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Prealgebra

3.2 Add Integers

Prealgebra3.2 Add Integers

Learning Objectives

By the end of this section, you will be able to:
  • Model addition of integers
  • Simplify expressions with integers
  • Evaluate variable expressions with integers
  • Translate word phrases to algebraic expressions
  • Add integers in applications

Be Prepared 3.2

Before you get started, take this readiness quiz.

  1. Evaluate x+8x+8 when x=6.x=6.
     If you missed this problem, review Example 2.13.
  2. Simplify: 8+2(5+1).8+2(5+1).
     If you missed this problem, review Example 2.8.
  3. Translate the sum of 33 and negative 77 into an algebraic expression.
     If you missed this problem, review Table 2.7

Model Addition of Integers

Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with integers.

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more difficult. This difficulty relates to the way the brain learns.

The brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of 2−1.2−1. Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.

Addition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.

We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.

This figure has a blue circle labeled positive and a red circle labeled negative.

If we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in Figure 3.17.

This figure has a blue circle over a red circle. Beside them is the statement 1 plus negative 1 equals 0.
Figure 3.17 A blue counter represents +1.+1. A red counter represents −1.−1. Together they add to zero.

Manipulative Mathematics

Doing the Manipulative Mathematics activity "Addition of signed Numbers" will help you develop a better understanding of adding integers.

We will model four addition facts using the numbers 5,−5and3,−3.5,−5and3,−3.

5+3−5+(−3)−5+35+(−3)5+3−5+(−3)−5+35+(−3)

Example 3.14

Model: 5+3.5+3.

Try It 3.27

Model the expression.

2+42+4

Try It 3.28

Model the expression.

2+52+5

Example 3.15

Model: −5+(−3).−5+(−3).

Try It 3.29

Model the expression.

−2+(−4)−2+(−4)

Try It 3.30

Model the expression.

−2+(−5)−2+(−5)

Example 3.14 and Example 3.15 are very similar. The first example adds 55 positives and 33 positives—both positives. The second example adds 55 negatives and 33 negatives—both negatives. In each case, we got a result of 8—either88—either8 positives or 88 negatives. When the signs are the same, the counters are all the same color.

Now let’s see what happens when the signs are different.

Example 3.16

Model: −5+3.−5+3.

Try It 3.31

Model the expression, and then simplify:

2+(−4)2+(−4)

Try It 3.32

Model the expression, and then simplify:

2+(−5)2+(−5)

Example 3.17

Model: 5+(−3).5+(−3).

Try It 3.33

Model the expression, and then simplify:

(−2)+4(−2)+4

Try It 3.34

Model the expression:

(−2)+5(−2)+5

Example 3.18

Modeling Addition of Positive and Negative Integers

Model each addition.

  1. ⓐ 4 + 2
  2. ⓑ −3 + 6
  3. ⓒ 4 + (−5)
  4. ