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chapter
2
INTEGERS AND
INTRODUCTION
INTRODUCTION
TO
ALGEBRA
© 2010 McGraw-Hill Companies
CHAPTER 2 OUTLINE
Section 2.1
Introduction to Integers page 121
Section 2.2
Addition of Integers page 132
Section 2.3
Subtraction of Integers page 139
Section 2.4
Multiplication of Integers page 145
Section 2.5
Division of Integers page 151
Section 2.6
Introduction to Algebra: Variables and
Expressions page 158
Section 2.7
Evaluating Algebraic Expressions page 167
Section 2.8
Simplifying Algebraic Expressions page 177
Section 2.9
Introduction to Linear Equations page 186
Section 2.10
The Addition Property of Equality page 192
2
Of all the sciences, meteorology may be both
the least precise and the most talked about.
Meteorologists study the weather and climate. Together with geologists, they provide
us with many predictions that play a part
in our everyday decisions. Among the things
we might look for are temperatures, rainfall,
water level, and tide level.
On a day with unusual weather, we often
become curious about the record for that
day or even the all-time record. Here are a
few of those records:
• The record high temperature for the United
States is 134F in 1913 in Death Valley,
California.
• The record low temperature for the U.S.
is –80F in 1971 in Prospect Creek Camp,
Alaska.
• The greatest one day temperature drop in
the U.S. happened on Christmas Eve, 1924,
in Montana. The temperature went from
63F during the day to –21F at night.
The use of negative numbers for temperatures below zero is common for such numbers, but there are many other applications.
For example, the tide range in Delaware
varies between 10 feet and 5 feet. We
examine these and several other climaterelated applications in Sections 2.1, 2.2, 2.3,
and 2.7.
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Pretest Chapter 2
Section
Date
This pretest will provide a preview of the types of exercises you will encounter in each section of this chapter. The answers for these exercises can be found in the back of the text. If
you are working on your own or are ahead of the class, this pretest can help you identify the
sections in which you should focus more of your time.
ANSWERS
[2.1]
1. See exercise
1. 6, 8, 4, 2, 10
2. 4, 2, 1, 0, 1, 5
3. Max: 7; Min: 5
4. 5
Represent the integers on the number line shown.
8
6
4
2
0
2
4
6
8
10
2. Place the following data set in ascending order: 5, 2, 4, 0, 1, 1.
5. 6
3. Determine the maximum and minimum of the following data set: 4, 1, 5,
6. 6
7, 3, 2.
7. 6
Evaluate:
8. 6
4. ⏐5⏐
5. ⏐6⏐
6. ⏐11 5⏐
7. ⏐11⏐⏐5⏐
8. ⏐4 5⏐⏐6 3⏐
9. 16
10.
23
11.
x8
12.
w
—
17x
13.
No
14.
Yes
11. 8 less than x
15.
10
12. the quotient when w is divided by the product of x and 17
16.
1
Identify which are expressions and which are not.
17.
5
18.
3
19.
19
20.
12
21.
0
22. 21
23.
7
24.
3
25.
55
26.
7
27.
8w 2t
28.
a2 4a 3
Find the opposite of each integer.
9. 16
Write each of the phrases using symbols.
13. 7x 5 11
[2.2 to 2.5]
[2.7]
14. 3x 2(x 1)
Perform the indicated operations.
15. 7 (3)
16. 8 (9)
17. (3) (2)
18. 8 11
19. 8 11
20. 9 (3)
21. 6 (6)
22. (7)(3)
23.
27 6
3
Evaluate each expression.
24. 5 42 3 6
25. (45 3 5) 52
26. If x 2, y 7, and w 4, evaluate the expression
[2.8]
Combine like terms.
27. 5w 2t 3w 2t
120
x2y
.
w
28. 4a2 3a 5 7a 2 5a2
© 2010 McGraw-Hill Companies
[2.6]
10. 23
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2.1
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Introduction to Integers
2.1 OBJECTIVES
1.
2.
3.
4.
5.
Represent integers on a number line
Place a set of integers in ascending order
Determine the extreme values of a data set
Find the opposite of a given integer
Evaluate expressions involving absolute value
When numbers are used to represent physical quantities (altitudes, temperatures, and
amounts of money are examples), it may be necessary to distinguish between positive and
negative quantities. The symbols and are used for this purpose. For instance, the altitude of Mount Whitney is 14,495 ft above sea level (14,495 ft).
14,495 ft
Mount Whitney
The altitude of Death Valley is 282 ft below sea level (282 ft).
282 ft
Death Valley
© 2010 McGraw-Hill Companies
On a given day the temperature in Chicago might be 10F below zero (10F).
110
100
90
8
7
6
5
4
3
2
1
0
–10
–20
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INTEGERS AND INTRODUCTION TO ALGEBRA
An account could show a gain of $100 (100) or a loss of $100 (100).
These numbers suggest the need to extend the whole numbers to include both positive
numbers (like 100) and negative numbers (like 282).
To represent the negative numbers, we extend the number line to the left of zero and
name equally spaced points.
Numbers corresponding to points to the right of zero are positive numbers. They are
written with a positive () sign or with no sign at all.
6 and 9 are positive numbers
Numbers corresponding to points to the left of zero are negative numbers. They are always written with a negative () sign.
3 and 20 are negative numbers
Read “negative 3.”
The positive and negative numbers, as well as zero, are called the real numbers.
Here is the number line extended to include positive and negative numbers, and zero.
NOTE On the number line, we
Zero is neither positive
nor negative
call zero the origin.
3 2 1
0
Negative numbers
to hold a collection of numbers.
We call the collection a set. The
dots are called ellipses and
indicate that the pattern
continues.
OBJECTIVE 1
2
3
Positive numbers
The numbers used to name the points shown on the number line are called the integers. The
integers consist of the natural numbers, their negatives, and the number zero. We can represent the set of integers by
{. . . , 3, 2, 1, 0, 1, 2, 3, . . .}
Example 1 Representing Integers on the Number Line
Represent the integers on the number line shown.
3, 12, 8, 15, 7
12
7 3
15 10 5
8
0
5
15
10
15
© 2010 McGraw-Hill Companies
NOTE Braces { and } are used
1
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INTRODUCTION TO INTEGERS
SECTION 2.1
123
CHECK YOURSELF 1
Represent the integers on a number line.
1, 9, 4, 11, 8, 20
15 10 5
0
5
10
15
20
The set of numbers on the number line is ordered. The numbers get smaller moving to
the left on the number line and larger moving to the right.
4 3 2
1
0
1
2
3
4
When a group of numbers is written from smallest to largest, the numbers are said to be in
ascending order.
OBJECTIVE 2
Example 2 Ordering Integers
Place each group of numbers in ascending order.
(a) 9, 5, 8, 3, 7
From smallest to largest, the numbers are
8, 5, 3, 7, 9
Note that this is the order in which the numbers appear on a number line
as we move from left to right.
(b) 3, 2, 18, 20, 13
From smallest to largest, the numbers are
20, 13, 2, 3, 18
CHECK YOURSELF 2
Place each group of numbers in ascending order.
(a) 12, 13, 15, 2, 8, 3
(b) 3, 6, 9, 3, 8
© 2010 McGraw-Hill Companies
The least and greatest numbers in a group are called the extreme values. The least
number is called the minimum, and the greatest number is called the maximum.
OBJECTIVE 3
Example 3 Labeling Extreme Values
For each group of numbers, determine the minimum and maximum values.
(a) 9, 5, 8, 3, 7
From our previous ordering of these numbers, we see that 8, the least number, is the minimum, and 9, the greatest number, is the maximum.
(b) 3, 2, 18, 20, 13
20 is the minimum and 18 is the maximum.
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CHECK YOURSELF 3
For each group of numbers, determine the minimum and maximum values.
(a) 12, 13, 15, 2, 8, 3
(b) 3, 6, 9, 3, 8
Each point on the number line corresponds to a real number.There are more points on the
number line than integers. The real numbers include decimals, fractions, and other numbers.
Example 4 Identifying Real Numbers that Are Integers
2
Which of the real numbers, (a) 145, (b) 28, (c) 0.35, and (d) , are also integers?
3
(a) 145 is an integer.
(b) 28 is an integer.
(c) 0.35 is not an integer.
2
(d) is not an integer.
3
CHECK YOURSELF 4
Which of the real numbers are also integers?
23
1,054
0.23
0
500
4
5
Sometimes we refer to the negative of a number as its opposite. For a nonzero number, this
corresponds to a point the same distance from the origin as the given number, but on the
other side of zero. Example 5 illustrates this.
OBJECTIVE 4
Example 5 Find the Opposite of Each Number
(a) 5
(b) 9
The opposite of 5 is 5.
The opposite of 9 is 9.
CHECK YOURSELF 5
Find the opposite of each number.
(b) 12
(a) 17
Definition:
Absolute Value
The absolute value of a number represents the distance of the point named by
the number from the origin on the number line.
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INTRODUCTION TO INTEGERS
SECTION 2.1
125
Because we think of distance as a positive quantity (or as zero), the absolute value of
a number is never negative.
5 units
5
5 units
0
5
The absolute value of 5 is 5. The absolute value of 5 is also 5.
As a consequence of the definition, the absolute value of a positive number or zero is
itself. The absolute value of a negative number is its opposite.
In symbols, we write
5 5
and
Read “the absolute
value of 5.”
5 5
Read “the absolute
value of negative 5.”
The absolute value of a number does not depend on whether the number is to the right
or to the left of the origin, but on its distance from the origin.
OBJECTIVE 5
Example 6 Simplifying Absolute Value Expressions
(a) ⏐7⏐ 7
(b) ⏐7⏐ 7
(c) ⏐7⏐ 7
This is the negative, or opposite, of the absolute value of negative 7.
CHECK YOURSELF 6
Evaluate.
(a) ⏐8⏐
(b) ⏐8⏐
(c) ⏐8⏐
To determine the order of operation for an expression that includes absolute values,
note that the absolute value bars are treated as a grouping symbol.
Example 7 Adding or Subtracting Absolute Values
© 2010 McGraw-Hill Companies
(a) ⏐10⏐⏐10⏐ 10 10 20
(b) ⏐8 3⏐⏐5⏐ 5
Absolute value bars, like parentheses, serve as a set of
grouping symbols, so do the operation inside first.
(c) ⏐8⏐⏐3⏐ 8 3 5
Evaluate the absolute values, then subtract.
CHECK YOURSELF 7
Evaluate.
(a) ⏐9⏐⏐4⏐
(b) ⏐9 4⏐
(c) ⏐9⏐⏐4⏐
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INTEGERS AND INTRODUCTION TO ALGEBRA
READING YOUR TEXT
The following fill-in-the-blank exercises are designed to assure that you understand
the key vocabulary used in this section. Each sentence comes directly from the
section. You will find the correct answers in Appendix C.
Section 2.1
(a) When numbers are used to represent physical quantities, it may be necessary
to distinguish between positive and
quantities.
(b) Numbers that correspond to points to the
are negative numbers.
of zero
(c) When a set of numbers is written from smallest to largest, the numbers are
said to be in
order.
(d) The absolute value of a number depends on its
the origin.
from
CHECK YOURSELF ANSWERS
1.
119
20 15 10 5
1
0
4
8
5
20
10
15
20
2. (a) 13, 8, 3, 2, 12, 15
(b) 9, 3, 3, 6, 8
3. (a) minimum is 13; maximum is 15; (b) minimum is 9; maximum is 8
4. 23, 1054, 0, and 500
5. (a) 17; (b) 12
6. (a) 8; (b) 8; (c) 8.
7. (a) 13; (b) 5; (c) 5
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Exercises
2.1
Represent each quantity with an integer.
• Practice Problems
• Self-Tests
• NetTutor
1. An altitude of 400 ft above sea level
• e-Professors
• Videos
Name
2. An altitude of 80 ft below sea level
Section
Date
3. A loss of $200
ANSWERS
4. A profit of $400
1. 400
5. A decrease in population of 25,000
2. 80
6. An increase in population of 12,500
3. 200
[Objective 1]
Represent the integers on the number lines shown.
7. 5, 15, 18, 8, 3
15
20
8
4. 400
35
10
0
5. 25,000
18
10
6. 12,500
20
7. See exercise
8. 18, 4, 5, 13, 9
18
20
5
10
4
9 13
0
10
8. See exercise
20
9. 5, 175, 234
Which numbers in the sets are integers?
9.
2
5, , 175, 234, 0.64
9
10.
3
45, 0.35, , 700, 26
5
[Objective 2]
Place each group of numbers in ascending order.
10.
45, 700, 26
11.
7, 5, 1, 0, 2, 3, 8
12.
8, 2, 1, 0, 1, 6, 7
13.
11, 6, 2, 1, 4, 5, 9
18, 15, 11, 5,
11. 3, 5, 2, 0, 7, 1, 8
12. 2, 7, 1, 8, 6, 1, 0
13. 9, 2, 11, 4, 6, 1, 5
14. 23, 18, 5, 11, 15, 14, 20
15.
16. 12, 13, 14, 14, 15, 15
16. 14, 15
14. 14, 20, 23
7, 6, 3, 3, 6, 7
15, 14, 13, 12,
© 2010 McGraw-Hill Companies
15. 6, 7, 7, 6, 3, 3
[Objective 3]
For each group of numbers, determine the maximum and minimum values.
