical, x 15 and x 12 27 are said to be equivalent equations.
EQUIVALENT EQUATIONS
Equations with the same solutions are called equivalent equations.
Solve equations that require
use of both the addition
principle and the division
principle.
Classify each pair as equivalent
expressions or equivalent equations.
1. a 5 3;
a2
It is important to be able to distinguish between equivalent expressions
and equivalent equations.
• 6a and 4a 2a are equivalent expressions because, for any replacement
of a, both expressions represent the same number.
• 3x 15 and 4x 20 are equivalent equations because any solution of
one equation is also a solution of the other equation.
EXAMPLE 1
Classify each pair as either equivalent equations or equivalent
expressions:
a) 5x 1;
2x 4 3x 5
b) x 7;
x 2 5.
a) First note that these are expressions, not equations. To see if they are
equivalent, we combine like terms in the second expression:
2x 4 3x 5 共2 3兲x 共4 5兲
Regrouping and using
the distributive law
2. a 9 6a; 7a 9
5x 1.
We see that 2x 4 3x 5 and 5x 1 are equivalent expressions.
b) First note that both x 7 and x 2 5 are equations. The solution of
x 7 is 7. We substitute to see if 7 is also the solution of x 2 5:
x 2 5
7 2 5
TRUE
Since x 7 and x 2 5 have the same solution, they are equivalent
equations.
ISBN 0-558-68755-5
Do Exercises 1 and 2.
There are principles that enable us to begin with one equation and create
an equivalent equation similar to x 15, for which the solution is obvious.
One such principle, the addition principle, is stated on the next page.
Suppose that a and b stand for the same number and some number c is
added to a. We get the same result if we add c to b, because a and b are equal.
Answers on page A-6
145
2.8 Solving Equations
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Solve.
3. x 5 19
THE ADDITION PRINCIPLE
For any numbers a, b, and c,
ab
EXAMPLE 2
is equivalent to
a c b c.
Solve: x 7 2.
We have
x 7 2
4. x 9 12
x 7 7 2 7
x05
Using the addition principle:
adding 7 to both sides
Adding 7 “undoes” the subtraction of 7.
x 5.
The solution appears to be 5. To check, we use the original equation.
Check:
Solve.
5. 42 x 17
x 7 2
5 7 ? 2
2
TRUE
The solution is 5.
Do Exercises 3 and 4.
6. a 8 6
Recall from Section 2.3 that we can subtract by adding the opposite of
the number being subtracted. Because of this, the addition principle allows
us to subtract the same number from both sides of an equation.
EXAMPLE 3
Solve: 23 t 7.
We have
23 t 7
23 7 t 7 7
Answers on page A-6
16 t 0
Using the addition principle to add 7
or to subtract 7 on both sides
Subtracting 7 “undoes” the addition of 7.
16 t .
The solution is 16. The check is left to the student.
Study Tips
USE A PENCIL
Do Exercises 5 and 6.
146
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
It is no coincidence that the
students who experience the
greatest success in this course
work in pencil. We all make
mistakes and by using pencil
and eraser we are more willing
to admit to ourselves that
something needs to be
rewritten. Please work with a
pencil and eraser if you aren’t
doing so already.
To visualize the addition principle, think of a jeweler’s balance. When
both sides of the balance hold equal amounts of weight, the balance is
level. If weight is added or removed, equally, on both sides, the balance
remains level.
The Division Principle
Solve.
7. 7x 42
In Section 1.7, we solved 8n 96 by dividing both sides by 8:
8 n 96
8 n 96
8
8
Dividing both sides by 8
n 12.
8 times n, divided by 8, is n. 96 8 is 12.
You can check that 8 n 96 and n 12 are equivalent. We can divide both
sides of an equation by any nonzero number in order to find an equivalent
equation.
THE DIVISION PRINCIPLE
For any numbers a, b, and c 共c 0兲,
ab
is equivalent to
8. 24 3t
a
b
.
c
c
In Chapter 3, after we have discussed multiplication of fractions, we will
use an equivalent form of this principle: the multiplication principle.
EXAMPLE 4
Solve: 9x 63.
We have
9x 63
9x 63
9
9
Using the division principle to
divide both sides by 9
Solve.
9. 63 7n
x 7.
Check:
9x 63
9 7 ? 63
63
TRUE
The solution is 7.
