What You Should Learn
1 䉴Add integers using a number line.
2䉴Add integers with like signs and with unlike signs.
3䉴Subtract integers with like signs and with unlike signs.
Why You Should Learn It Real numbers are used to represent many real-life quantities. For instance, in Exercise 107 on page 18, you will use real numbers to find the change in digital camera sales.
Nancy R. Cohen/Getty Images
Adding Integers Using a Number Line
In this and the next section, you will study the four operations of arithmetic (addition, subtraction, multiplication, and division) on the set of integers. There are many examples of these operations in real life. For example, your business had a gain of $550 during one week and a loss of $600 the next week. Over the two-week period, your business had a combined profit of
which represents an overall loss of $50.
The number line is a good visual model for demonstrating addition of integers.
To add two integers, using a number line, start at 0. Then move right or left aunits depending on whether ais positive or negative. From that position, move right or left bunits depending on whether bis positive or negative. The final posi- tion is called the sum.
EXAMPLE 1 Adding Integers with Like Signs Using a Number Line
Find each sum.
a. b.
Solution
a. Start at zero and move five units to the right. Then move two more units to the right, as shown in Figure 1.20. So,
Figure 1.20
b. Start at zero and move three units to the left. Then move five more units to the left, as shown in Figure 1.21. So,
Figure 1.21 Now try Exercise 3.
−3
−5
−1
−2
−3
−4
−5
−6
−7
−8 0 1 2 3 4 5 6
3共5兲 8.
5 2
0 1 2 3 4 5 6 7 8
−1
−2
−3
−4
−5
−6
527.
3共5兲 52
ab,
550共600兲 50
1 䉴Add integers using a number line.
CHECKPOINT
Addition of Integers
1. To addtwo integers with likesigns, add their absolute values and attach the common sign to the result.
2. To addtwo integers with unlikesigns, subtract the smaller absolute value from the larger absolute value and attach the sign of the integer with the larger absolute value.
2 䉴Add integers with like signs and with unlike signs.
EXAMPLE 2 Adding Integers with Unlike Signs Using a Number Line
Find each sum.
a. b. c.
Solution
a. Start at zero and move five units to the left. Then move two units to the right, as shown in Figure 1.22. So,
Figure 1.22
b. Start at zero and move seven units to the right. Then move three units to the left, as shown in Figure 1.23. So,
Figure 1.23
c. Start at zero and move four units to the left. Then move four units to the right, as shown in Figure 1.24. So,
Figure 1.24 Now try Exercise 7.
In Example 2(c), notice that the sum of and 4 is 0. Two numbers whose sum is zero are called opposites(or additive inverses) of each other, So, is the opposite of 4 and 4 is the opposite of
Adding Integers Algebraically
Examples 1 and 2 illustrated a graphical approach to adding integers. It is more common to use an algebraicapproach to adding integers, as summarized below.
4.
4 4
−4 4
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−7
−8 0 1 2 3 4 5 6 7 8
440.
7 −3
−1
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−7
−8 0 1 2 3 4 5 6 7 8
7共3兲4.
−1
−2
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−8
−5 2
0 1 2 3 4 5 6 7 8
52 3.
44 7共3兲
52
CHECKPOINT CHECKPOINT CHECKPOINT
EXAMPLE 3 Adding Integers
a. Like signs:
b. Unlike signs:
c. Unlike signs:
Now try Exercise 17.
There are different ways to add three or more integers. You can use the carrying algorithm with a vertical format with nonnegative integers, as shown in Figure 1.25, or you can add them two at a time, as illustrated in Example 4.
EXAMPLE 4 Account Balance
At the beginning of a month, your account balance was $28. During the month, you deposited $60 and withdrew $40. What was your balance at the end of the month?
Solution
Balance Now try Exercise 103.
Subtracting Integers Algebraically
Subtraction can be thought of as “taking away.” For instance, can be thought of as “8 take away 5,” which leaves 3. Moreover, note that
which means that
In other words, can also be accomplished by “adding the opposite of 5 to 8.”
EXAMPLE 5 Subtracting Integers
a. Add opposite of 8.
b. Add opposite of
c. Add opposite of 12.
Now try Exercise 47.
512 5共12兲 17
13.
10共13兲101323
383共8兲 5
85 858共5兲.
8共5兲3,
85 $48
$88共$40兲
$28$60共$40兲共$28$60兲共$40兲
2284共1417兲ⱍ 共22ⱍⱍ84ⱍ ⱍ17ⱍ14ⱍ ⱍ兲 22共8417145兲 70
18共62兲 共 ⱍ18ⱍⱍ62ⱍ 兲 共1862兲 80
Subtraction of Integers
To subtractone integer from another, add the opposite of the integer being subtracted to the other integer. The result is called the differenceof the two integers.
1 1 5 7 1 4 6 3 4 8
2 6 6
Figure 1.25 Carrying Algorithm
3䉴Subtract integers with like signs and with unlike signs.