17. 5, 6, 0, 10, 3, 15, 1, 8
18. 9, 1, 3, 11, 4, 2, 5, 2
19. 21, 15, 0, 7, 9, 16, 3, 11
20. 22, 0, 22, 31, 18, 5, 3
21. 3, 0, 1, 2, 5, 4, 1
22. 2, 7, 3, 5, 10, 5
17.
Max: 15; Min: 6
18.
Max: 11; Min: 4
19.
Max: 21; Min: 15
20.
Max: 22; Min: 31
21.
Max: 5; Min: 2
22.
Max: 7; Min: 10
SECTION 2.1
127
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ANSWERS
23.
15
24.
18
[Objective 4]
Find the opposite of each number.
25.
11
23. 15
24. 18
26.
34
25. 11
26. 34
27. 19
28. 5
29. 7
30. 54
27.
19
28.
5
29.
7
30.
54
31.
17
32.
28
33.
10
34.
7
35.
3
36.
5
37.
8
38.
13
39.
5
40.
7
41.
18
42.
22
43.
0
44.
0
45.
7
46.
8
47.
7
48.
8
49.
11
50.
11
Label each statement as true or false.
51.
1
52.
5
53. All whole numbers are integers.
53.
True
54.
True
128 SECTION 2.1
31. ⏐17⏐
32. ⏐28⏐
33. ⏐10⏐
34. ⏐7⏐
35. ⏐3⏐
36. ⏐5⏐
37. ⏐8⏐
38. ⏐13⏐
39. ⏐2⏐⏐3⏐
40. ⏐4⏐⏐3⏐
41. ⏐9⏐⏐9⏐
42. ⏐11⏐⏐11⏐
43. ⏐4⏐⏐4⏐
44. ⏐5⏐⏐5⏐
45. ⏐15⏐⏐8⏐
46. ⏐11⏐⏐3⏐
47. ⏐15 8⏐
48. ⏐11 3⏐
49. ⏐9⏐⏐2⏐
50. ⏐7⏐⏐4⏐
51. ⏐8⏐⏐7⏐
52. ⏐9⏐⏐4⏐
54. All nonzero integers are real numbers.
© 2010 McGraw-Hill Companies
[Objective 5]
Evaluate.
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ANSWERS
55. All integers are whole numbers.
56. All real numbers are integers.
57. All negative integers are whole numbers.
55.
False
56.
False
57.
False
58.
True
59.
⏐6⏐ 2 8
60.
⏐8⏐⏐3⏐ 11
61.
6 ⏐2⏐ 8
62.
8 ⏐(3)⏐ 11
63.
5 cm
64.
2.5 cm
65.
50 dollars
66.
200 dollars
67.
10
58. Zero is neither positive nor negative.
Place absolute value bars in the proper location on the left side of the expression so that the
equation is true.
59. (6) 2 8
60. (8) (3) 11
61. 6 (2) 8
62. 8 (3) 11
Represent each quantity with a real number.
63. Science and Medicine
The erosion of 5 centimeters (cm) of topsoil from an
Iowa cornfield.
64. Science and Medicine
The formation of 2.5 cm of new topsoil on the African
savanna.
65. Business and Finance
The withdrawal of $50 from a checking account.
66. Business and Finance
The deposit of $200 in a savings account.
67. Science and Medicine
The temperature change pictured.
© 2010 McGraw-Hill Companies
2
60°F
50°F
–
–
–
–
1:00 P.M.
2:00 P.M.
SECTION 2.1
129
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ANSWERS
68.
10°
69.
20°
70.
30°
71.
90,000,000
72.
60,000,000
68. Science and Medicine
The temperature change indicated.
2
90
80
70
60
50
40
30
20
10
0
–10
–
–30
–40
90
80
40
30
20
10
–20°F
–30°F
–
1:00 P.M.
69. Science and Medicine
2:00 P.M.
The temperature change indicated.
2
90
80
70
60
50
40
30
20
10
0
–10
–
–30
–40
90
80
40
30
20
10
0°F
–30°F
–
1:00 P.M.
70. Science and Medicine
2:00 P.M.
The temperature change indicated.
2
90
80
40
30
20
10
F
–30°F
–
1:00 P.M.
2:00 P.M.
A country exported $90,000,000 more than it imported,
creating a positive trade balance.
71. Business and Finance
A country exported $60,000,000 less than it imported,
creating a negative trade balance.
72. Business and Finance
130 SECTION 2.1
© 2010 McGraw-Hill Companies
90
80
70
60
50
40
30
20
10
0
–10
–
–30
–40
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ANSWERS
For each group of numbers given in exercises 73 to 76, answer questions (a) to (d):
(a)
(b)
(c)
(d)
Which number is smallest?
Which number lies farthest from the origin?
Which number has the largest absolute value?
Which number has the smallest absolute value?
73. 6, 3, 8, 7, 2
74. 8, 3, 5, 4, 9
75. 2, 6, 1, 0, 2, 5
76. 9, 0, 2, 3, 6
73.
6; 8; 8; 2
74.
8; 9; 9; 3
75.
2; 6; 6; 0
76.
9; 9; 9; 0
77.
77. Simplify each of the following:
(7)
((7))
(((7)))
Based on your answers, generalize your results.
Answers
3. 200
1. 400 or (400)
15
7.
10
35
0
5. 25,000
18
10
20
9. 5, 175, 234
7, 5, 1, 0, 2, 3, 8
13. 11, 6, 2, 1, 4, 5, 9
7, 6, 3, 3, 6, 7
17. Max: 15; Min: 6
19. Max: 21; Min: 15
Max: 5; Min: 2
23. 15
25. 11
27. 19
29. 7
17
33. 10
35. 3
37. 8
39. 5
41. 18
43. 0
7
47. 7
49. 11
51. 1
53. True
55. False
57. False
⏐6⏐ 2 8
61. 6 ⏐2⏐ 8
63. 5 cm
65. 50 dollars
10
69. 20
71. 90,000,000
73. 6; 8; 8; 2
2; 6; 6; 0
77.
© 2010 McGraw-Hill Companies
11.
15.
21.
31.
45.
59.
67.
75.
20
8
SECTION 2.1
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2.2
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Addition of Integers
2.2 OBJECTIVES
1. Add two integers with the same sign
2. Add two integers with opposite signs
3. Solve applications involving integers
In Section 2.1 we introduced the idea of negative numbers. Here we examine the four arithmetic operations (addition, subtraction, multiplication, and division) and see how those operations are performed when integers are involved. We start by considering addition.
An application may help. We will represent a gain of money as a positive number and
a loss as a negative number.
If you gain $3 and then gain $4, the result is a gain of $7:
347
If you lose $3 and then lose $4, the result is a loss of $7:
3 (4) 7
If you gain $3 and then lose $4, the result is a loss of $1:
3 (4) 1
If you lose $3 and then gain $4, the result is a gain of $1:
3 4 1
The number line can be used to illustrate the addition of integers. Starting at the origin, we move to the right for positive integers and to the left for negative integers.
OBJECTIVE 1
Example 1 Adding Integers
(a) Add 3 4.
4
3
0
3
7
Start at the origin and move 3 units to the right. Then move 4 more units to the right to find
the sum. From the number line, we see that the sum is
(b) Add (3) (4).
4
7
3
3
0
Start at the origin and move 3 units to the left. Then move 4 more units to the left to find the
sum. From the number line, we see that the sum is
(3) (4) 7
132
© 2010 McGraw-Hill Companies
347
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ADDITION OF INTEGERS
SECTION 2.2
133
CHECK YOURSELF 1
Add.
(a) (4) (5)
(c) (5) (15)
(b) (3) (7)
(d) (5) (3)
You have probably noticed a helpful pattern in the previous example. This pattern will
allow you to do the work mentally without having to use the number line. Look at the
following rule.
Property:
NOTE This means that the sum
of two positive integers is
positive and the sum of two
negative integers is negative.
Adding Integers Case 1: Same Sign
If two integers have the same sign, add their absolute values. Give the result
the sign of the original integers.
We can use the number line to illustrate the addition of two integers. This time the
integers will have different signs.
OBJECTIVE 2
Example 2 Adding Integers
(a) Add 3 (6).
6
3
3
0
3
First move 3 units to the right of the origin. Then move 6 units to the left.
3 (6) 3
(b) Add 4 7.
7
4
© 2010 McGraw-Hill Companies
4
0
3
This time move 4 units to the left of the origin as the first step. Then move 7 units to the
right.
4 7 3
CHECK YOURSELF 2
Add.
(a) 7 (5)
(b) 4 (8)
(c) 1 16
(d) 7 3
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INTEGERS AND INTRODUCTION TO ALGEBRA
You have no doubt noticed that, in adding a positive integer and a negative integer,
sometimes the sum is positive and sometimes it is negative. This depends on which of the
integers has the larger absolute value. This leads us to the second part of our addition rule.
Property:
RECALL We first encountered
absolute values in Section 2.1.
Adding Integers Case 2: Different Signs
If two integers have different signs, subtract their absolute values, the smaller
from the larger. Give the result the sign of the integer with the larger absolute
value.
Example 3 Adding Integers
(a) 7 (19) 12
Because the two integers have different signs, subtract the absolute values (19 7 12).
The sum of 7 and 19 has the sign () of the integer with the larger absolute value, 19.
(b) 13 7 6
Subtract the absolute values (13 7 6). The sum of 13 and 7 has the sign () of the
integer with the larger absolute value, 13.
CHECK YOURSELF 3
Add mentally.
(a) 5 (14)
(c) 8 15
NOTE The opposite of a
number is also called the
additive inverse of that
number.
NOTE 3 and 3 are opposites.
(b) 7 (8)
(d) 7 (8)
In Section 1.2, we discussed the commutative, associative, and additive identity properties. There is another property of addition that we should mention.
Recall that every number has an opposite. It corresponds to a point the same distance
from the origin as the given number but in the opposite direction.
3
3
3
0
3
The additive inverse property states that the sum of any number and its opposite is 0.
Property:
Additive Inverse Property
NOTE Here a represents the
For any number a, there exists a number a such that
opposite of the number a. If a is
positive, a is negative. If a is
negative, a is positive.
a (a) (a) a 0
The sum of any number and its opposite, or additive inverse, is 0.
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The opposite of 9 is 9.
The opposite of 15 is 15.
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ADDITION OF INTEGERS
SECTION 2.2
135
Example 4 Adding Integers
NOTE Later, we will show that
0 0; therefore, the opposite
of 0 is 0.
(a) 9 (9) 0
(b) 15 15 0
CHECK YOURSELF 4
Add.
(a) (17) 17
(b) 12 (12)
When solving an application of integer arithmetic, the first step is to translate the phrase or
statement using integers. Example 5 illustrates this step.
OBJECTIVE 3
Example 5 An Application of the Addition of Integers
Shanique has $250 in her checking account. She writes a check for $120 and makes a
deposit of $90. What is the resulting balance?
First, translate the phrase using integers. Such problems will usually include something that is represented by negative integers and something that is represented by positive
integers. In this case, a check can be represented as a negative integer and a deposit as a
positive integer. We have
250 (120) 90
This expression can now be evaluated.
250 (120) 90
130 90
220
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The resulting balance is $220.
CHECK YOURSELF 5
Translate the problem into an integer expression and then answer the question.
When Kirin awoke, the temperature was twelve degrees below zero, Fahrenheit.
Over the next six hours, the temperature increased by seventeen degrees. What was
the temperature at that time?
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INTEGERS AND INTRODUCTION TO ALGEBRA
READING YOUR TEXT
The following fill-in-the-blank exercises are designed to assure that you understand
the key vocabulary used in this section. Each sentence comes directly from the
section. You will find the correct answers in Appendix C.
Section 2.2
(a) In Section 2.1, we introduced the idea of
(b) To add two numbers with different signs,
numbers.
their absolute values.
(c) The sum of any number and its opposite, or additive
(d) The opposite of zero is
, is 0.
.
CHECK YOURSELF ANSWERS
1. (a) 9; (b) 10; (c) 20; (d) 8
2. (a) 2; (b) 4; (c) 15; (d) 4
3. (a) 9; (b) 15; (c) 7; (d) 1
4. (a) 0; (b) 0
5. 12 17; the temperature was 5F.
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Exercises
[Objectives 1 and 2]
Add.
1. 3 6
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2. 5 9
Name
3. 11 5
4. 8 7
Section
Date
ANSWERS
5. (2) (3)
6. (1) (9)
7. 9 (3)
8. 10 (4)
1. 9
2. 14
3. 16
9. 9 0
10. 15 0
4. 15
5. 5
11. 7 (7)
12. 12 (12)
13. 7 (9) (5) 6
14. (4) 6 (3) 0
15. 7 (3) 5 (11)
16. 6 (13) 16
6. 10
7. 6
9. 9
10.
15
[Objective 3]
In exercises 17 to 22, restate the problem using an expression involving integers and then
answer the question.
11.
0
12.
0
Amir has $100 in his checking account. He writes a
check for $23 and makes a deposit of $51. What is his new balance?
13.
1
14.
1
15.
2
16.
3
17.
$128
18.
$225
19.
120 lb/in.2
17. Business and Finance
Olga has $250 in her checking account. She deposits $52
and then writes a check for $77. What is her new balance?
18. Business and Finance
© 2010 McGraw-Hill Companies
8. 6
A pneumatic actuator is operated by a pressurized air
reservoir. At the beginning of an operator’s shift, the pressure in the reservoir was
126 pounds per square inch (lb/in.2). At the end of each hour, the operator recorded
the reservoir’s change in pressure. The values recorded (in lb/in.2) were a drop of
12, a drop of 7, a rise of 32, a drop of 17, a drop of 15, a rise of 31, a drop of 4, and
a drop of 14. What was the pressure in the tank at the end of the shift?
19. Mechanical Engineering
SECTION 2.2
137
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ANSWERS
20.
15 qt
21.
4°F
22.
3°F
23.
A diesel engine for an industrial shredder has an
18-quart (qt) oil capacity. When a maintenance technician checked the oil, it was
7 qt low. Later that day, she added 4 qt to the engine. What was the oil level after
the 4 qt were added?
20. Mechanical Engineering
The lowest one-day temperature in Helena, Montana,
was 21F at night. The temperature increased by 25 degrees by noon. What was
the temperature at noon?
21. Science and Medicine
2
At 7 A.M., the temperature was 15F. By 1 P.M., the temperature had increased by 18 degrees. What was the temperature at 1 P.M.?
22. Science and Medicine
2
23. In this chapter, it is stated that “every number has an opposite.” The opposite of 9
is 9. This corresponds to the idea of an opposite in English. In English, an opposite is often expressed by a prefix, for example, un- or ir-.
(a) Write the opposite of these words: unmentionable, uninteresting, irredeemable, irregular, uncomfortable.
(b) What is the meaning of these expressions: not uninteresting, not irredeemable,
not irregular, not unmentionable?
(c) Think of other prefixes that negate or change the meaning of a word to its opposite. Make a list of words formed with these prefixes and write a sentence
with three of the words you found. Make a sentence with two words and
phrases from parts (a) and (b).
What is the value of [(5)]? What is the value of (6)? How does
this relate to the given examples? Write a short description about this relationship.
Answers
3. 16
5. 5
7. 6
9. 9
11. 0
17. $128
19. 120 lb/in.2
21. 4F
23.
13. 1
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1. 9
15. 2
138 SECTION 2.2
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Subtraction of Integers
2.3 OBJECTIVES
1. Find the difference of two integers
2. Solve applications involving the subtraction of integers
To begin our discussion of subtraction when integers are involved, we can look back at a
problem using natural numbers. We know that
853
From our work in adding integers, we know that it is also true that
8 (5) 3
Comparing these equations, we see that the results are the same. This leads us to an important pattern. Any subtraction problem can be written as a problem in addition. Subtracting 5 is the same as adding the opposite of 5, or 5. We can write this fact as follows:
8 5 8 (5) 3
This leads us to the following rule for subtracting integers.
Property:
Subtracting Integers
1. To rewrite the subtraction problem as an addition problem:
a. Change the subtraction operation to addition.
b. Replace the integer being subtracted with its opposite.
2. Add the resulting integers as before.
In symbols,
a b a (b)
Example 1 illustrates the use of this property while subtracting.
OBJECTIVE 1
Example 1 Subtracting Integers
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Change the subtraction symbol ()
to an addition symbol ().
(a) 15 7 15 (7)
8
Replace 7 with its opposite, 7.
(b) 9 12 9 (12) 3
(c) 6 7 6 (7) 13
(d) Subtract 5 from 2. We write the statement as 2 5 and proceed as before:
2 5 2 (5) 7
139
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INTEGERS AND INTRODUCTION TO ALGEBRA
CHECK YOURSELF 1
Subtract.
(a) 18 7
(b) 5 13
(c) 7 9
(d) 2 7
The subtraction rule is used in the same way when the integer being subtracted is negative. Change the subtraction to addition. Replace the negative integer being subtracted
with its opposite, which is positive. Example 2 illustrates this principle.
Example 2 Subtracting Integers
Change the subtraction
to addition.
(a) 5 (2) 5 (2) 5 2 7
Replace 2 with its opposite, 2 or 2.
(b) 7 (8) 7 (8) 7 8 15
(c) 9 (5) 9 5 4
(d) Subtract 4 from 5. We write
5 (4) 5 4 1
CHECK YOURSELF 2
Subtract.
(a) 8 (2)
(c) 7 (2)
OBJECTIVE 2
(b) 3 (10)
(d) 7 (7)
Example 3 An Application of the Subtraction of Integers
Susanna’s checking account shows a balance of $285. She has discovered that a deposit for
$47 was accidently recorded as a check for $47. Write an integer expression that represents
the correction on the balance. Then find the corrected balance.
285 (47) (47)
Subtract the check and then add the deposit.
285 (47) (47) 285 47 47 379
The corrected balance is $379.
CHECK YOURSELF 3
It appears that Marshal, a running back, gained 97 yards in the last game. A closer
inspection of the statistics revealed that a 9-yard gain had been recorded as a
9-yard loss. Write an integer expression that represents the corrected yards gained
and then find that number.
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SUBTRACTION OF INTEGERS
SECTION 2.3
141
Using Your Calculator to Add
and Subtract Integers
Your scientific (or graphing) calculator has a key that makes a number negative. This key
is different from the “subtraction” key. The negative key is marked as either or () .
With a scientific calculator, this key is pressed after the number you wish to make negative is
entered. All of the instructions in this section assume that you have a scientific calculator.
Example 4 Entering a Negative Integer into the Calculator
Enter each of the following into your calculator.
(a) 24
24 NOTE The 12 changes between
positive and negative in the
display. The final display is 12,
because there are an even
number of negative signs in
front of the 12.
(b) (((12)))
12 CHECK YOURSELF 4
Enter each number into your calculator.
(a) 36
(b) ((6))
Example 5 Adding Integers
Find the sum for each pair of integers.
(a) 256 (297)
256 297 Your display should read 41.
(b) 312 (569)
312 569 Your display should read 881.
CHECK YOURSELF 5
© 2010 McGraw-Hill Companies
Find the sum for each pair of integers.
(a) 368 547
Example 6 Subtracting Integers
Find the difference for 356 (469).
356 469 Your display should read 113.
(b) 596 (834)
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INTEGERS AND INTRODUCTION TO ALGEBRA
CHECK YOURSELF 6
Find each difference.
(a) 349 (49)
(b) 294 (137)
READING YOUR TEXT
The following fill-in-the-blank exercises are designed to assure that you understand
the key vocabulary used in this section. Each sentence comes directly from the
section. You will find the correct answers in Appendix C.
Section 2.3
(a) Any subtraction problem can be written as a problem in
.
(b) To rewrite a subtraction problem as an addition problem, change the subtraction operation to addition and replace the integer being subtracted with
its
.
(c) The opposite of 2 is
.
(d) The calculator key that makes a number negative is different from the
key.