Do Exercises 7 and 8.
EXAMPLE 5
Solve: 48 8n .
It is important to distinguish between an opposite, as we have in 8n,
and subtraction, as we had in x 5 19 (margin exercise 3). To undo multiplication by 8, we use the division principle:
10. 6x 72
48 8n
8n
48
8
8
Dividing both sides by 8
6 n .
ISBN 0-558-68755-5
Check:
48 8n
48 ? 8共6兲
48
The solution is 6.
TRUE
Answers on page A-6
147
2.8 Solving Equations
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Solve.
11. x 23
Be sure that you understand why the addition principle is used in Example 2
and the division principle is used in Example 5.
Do Exercises 9 and 10 on the previous page.
Equations like x 7 or t 3 often give students difficulty. One
way to handle problems of this sort is to multiply both sides of the equation by 1.
EXAMPLE 6
Solve: x 7.
To solve an equation like x 7, remember that when an expression is
multiplied or divided by 1, its sign is changed. Here we multiply on both
sides by 1 to change the sign of x:
x 7
共1兲 共x兲 共1兲 7
Multiplying both sides by 1
x 7.
Note that 共1兲 共x兲 is the same as 共1兲 共1兲x.
x 7
Check:
共7兲 ? 7
7
TRUE
The solution is 7.
Another way to solve Example 6 is to note that x 1 x. Then we can
divide both sides by 1:
12. t 3
x 7
1 x 7
1 x
7
1
1
x 7.
Do Exercises 11 and 12.
Selecting the Correct Approach
It is important for you to be able to determine which principle should be used
to solve a particular equation.
EXAMPLES
Solve.
7. 39 3 t
Note that 3 is added to t. To undo addition of 3, we subtract 3 or simply add 3 on both sides:
3 39 3 共3兲 t
Using the addition principle
42 0 t
42 t .
Check:
39 3 t
TRUE
Answers on page A-6
The solution is 42.
148
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
39 ? 3 42
39
8. 39 3t
Solve.
To undo multiplication by 3, we divide by 3 on both sides:
13. 2x 52
39 3t
3t
39
3
3
Using the division principle
13 t .
Check:
39 3t
39 ? 3共13兲
39
TRUE
14. 2 x 52
The solution is 13.
Do Exercises 13–15.
Using the Principles Together
Suppose we want to determine whether 7 is the solution of 5x 8 27. To
check, we replace x with 7 and simplify.
Check:
15. x 7 28
5x 8 27
5 7 8 ? 27
35 8
27
TRUE
This shows that 7 is the solution.
Do Exercises 16 and 17.
In the check above, note that the rules for order of operations require that
we multiply before we subtract (or add).
The rules for order of operations dictate that unless grouping symbols indicate otherwise, multiplication and division are performed before any addition or subtraction. Thus, to evaluate 5x 8,
we select a value:
x
then multiply by 5:
5x
and then subtract 8:
5x 8.
16. Determine whether 9 is the
solution of 7x 8 55.
In Example 9, which follows, these steps are reversed to solve for x:
We will add 8:
5x 8 8,
then divide by 5:
5x
5
and isolate x:
x.
17. Determine whether 6 is the
solution of 4x 3 25.
ISBN 0-558-68755-5
In general, the last step performed when calculating is the first step to be reversed when finding a solution.
Answers on page A-6
149
2.8 Solving Equations
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
18. Solve: 2x 9 43
EXAMPLE 9
Solve: 5x 8 27.
We first note that the term containing x is 5x. To isolate 5x, we add 8 on
both sides:
5x 8 27
5x 8 8 27 8
Using the addition principle
5x 0 35
Try to do this step mentally.
5x 35.
Next, we isolate x by dividing by 5 on both sides:
5x 35
5x 35
5
5
Using the division principle
1x 7
Try to do this step mentally.
x 7.
The check was performed on the previous page. The solution is 7.
Do Exercise 18.
EXAMPLE 10
Solve: 38 9x 2.
We first isolate 9x by subtracting 2 on both sides:
38 9x 2
38 2 9x 2 2
Subtracting 2 (or adding 2) from both sides
36 9x 0
19. Solve: 3x 2 47
Try to do this step mentally.
36 9x.