CHECKPOINT CHECKPOINT
Be sure you understand that the terminology of subtraction is not the same as that used for negative numbers. For instance, is read as “negative 5,” but is read as “8 subtract 5.” It is important to distinguish between the operation and the signs of the numbers involved. For instance, in the operation is subtraction and the numbers are and 5.
For subtraction problems involving two nonnegative integers, you can use the borrowing algorithm shown in Figure 1.26.
EXAMPLE 6 Subtracting Integers
a. Subtract 10 from means:
b. subtract means:
Now try Exercise 77.
To evaluate an expression that contains a series of additions and subtractions, write the subtractions as equivalent additions and simplify from left to right, as shown in Example 7.
EXAMPLE 7 Evaluating Expressions
Evaluate each expression.
a. b.
c. d.
Solution
a. Add opposites.
Add and
Add and 11.
Add.
b. Add opposites.
Add 5 and 9.
Add 14 and Add.
c. Add opposites.
Add and Add and Add.
d. Add opposites.
Add 5 and Add 4 and
Add and 7.
Add.
Now try Exercise 87.
13
4
310
8.
4710
1.
4共8兲710
5187共10兲5共1兲共8兲710 2
4.
4
86
3.
1
4共4兲6
1346 1共3兲共4兲6 4
12.
22
14共12兲2 5共9兲12259共12兲2
5
20
94
7.
13
20114
13711共4兲 13共7兲114 5187共10兲 1346
5共9兲122 13711共4兲
3共8兲 385.
8 3
410 4共10兲 14.
4
3
35 85
5
3 4 2 1
10 1 7 3
15 5 6 9
Figure 1.26 Borrowing Algorithm
CHECKPOINT CHECKPOINT
EXAMPLE 8 Temperature Change
The temperature in Minneapolis, Minnesota at 4 P.M. was By midnight, the temperature had decreased by What was the temperature in Minneapolis at midnight?
Solution
To find the temperature at midnight, subtract 18 from 15.
The temperature in Minneapolis at midnight was
Now try Exercise 97.
This text includes several examples and exercises that use a calculator.
As each new calculator application is encountered, you will be given general instructions for using a calculator. These instructions, however, may not agree precisely with the steps required by yourcalculator, so be sure you are familiar with the use of the keys on your own calculator.
For each of the calculator examples in the text, two possible keystroke sequences are given: one for a standard scientificcalculator and one for a graphing calculator.
EXAMPLE 9 Evaluating Expressions with a Calculator
Evaluate each expression with a calculator.
a. b.
Keystrokes Display
a. 4 5 Scientific
4 5 Graphing
Keystrokes Display
b. 2 3 9 14 Scientific
2 3 9 14 Graphing
Now try Exercise 93.
ENTER ⴙ
冇ⴚ冈 冈 ⴚ 冇
ⴝ ⴙ ⴙⲐⴚ 冈 ⴚ 冇
9
ENTER ⴚ
冇ⴚ冈
9
ⴝ ⴚ ⴙⲐⴚ
2共3兲9 45
3F.
3
151815共18兲 18.
15F.
The keys and change a number to its opposite, and is the subtraction key. For instance, the keystrokes 4 5 will not produce the result shown in Example 9(a).
ENTER ⴚ
ⴚ
ⴚ ⴙⲐⴚ 冇ⴚ冈
Technology: Tip
In Exercises 1–8, find the sum and demonstrate the addition on the real number line.See Examples 1 and 2.
1. 2.
3. 4.
5. 6.
7. 8.
In Exercises 9– 42, find the sum.See Example 3.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
In Exercises 43–76, find the difference.See Example 5.
43. 44.
45. 46.
47. 48.
49. 50.
51. 52.
53. 54.
55. 56.
57. 58.
59. 60.
61. 62.
63. 64.
65. 66.
67. 68.
69. 70.
71. 72.
73. 74.
75. 110共30兲 76. 2500共600兲 120142 210400
84106 7132
12共7兲 10共4兲
942共942兲 4共4兲
63共8兲 135共114兲
2634 1324
315 211
48222 22131
5374 6185
1832 2757
12共5兲 19共31兲
62共28兲 18共18兲
8共31兲 15共10兲
7共8兲 1共4兲
3737 5125
4712 2118
770共383兲492 890共90兲62 1200共1300兲共275兲 312共564兲共100兲
4365共2145兲共1873兲40,084 803共104兲共613兲214 32共32兲共16兲
75共75兲共15兲
312015 13124 241共19兲 162共7兲 2共51兲13 9共18兲4 82共36兲82 15共3兲8 7共4兲1 10共6兲34 18共26兲 9共14兲
5468 75100
34共16兲 18共12兲
32共16兲 23共4兲
2019 1413
2323 4545
10共10兲 14共14兲
83 610
125 64
14共8兲 10共3兲
4共7兲 8共3兲
39 27
Developing Skills 1.2 EXERCISES Go to pages 58–59 to
record your assignments.