CHECK YOURSELF ANSWERS
1. (a) 11; (b) 8; (c) 16; (d) 9
2. (a) 10; (b) 13; (c) 5; (d) 14
3. 97 (9) 9 115 yards
4. (a) 36; (b) 6
5. (a) 179; (b) 1,430
6. (a) 398; (b) 157
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Exercises
[Objective 1]
Subtract.
1. 21 13
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2. 36 22
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3. 82 45
Name
4. 103 56
5. 8 10
6. 14 19
7. 24 45
8. 136 352
9. 5 3
10. 15 8
11. 9 14
12. 8 12
13. 5 (11)
14. 7 (5)
15. 7 (12)
16. 3 (10)
17. 36 (24)
18. 28 (11)
19. 19 (27)
20. 11 (16)
Section
ANSWERS
1. 8
2. 14
3. 37
4. 47
[Objective 2]
For exercises 21 to 23, write an integer expression that describes the situation. Then answer
the question.
The temperature at noon on a June day was 82F. It fell
by 12 degrees in the next 4 h. What was the temperature at 4:00 P.M.?
21. Science and Medicine
2
Jason’s checking account shows a balance of $853. He
has discovered that a deposit of $70 was accidently recorded as a check for $70.
What is the corrected balance?
22. Business and Finance
Ylena’s checking account shows a balance of $947. She
has discovered that a check for $86 was recorded as a deposit of $86. What is the
corrected balance?
23. Business and Finance
How long ago was the year 1250 B.C.E.? What year was 3,300 years
ago? Make a number line and locate the following events, cultures, and objects on
it. How long ago was each item in the list? Which two events are the closest to each
other? You may want to learn more about some of the cultures in the list and the
mathematics and science developed by each culture.
24. Technology
© 2010 McGraw-Hill Companies
Date
Inca culture in Peru—1400 A.D.
The Ahmes Papyrus, a mathematical text from Egypt—1650 B.C.E.
Babylonian arithmetic develops the use of a zero symbol—300 B.C.E.
First Olympic Games—776 B.C.E.
Pythagoras of Greece dies—500 B.C.E.
Mayans in Central America independently develop use of zero—500 A.D.
The Chou Pei, a mathematics classic from China—1000 B.C.E.
The Aryabhatiya, a mathematics work from India—499 A.D.
Trigonometry arrives in Europe via the Arabs and India—1464 A.D.
Arabs receive algebra from Greek, Hindu, and Babylonian sources and develop
it into a new systematic form—850 A.D.
Development of calculus in Europe—1670 A.D.
Rise of abstract algebra—1860 A.D.
Growing importance of probability and development of statistics—1902 A.D.
5. 2
6. 5
7. 21
8. 216
9. 8
10.
23
11.
23
12.
20
13.
16
14.
12
15.
19
16.
13
17.
12
18.
17
19.
8
20.
5
21.
82° 12° 70°F
853 (70) 70
22. $993 is the balance
947 86 (86)
23. $775 is the balance
24.
SECTION 2.3
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ANSWERS
25. Complete the following statement: “3 (7) is the same as ____ because . . .”
25.
Write a problem that might be answered by doing this subtraction.
26.
26. Explain the difference between the two phrases: “a number subtracted from 5”
27.
84°F
28.
76°F
29.
15
30.
9
31.
917
32.
1,477
33.
780
34.
779
35.
72
36.
342
37.
690
38.
535
and “a number less than 5.” Use algebra and English to explain the meaning of
these phrases. Write other ways to express subtraction in English. Which ones are
confusing?
The greatest one-day temperature drop in the United
States happened on Christmas Eve, 1924, in Montana. The temperature went from
63F during the day to 21F at night. What was the total temperature drop?
27. Science and Medicine
2
A similar one-day temperature drop happened in Alaska.
The temperature went from 47F during the day to 29F at night. What was the
total temperature drop?
28. Science and Medicine
2
The tide at the mouth of the Delaware River tends to vary
between a maximum of 10 ft and a minimum of 5 ft. What is the difference in
feet between the high tide and the low tide?
29. Science and Medicine
2
The tide at the mouth of the Sacramento River tends to
vary between a maximum of 7 ft and a minimum of 2 ft. What is the difference
in feet between the high tide and the low tide?
30. Science and Medicine
2
Calculator Exercises
31. 789 (128)
32. 910 (567)
33. 349 (431)
34. 412 (367)
35. 47 (25)
36. 123 (219)
37. 234 (456)
38. 412 (123)
Answers
1. 8
3. 37
5. 2
7. 21
9. 8
11. 23
13. 16
15. 19
17. 12
19. 8
21. 82° 12° 70F
23. 947 86 (86); $775 is the balance
25.
27. 84°F
29. 15
144 SECTION 2.3
31. 917
33. 780
35. 72
37. 690
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Use your calculator to perform the following operations.
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Multiplication of Integers
2.4 OBJECTIVES
1. Find the product of two or more integers
2. Use the order of operations with integers
When you first considered multiplication in arithmetic, it was thought of as repeated
addition. Now we look at what our work with the addition of integers can tell us about multiplication when integers are involved. For example,
3 4 4 4 4 12
We interpret multiplication as repeated
addition to find the product, 12.
Now, consider the product (3)(4):
(3)(4) (4) (4) (4) 12
Looking at this product suggests the first portion of our rule for multiplying integers.
The product of a positive integer and a negative integer is negative.
Property: Multiplying Integers Case 1: Different Signs
The product of two integers with different signs is negative.
To use this rule in multiplying two integers with different signs, multiply their absolute
values and attach a negative sign.
OBJECTIVE 1
Example 1 Multiplying Integers
Multiply.
(a) (5)(6) 30
© 2010 McGraw-Hill Companies
The product is negative.
(b) (10)(10) 100
(c) (8)(12) 96
CHECK YOURSELF 1
Multiply.
(a) (7)(5)
(b) (12)(9)
(c) (15)(8)
145
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The product of two negative integers is harder to visualize. The following pattern may
help you see how we can determine the sign of the product.
(3)(2) 6
(2)(2) 4
(1)(2) 2
NOTE This number is decreasing
by 1.
Do you see that the product is
increasing by 2 each time as you go down?
(0)(2) 0
of 2.
(1)(2) 2
What should the product (2)(2) be? Continuing the pattern shown, we see that
(2)(2) 4
This suggests that the product of two negative integers is positive, which is the case. We can
extend our multiplication rule.
Property: Multiplying Integers Case 2: Same Sign
The product of two integers with the same sign is positive.
Example 2 Multiplying Integers
Multiply.
(a) 9 # 7 63
The product of two positive numbers
(same sign, ) is positive.
(b) (8)(5) 40
The product of two negative numbers
(same sign, ) is positive.
CHECK YOURSELF 2
Multiply.
(a) 10 12
(b) (8)(9)
The multiplicative identity property and multiplicative property of zero studied in Section 1.5 can be applied to integers, as illustrated in Example 3.