Now that we have isolated 9x on one side of the equation, we can divide by
9 to isolate x:
36 9x
Check:
9x
36
9
9
Dividing both sides by 9
4 x.
Simplifying
38 9x 2
38 ? 9 共4兲 2
36 2
38
TRUE
The solution is 4.
Do Exercise 19.
150
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
Answers on page A-6
2.8
EXERCISE SET
For Extra Help
MathXL
MyMathLab
InterAct
Math
Math Tutor
Center
Video
Lectures
on CD
Disc 1
Student’s
Solutions
Manual
Classify each pair as either equivalent expressions or equivalent equations.
1. 2x 10; 5x 25
2. 4x 1; 6 4x 5
Equivalent equations
4. 7t 14; 4t 8
Equivalent expressions
6. 2r 7; r 7 r
Equivalent expressions
Equivalent expressions
8. t 4 19; t 6 9
Equivalent equations
10. 2x 14; x 2 9
Equivalent expressions
5. 4r 3; 8 4r 5
Equivalent equations
7. x 9 8; x 3 20
3. 7a 3; 4a 3 3a
9. 3共t 2兲; 5 3t 1
Equivalent equations
Equivalent expressions
11. x 4 8; 2x 24
Equivalent equations
12. 4共x 7兲; 3x 28 x
Equivalent equations
Equivalent expressions
Solve.
13. x 6 9 3
14. x 5 7 2
15. x 4 12 8
16. x 7 5 12
17. a 7 25 18
18. x 9 3 12
19. x 8 6 14
20. t 5 13 8
21. 24 t 8 32
22. 9 x 3 12
23. 12 x 5 17
24. 17 n 6 23
25. 5 a 12 17
26. 3 17 x 14
27. 8 8 t
28. 7 t 7 0
29. 6x 24 4
30. 8t 40 5
31. 3t 42 14
32. 3x 24 8
33. 7n 35 5
34. 64 2t
35. 0 6x 0
36. 5n 65 13
37. 55 5t
38. x 83 83
39. x 56 56
40. 2x 0 0
42. x 475 475
43. x 390 390
44. n共6兲 42 7
0
Solve.
11
ISBN 0-558-68755-5
41. n共4兲 48 12
32
151
Exercise Set 2.8
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Solve.
45. t 6 2 4
46. 3t 45 15
47. 6x 54 9
48. x 9 15 24
49. 15 x 15
50. 13 x 4 9
51. 21 x 5 26
52. 42 x 42
53. 35 7t
54. 7 t 18 25
55. 17x 68 4
56. 34 x 10 44
58. 48 t共12兲 4
59. 27 x 23 50
60. 135 9t
61. 5x 1 34 7
62. 7x 3 25 4
63. 4t 2 14 3
64. 3t 5 26 7
65. 6a 1 17 3
66. 8a 3 37 5
67. 2x 9 23 7
68. 3x 5 35 10
69. 2x 1 17 8
70. 4t 3 17 5
71. 8t 3 67 8
72. 7x 4 46 6
73. x 9 15 24
74. x 6 8 14
75. 7 2x 5 6
76. 9 4x 7 4
77. 13 3 2x 5
78. 33 5 4x 7
79. 13 5 x 8
80. 12 7 x 5
5
57. 18 t 160 178
15
Solve.
81.
DW To solve 5x 13, Eva decides to add 5 to both
sides of the equation. Is there anything wrong with
her doing this? Why or why not?
82.
DW Gary decides to solve x 9 5 by adding 5 to
both sides of the equation. Is there anything wrong
with his doing this? Why or why not?
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
152
SKILL MAINTENANCE
i
VOCABULARY REINFORCEMENT
In each of Exercises 83–90, fill in the blank with the correct term from the given list. Some of the choices may not
be used and some may be used more than once.
83. A(n)
polygon
is a closed geometric figure. [2.7b]
similar
84. Terms are
factor(s). [2.7a]
if they have the same variable
factors
85. Numbers we multiply together are called
[1.5a]
86. Equations are
solutions. [2.8a]
equivalent
if they have the same
87. The result of an addition is a(n)
variable
88. A(n)
numbers. [2.6a]
.
sum
absolute value
substitute
opposite
polygon
constant
perimeter
variable
equivalent
factors
similar
addends
sum
evaluate
product
. [1.2a]
is a letter that can stand for various
89. The absolute value of a number is its distance from zero
on a number line. [2.1c]
substitute
90. We
a number. [2.6a]
for a variable when we replace it with
SYNTHESIS
91.