Concept Check
1. Explain how to use a number line to add three negative integers.
2. In your own words, write the rule for adding two integers with opposite signs. How do you determine the sign of the sum?
3. Explain how to find the difference of two integers.
4. When is the difference of two integers equal to zero?
In Exercises 77–82, find the difference.See Example 6.
77. Subtract 15 from 78. Subtract 24 from
79. Subtract from 380.
80. Subtract from 140.
81. subtract 82. subtract
83. Think About It What number must be added to 10 to obtain
84. Think About It What number must be added to 36 to obtain
85. Think About It What number must be subtracted from to obtain 24?
86. Think About It What number must be subtracted from to obtain 15?
In Exercises 87–92, evaluate the expression. See Example 7.
87.
88.
89.
90.
91.
92.
In Exercises 93–96, write the keystrokes used to evaluate the expression with a calculator (either scientific or graphing). Then evaluate the expression.
See Example 9.
93. 94.
95. 65共7兲 96. 43共9兲 9共2兲 37
15共2兲46 共5兲7184 32209 67125 1263共8兲 13共4兲10
20 12
12?
5?
110 77
22 43
80 120
17.
6.
97. Temperature Change The temperature at 6 A.M. was By noon, the temperature had increased by What was the temperature at noon?
98. Account Balance A credit card owner charged
$142 worth of goods on her account that had an initial balance of $0. Find the balance after a payment of $87 was made.
99. Outdoor Recreation A hiker descended 847 meters into the Grand Canyon. He climbed back up 385 meters and then rested. Find the distance between where the hiker rested and where he started his descent.
100. Outdoor Recreation A fisherman dropped his line 27 meters below the surface of the water.
Because the fish were not biting, he raised his line by 8 meters. How far below the surface of the water was his line?
101. Profit A telephone company lost $650,000 during the first half of the year. By the end of the year, the company had an overall profit of $362,000. What was the company’s profit during the second half of the year?
102. Altitude An airplane flying at a cruising altitude of 31,000 feet is instructed to descend as shown in the diagram below. How many feet must the airplane descend?
24,000 ft 31,000 ft
Not drawn to scale
John Burcham/Getty Images
The Grand Canyon, located in Arizona, is 277 river miles long.
22F.
10F.
Solving Problems
The symbol indicates an exercise in which you are instructed to use a graphing calculator.
103. Account Balance At the beginning of a month, your account balance was $2750. During the month, you withdrew $350 and $500, deposited
$450, and earned $6.42 in interest. What was your balance at the end of the month?
104. Account Balance At the beginning of a month, your account balance was $1204. During the month, you withdrew $425 and $621, deposited $150 and
$80, and earned $2.02 in interest. What was your balance at the end of the month?
105. Temperature Change When you left for class in the morning, the temperature was By the time class ended, the temperature had increased by While you studied, the temperature increased by During your soccer practice, the temperature decreased by What was the temperature after your soccer practice?
106. Temperature Change When you left for class in the morning, the temperature was By the time class ended, the temperature had increased by While you studied, the temperature decreased by
During your club meeting, the temperature decreased by What was the temperature after your club meeting?
107. Digital Cameras The bar graph shows the factory sales (in millions of dollars) of digital cameras in the United States for the years 2000 to 2005.
(Source: Consumer Electronics Association) (a) Find the change in factory sales of digital
cameras from 2000 to 2001.
(b) Find the change in factory sales of digital cameras from 2004 to 2005.
Figure for 107
108. Population The bar graph shows the estimated populations (in thousands) of Cleveland, OH for the years 2000 to 2006. (Source: U.S. Census Bureau) (a) Find the change in population of Cleveland
from 2000 to 2003.
(b) Find the change in population of Cleveland from 2005 to 2006.
440 445 450 455 460 465 470 475 480
2000 2001 2002 2003 2004 2005 2006
Population (in thousands)
Year 477
472 468
463 457
451 444
Cleveland Ohio 1000
2000 3000 4000 5000 6000 7000 8000
2000 1825
2001 1972
2002 2794
2003 3921
2004 4739
2005 7468
Year Factory sales (in millions of dollars)
7.1 MEGAPIXELS
Digital ClearPic 1000Made in U.S.A.
6. 5.
13. 40F.
9.
3. 4. 25C.
In Exercises 109 and 110, an addition problem is shown visually on the real number line. (a) Write the addition problem and find the sum. (b) State the rule for the addition of integers demonstrated.
109.
110.
111. Explain why the sum of two negative integers is a negative integer.
112. When is the sum of a positive integer and a negative integer a positive integer?
113. Is it possible that the sum of two positive integers is a negative integer? Explain.
114. Is it possible that the difference of two negative integers is a positive integer? Explain.
0 1 2 3 4 5
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0 1 2 3 4 5 6
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Explaining Concepts