Example 3 Multiplying Integers by One and Zero
Find each product.
(a) (1)(7) 7
(b) (15)(1) 15
(c) (7)(0) 0
(d) 0 # 12 0
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NOTE (1)(2) is the opposite
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SECTION 2.4
147
CHECK YOURSELF 3
Multiply.
(a) (10)(1)
(b) (0)(17)
We can now extend the rules for the order of operations learned in Section 1.8 to
simplify expressions containing integers. First, we will work with integers raised to a
power.
OBJECTIVE 2
Example 4 Integers with Exponents
Evaluate each expression.
NOTE In part (b) of Example 4,
we have a negative integer
raised to a power.
In part (c), only the 3 is
raised to a power. We have
the opposite of 3 squared.
(a) (3)2 (3)(3) 9
(b) (3)3 (3)(3)(3) 27
(c) 32 (3 3) 9
Note that the negative is not squared.
CHECK YOURSELF 4
Evaluate each expression.
(a) (4)2
(b) (4)3
(c) 42
In Example 5 we will apply the order of operations.
Example 5 Using Order of Operations with Integers
© 2010 McGraw-Hill Companies
Evaluate each expression.
(a) 7(9 12)
7(3) 21
Evaluate inside the parentheses first.
(b) (8)(7) 40
56 40
16
Multiply first, then subtract.
(c) (5)2 3
(5)(5) 3
25 3
22
Evaluate the power first.
(d) 52 3
25 3
28
Note that 52 25. The power applies only to the 5.
Note that (5)2 (5)(5)
25
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CHECK YOURSELF 5
Evaluate each expression.
(a) 8(9 7)
(c) (4)2 (4)
(b) (3)(5) 7
(d) 42 (4)
READING YOUR TEXT
The following fill-in-the-blank exercises are designed to assure that you understand
the key vocabulary used in this section. Each sentence comes directly from the
section. You will find the correct answers in Appendix C.
Section 2.4
(a) The product of two integers with different signs is
.
(b) The product of two integers with the same sign is
.
(c) Given the expression 32, the
is not squared.
(d) The rules for order of operations were learned in Section
.
CHECK YOURSELF ANSWERS
1. (a) 35; (b) 108; (c) 120
2. (a) 120; (b) 72
3. (a) 10; (b) 0
4. (a) 16; (b) 64; (c) 16
5. (a) 16; (b) 22; (c) 20; (d) 12
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[Objective 1]
Multiply.
1. 4 10
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2. 3 14
Name
3. (5)(12)
4. (10)(2)
Section
5. (8)(9)
6. (12)(3)
ANSWERS
7. (8)(7)
8. (9)(8)
Date
1. 40
2. 42
9. (5)(12)
11. (0)(18)
10. (7)(3)
12. (17)(0)
3. 60
4. 20
5. 72
13. (15)(0)
14. (0)(25)
6. 36
7. 56
[Objective 2]
Do the indicated operations. Remember the rules for the order of operations.
© 2010 McGraw-Hill Companies
15. 5(7 2)
16. 7(8 5)
17. 2(5 8)
18. 6(14 16)
19. 3(9 7)
20. 6(12 9)
21. 3(2 5)
22. 2(7 3)
23. (2)(3) 5
24. (6)(8) 27
25. 4(7) 5
26. (3)(9) 11
27. (5)(2) 12
28. (7)(3) 25
29. (3)(7) 20
30. (2)(6) 8
31. 4 (3)(6)
32. 5 (2)(3)
33. 7 (4)(2)
34. 9 (2)(7)
35. (7)2 17
36. (6)2 20
8. 72
9. 60
10.
21
11.
0
12.
0
13.
0
14.
0
15.
25
16.
21
17.
6
18.
12
19.
6
20.
18
21.
21
22.
20
23.
11
24.
75
25.
33
26. 16
27. 2
28. 4
29. 1
30. 4
31. 22
32. 11
33. 1
34. 5
35. 32
36. 16
SECTION 2.4
149
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ANSWERS
37. 43
37. (5)2 18
38. (2)2 10
39. 62 4
40. 52 3
41. (4)2 (2)(5)
42. (3)3 (8)(2)
43. (8)2 52
44. (6)2 42
45. (6)2 (3)2
46. (8)2 (4)2
47. 82 52
48. 62 32
50. 92 (6)2
38. 14
39. 40
40.
28
41.
6
42.
43
43.
39
44.
20
45.
27
46.
48
49. 82 (5)2
47.
89
51. Business and Finance
48.
45
49.
89
50.
117
51.
$17,086
52.
$540
53.
22F
Stores occasionally sell products at a loss in order to draw
in customers or to reward good customers. The theory is that customers will buy
other products along with the discounted item and the store will ultimately profit.
Beguhn Industries sells five different products. The company makes $18 on
each product-A item sold, loses $4 on product-B items, earns $11 on product C,
makes $38 on product D, and loses $15 on product E.
One month, Beguhn Industries sold 127 units of product A, 273 units of product B, 201 units of product C, 377 units of product D, and 43 units of product E.
What was their profit or loss that month?
In Atlantic City, Nick played the slot machines for 12 h. He lost $45 an
hour. Use integers to represent the change in Nick’s financial status at the end of
the 12 h.
52. Statistics
The temperature is 6F at 5:00 in the evening. If the temperature drops 2 degrees every hour, what is the temperature at 1:00 A.M.?
53. Science and Medicine
2
1. 40
3. 60
5. 72
15. 25
17. 6
19. 6
27. 2
29. 1
31. 22
39. 40
41. 6
43. 39
51. $17,086
53. 22F
150 SECTION 2.4
7. 56
9. 60
11. 0
13. 0
21. 21
23. 11
25. 33
33. 1
35. 32
37. 43
45. 27
47. 89
49. 89
© 2010 McGraw-Hill Companies
Answers
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2.5
Page 151
Division of Integers
2.5 OBJECTIVES
1. Find the quotient of two integers
2. Use the order of operations with integers
You know from your work in arithmetic that multiplication and division are related operations. We can use that fact, and our work of Section 2.4, to determine rules for the division
of integers. Every division problem can be stated as an equivalent multiplication problem.
For instance,
15
3
5
because
15 5 # 3
24
4
6
because
24 (6)(4)
30
6
5
because
30 (5)(6)
These examples illustrate that because the two operations are related, the rule of signs
that we stated in Section 2.4 for multiplication is also true for division.
Property:
Dividing Integers
1. The quotient of two integers with different signs is negative.
2. The quotient of two integers with the same sign is positive.
Again, the rule is easy to use. To divide two integers, divide their absolute values. Then
attach the proper sign according to the rule.
OBJECTIVE 1
Example 1 Dividing Integers
Divide.