DW Explain how equivalent expressions can be used
92.
to write equivalent equations.
DW To solve 2x 8 24, Wilma divides both sides
by 2. Can this first step lead to a solution? Why or
why not?
Solve.
93. 2x 7x 40 8
96. 共9兲2 23t 共3 6 1兲t
99.
ISBN 0-558-68755-5
102.
3
94. 9 x 5 23 19
95. 17 32 4 t 52 29
97. 共7兲2 5 t 43 20
98.
x 共19兲3 183 1027
100.
232 x 222 45
248 24 32x 7
103.
529 143x 1902 17
101.
共42兲3 142t
353 125t
378
343
153
Exercise Set 2.8
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
2
Summary and Review
The review that follows is meant to prepare you for a chapter exam. It consists of three parts. The first part, Concept
Reinforcement, is designed to increase understanding of the concepts through true/false exercises. The second part is
a list of important properties and formulas. The third part is the Review Exercises. These provide practice exercises for
the exam, together with references to section objectives so you can go back and review. Before beginning, stop and
look back over the skills you have obtained. What skills in mathematics do you have now that you did not have before
studying this chapter?
i CONCEPT REINFORCEMENT
Determine whether the statement is true or false. Answers are given at the back of the book.
1. The absolute value of a number is always nonnegative. True
2. The opposite of the opposite of a number is the original number. True
3. The product of an even number of negative numbers is positive. True
4. The expression 2共x 3兲 is equivalent to the expression 2 x 3. False
5. 3 x 4x and 5x 3 are equivalent equations. False
6. Collecting like terms is based on the distributive law. True
IMPORTANT PROPERTIES AND FORMULAS
For any integers a, b, and c :
a 共a兲 0; a b a 共b兲;
a 0 0; a共b c兲 ab ac
Perimeter of a Rectangle:
P 2l 2w, or
Perimeter of a Square:
P 4s
P 2共l w兲
The Addition Principle:
a b is equivalent to
The Division Principle:
For c 0, a b is equivalent to
a c b c.
b
a
.
c
c
Review Exercises
1. Tell which integers correspond to this situation: [2.1a]
Bonnie has $527 in her campus account and Roger is
$53 in debt.
527; 53
Use either or for
2. 0
5
to form a true statement. [2.1b]
3. 7
6
4. 4
19
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
154
Find the absolute value. [2.1c]
5. 兩39兩 39
26. 7 12 共3兲 4 25
6. 兩23兩 23
27. 共3兲 兩4 32兩 5 20
7. 兩0兩 0
8. Find x when x 72. [2.1d] 72
28. Evaluate 3a b for a 4 and b 5.
[2.6a] 7
9. Find 共x兲 when x 59. [2.1d] 59
29. Evaluate
x
x x
,
, and for x 30 and y 5.
y y
y
[2.6a] 6, 6, 6
Compute and simplify. [2.2a], [2.3a], [2.4a, b], [2.5a, b]
10. 14 5 9
12. 14 共8兲 6
11. 5 共6兲 11
13. 0 共24兲 24
Use the distributive law to write an equivalent expression.
[2.6b]
30. 4共5x 9兲
31. 3共2a 4b 5兲
20x 36
14. 17 29 12
6a 12b 15
15. 9 共14兲 23
Combine like terms. [2.7a]
32. 5a 12a 17a
16. 8 共7兲 1
17. 3 共10兲 7
18. 3 7 共8兲 4
19. 8 共9兲 7 2 12
20. 23 共4兲 92
21. 7共12兲 84
33. 7x 13x 6x
34. 9m 14 12m 8 3m 6
35. Find the perimeter of a rectangular frame that is 8 in. by
10 in. [2.7b] 36 in.
22. 2共4兲 共5兲 共1兲 40
ISBN 0-558-68755-5
24.
55
5
11
23. 15 共5兲 3
25.
0
7
36. Find the perimeter of a square pane of glass that is
25 cm on each side. [2.7b] 100 cm
0
155
Summary and Review: Chapter 2
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Solve.
[2.8a, b, c, d]
43.
37. x 9 17 8
DW Explain the difference between equivalent
expressions and equivalent equations.
[2.8a]
Equivalent expressions are expressions that have the same
value when evaluated for various replacements of the
variable(s). Equivalent equations are equations that have
the same solution(s).
44.
DW Is a number’s absolute value ever less than the
number itself? Why or why not? [2.1c]
A number’s absolute value is the number itself if the
number is nonnegative, and the opposite of the number if
the number is negative. In neither case is the result less
than the number itself, so “no,”a number’s absolute value is
never less than the number itself.
38. 4t 36 9
SYNTHESIS
45.
DW A classmate insists on reading x as “negative x.”
When asked why, the response is “because x is
negative.” What mistake is this student making? [2.1d]
The notation “x” means “the opposite of x.” If x is a
negative number, then x is a positive number. For
example, if x 2, then x 2.
39. 13 x 13
46.
DW Are 共a b兲2 and 共b a兲2 equivalent for all choices
of a and b? Why or why not? Experiment with different
replacements for a and b. [2.6a]
The expressions 共a b兲2 and 共b a兲2 are equivalent
for all choices of a and b because a b and b a are
opposites. When opposites are raised to an even power,
the results are the same.
40. 56 6x 10 11
Simplify.
[2.5b]
47.
87 3 29 3 共6兲6 1957 662,582
48.
1969 共8兲5 17 153 88,174
49.
113 173
240
15 83 507
41. x 3 12 15
42. 18 4 2x 7
50. For what values of x will 8 x 3 be negative? [2.6a]
x 2
51. For what values of x is 兩x兩 x? [2.1b, c] x 0
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
156
2
Chapter Test
For Extra Help
Work It Out!
Chapter Test Video
on CD
1. Tell which integers correspond to this situation: The
Tee Shop sold 542 fewer muscle shirts than expected in
January and 307 more than expected in February.
2. Use either or for
3. Find the absolute value: 兩739兩.
4. Find 共x兲 when x 19.
14
to form a true statement.
21
Compute and simplify.
5. 6 共17兲
6. 9 共12兲
8. 0 12
9. 7 22
10. 5 19
11. 8 共27兲
12. 31 共3兲 5 9
13. 共4兲3
14. 27共10兲
15. 9 0
16. 72 共9兲
18. 8 2 2 32
19. 29 共3 5兲2
17.
ISBN 0-558-68755-5
7. 8 17
56
7
157
Test: Chapter 2
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Africa
lan
At
Antarctica
Ocean
ian
South
America
cean
t ic O
d
In
20. Antarctica Highs and Lows. The continent of
Antarctica, which lies in the southern hemisphere,
experiences winter in July. The average high
temperature is 67
F and the average low temperature
is 81
F. How much higher is the average high than the
average low?
Pa c
Source: National Climatic Data Center
ific
O
ce
Australia
an
ab
for a 8 and b 10.
6
21. Jeannie rewound a tape in her video camera from the
8 minute mark to the 15 minute mark. How many
minutes of tape were rewound?
22. Evaluate
23. Use the distributive law to write an equivalent
expression.
7共2x 3y 1兲
24. Combine like terms.
9x 14 5x 3
Solve.
25. 7x 35
26. a 9 3
SYNTHESIS
27. Monty plans to attach trim around the doorway and along the base of the walls in a 12-ft by 14-ft room. If the doorway is
3 ft by 7 ft, how many feet of trim are needed? (Only three sides of a doorway get trim.)
Simplify.
30.
49 143 74 19262 62
29. 15x 3共2x 7兲 9共4 5x兲
31.
3487 16 4 4 28 144
158
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
28. 9 5关x 2共3 4x兲兴 14
CHAPTERS
1––2
Cumulative Review
1. Write standard notation for the number written in
words in the following sentence: In 2003 there were
about one hundred eighty-one million, five hundred
ninety-nine thousand, nine hundred telephone lines in
use in the United States. [1.1c] 181,599,900
2. Write a word name for 5,380,001,437.
[1.1c] Five billion, three hundred eighty million, one
thousand, four hundred thirty-seven
Add.
3.
1 5,8 9 2
2,9 3 5
4.
[1.2b] 18,827
7 9 8 9
7 8 9
7 9
[1.2b] 8857
Subtract.
5.
8 2 7 6
4 3 0
6.
[1.3d] 7846
3 0 0 6
5 7 8
[1.3d] 2428
Multiply.