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(a)
(b)
(c)
(d)
Positive
Positive
Negative
Negative
Negative
Positive
Positive
Negative
28
4
7
Positive
36
9
4
Positive
42
6
7
Negative
75
25
3
Negative
151
152
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INTEGERS AND INTRODUCTION TO ALGEBRA
CHECK YOURSELF 1
Divide.
55
11
48
(c)
8
80
20
144
(d)
12
(a)
(b)
As discussed in Section 1.6, we must be very careful when 0 is involved in a division
problem. Remember that 0 divided by any nonzero number is just 0. This rule can be
extended to include integers, so that
0
0
7
because
0 (7)(0)
However, if zero is the divisor, we have a special problem. Consider
9
?
0
This means that 9 0 ?.
Can 0 times a number ever be 9? No, so there is no solution.
9
Because
cannot be replaced by any number, we agree that division by 0 is not
0
allowed. We say that division by 0 is undefined.
Example 2 Dividing Integers
Divide, if possible.
(a)
7
is undefined.
0
(b)
9
is undefined.
0
(c)
0
0
5
(d)
0
0
8
0
is called an indeterminate form. You will learn more about this
0
in later mathematics classes.
Note: The expression
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DIVISION OF INTEGERS
SECTION 2.5
153
CHECK YOURSELF 2
Divide, if possible.
(a)
0
3
(b)
5
0
(c)
7
0
(d)
0
9
The fraction bar, like parentheses and the absolute value bars, serves as a grouping symbol. This means that all operations in the numerator and denominator should be performed
separately. Then the division is done as the last step. Example 3 illustrates this property.
OBJECTIVE 2
Example 3 Using Order of Operations
Evaluate each expression.
(a)
(6)(7)
42
14
3
3
Multiply in the numerator, then
divide.
(b)
3 (12)
9
3
3
3
Add in the numerator, then
divide.
(c)
4 (2)(6)
4 (12)
6 2
6 2
Multiply in the numerator. Then
add in the numerator and
subtract in the denominator.
16
2
8
Divide as the last step.
CHECK YOURSELF 3
Evaluate each expression.
(a)
4 (8)
6
(b)
3 (2)(6)
5
(c)
(2)(4) (6)(5)
(2)(11)
© 2010 McGraw-Hill Companies
Using Your Calculator to
Multiply and Divide Integers
Finding the product of two integers using a calculator is relatively straightforward.
Example 4 Multiplying Integers
Find the product. (457) (734)
457 734 Your display should read 335,438.
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INTEGERS AND INTRODUCTION TO ALGEBRA
CHECK YOURSELF 4
Find the products.
(a) (36) (91)
(b) (12) (284)
Finding the quotient of integers is also straightforward.
Example 5 Dividing Integers
Find the quotient.
384
16
384 16 Your display should read 24.
CHECK YOURSELF 5
Find the quotient.
(7,865) (242)
We can also use the calculator to raise an integer to a power.
Example 6 Raising a Number to a Power
Evaluate.
(3)6
( 3 ) yx 6 that the negative is attached to
the 3 before it is raised to a
power.
or, on some calculators
( () 3 ) ^ 6 Enter
Either way, your display should read 729.
CHECK YOURSELF 6
Evaluate.
(2)9
© 2010 McGraw-Hill Companies
NOTE The parentheses ensure
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DIVISION OF INTEGERS
SECTION 2.5
155
READING YOUR TEXT
The following fill-in-the-blank exercises are designed to assure that you understand
the key vocabulary used in this section. Each sentence comes directly from the
section. You will find the correct answers in Appendix C.
Section 2.5
(a) The quotient of two integers with different signs is
.
(b) The quotient of two integers with the same sign is
.
(c) Division by
is not allowed.
(d) The fraction bar serves as a
symbol.
CHECK YOURSELF ANSWERS
© 2010 McGraw-Hill Companies
1. (a) 5; (b) 4; (c) 6; (d) 12
2. (a) 0; (b) undefined; (c) undefined; (d) 0
3. (a) 2; (b) 3; (c) 1
4. (a) 3,276; (b) 3,408
5. 32.5
6. 512
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2.5
• e-Professors
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Name
Section
9:56 AM
Exercises
[Objective 1]
Divide.
1.
20
4
2.
70
14
3.
48
6
4.
24
8
5.
50
5
6.
32
8
7.
52
4
8.
56
7
9.
75
3
10.
60
15
11.
0
8
12.
125
25
13.
9
1
14.
10
0
15.
96
8
16.
20
2
17.
18
0
18.
0
8
19.
17
1
20.
27
1
21.
144
16
22.
150
6
Date
ANSWERS
1. 5
2. 5
3. 8
4. 3
5. 10
6. 4
7. 13
8. 8
9. 25
10.
4
11.
0
12.
5
14.
Undefined
13.
16.
9
10
15.
12
17.
Undefined
18.
0
19.
17
20.
21.
9
22. 25
23.
9
24.
15
25.
4
26.
4
27.
2
28. 9
29.
8
30.
[Objective 2]
Perform the indicated operations.
2
33.
Undefined
34.
Undefined
32.
156 SECTION 2.5
23.
(6)(3)
2
24.
(9)(5)
3
25.
(8)(2)
4
26.
(7)(8)
14
27.
24
4 8
28.
36
7 3
29.
12 12
3
30.
14 4
6
31.
55 19
12 6
32.
11 7
14 8
33.
75
22
34.
10 6
44
3
3
© 2010 McGraw-Hill Companies
31.
27
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ANSWERS
For exercises 35 to 37, use integers to write an expression that represents the situation. Then
answer the question.
Patrick worked all day mowing lawns and was paid
$9 per hour. If he had $125 at the end of a 9-h day, how much did he have before
he started working?
35.
125 9 9 $44
36.
42
— 14 weeks
3
35. Business and Finance
36. Social Science
A woman lost 42 lb. If she lost 3 lb each week, how long has she
been dieting?
Suppose that you and your two brothers bought equal
shares of an investment for a total of $20,000 and sold it later for $16,232. How
much did each person lose?
37.
20,000 16,232
—— $1,256
3
38.
675
39.
936
40.
1,736
41.
952
42.
1,349
43.
2
44.
625
45.
1,024
37. Business and Finance
Calculator Exercises
Use your calculator to multiply and divide.
38. (15) (45)
39. (78) (12)
40. (56) (31)
41. (34) (28)
42. (71) (19)
43.
44. (5)4
45. (4)5
28
14
© 2010 McGraw-Hill Companies
Answers
1. 5
15. 12
27. 2
3. 8
5. 10
7. 13
9. 25
17. Undefined
19. 17
21. 9
29. 8
31. 2
33. Undefined
20,000 16,232
$1,256
3
45. 1,024
37.
39. 936
11. 0
13. 9
23. 9
25. 4
35. 125 9 9 $44
41. 952
43. 2
SECTION 2.5
157