7.
6 2 1
2 7
8.
[1.5a] 16,767
2 5 0 5
3 3 0 0
9. 43 共8兲
10. 12共6兲
[2.4a] 344
[2.4a] 72
[1.5a] 8,266,500
Divide.
11. 6 3 兲 6 5 5 2
12. 6 2 兲 3 8 4 4
[1.6c] 104
13. 0 共67兲
14. 60 共12兲
[2.5a] 0
[2.5a] 5
[1.6c] 62
15. Round 427,931 to the nearest thousand.
16. Round 5309 to the nearest hundred.
[1.4a] 428,000
[1.4a] 5300
Estimate each sum or product by rounding to the nearest hundred. Show your work.
17.
7 4 9,5 5 9
3 0 1,3 6 2
18.
[1.4b] 749,600 301,400 1,051,000
19. Use or for
ISBN 0-558-68755-5
26
to form a true sentence:
19.
7 4 9
5 3 1
[1.5b] 700 500 350,000
20. Find the absolute value: 兩279兩.
[2.1c] 279
[2.1b] 159
Cumulative Review: Chapters 1–2
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
Simplify.
21. 35 25 5 2 3
22. 兵17 关8 共5 2 2兲兴其 共3 12 6兲
[1.9c], [2.5b] 36
[1.9d], [2.5b] 2
23. 10 1共5兲 62
24. 53
[1.9c], [2.5b] 86
25. Evaluate
[1.9b] 125
xy
for x 11 and y 4.
5
26. Evaluate 7x 2 for x 2.
[2.6a] 28
[2.6a] 3
Use the distributive law to write an equivalent expression.
27. 2共x 5兲
28. 6共3x 2y 4兲
[2.6b] 2x 10
[2.6b] 18x 12y 24
Simplify.
29. 12 共14兲
30. 共3兲 共10兲
[2.2a] 26
[2.4a] 30
33. 12 共25兲
34. 共2兲 共3兲 共5兲
[2.3a] 13
[2.4a] 30
31. 23 38
[2.3a] 15
35. 3 共8兲 2 共3兲
32. 64 共2兲
[2.5a] 32
36. 16 2共8兲 7
[2.3a] 16
[2.5b] 57
39. 6 x 9
40. 39 4x 7
[2.8a] 15
[2.8d] 8
Solve.
37. x 8 35
38. 12t 36
[1.7b], [2.8a] 27
[2.8b] 3
Solve.
41. In the movie Little Big Man, Dustin Hoffman plays a
character who ages from 17 to 121. This represents the
greatest age range depicted by one actor in one film.
How many years did Hoffman’s character age?
Source: Guinness Book of World Records
[1.8a] 104 yr
42. The ten largest hotels in the United States are in Las
Vegas. Of these, the four largest are the MGM Grand, the
Luxor, the Excalibur, and the Circus Circus. These have
5034 rooms, 4408 rooms, 4008 rooms, and 3770 rooms,
respectively. What is the total number of rooms in these
four hotels?
Source: http://govegas.about.com/cs/hotels/tp/largesthotels.htm
[1.8a] 17,220 rooms
43. Amanda is offered a part-time job paying $4940 a year.
How much is each weekly paycheck?
[1.8a] $95
44. Eastside Appliance sells a refrigerator for $600 and $30
tax with no delivery charge. Westside Appliance sells the
same model for $560 and $28 tax plus a $25 delivery
charge. Which is the better buy?
[1.8a] Westside Appliance
45. Write an equivalent expression by combining like terms: 7x 9 3x 5.
[2.7a] 10x 14
SYNTHESIS
47. Simplify: a 兵3a 关4a 共2a 4a兲兴其. [2.5b] 4a
48.
Simplify: 37 64 42 2 共73 共4兲5 兲.
[2.5b] 1071
49. Find two solutions of 5兩x兩 2 13. [2.8d] 3
160
CHAPTER 2: Introduction to Integers
and Algebraic Expressions
Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.
ISBN 0-558-68755-5
46. A soft-drink distributor has 166 loose cans of cola. The
distributor wishes to form as many 24-can cases as
possible and then, with any remaining cans, as many
six-packs as possible. How many cases will be filled?
How many six-packs? How many loose cans will
remain? [1.8a] Cases: 6; six-packs: 3; loose cans: 4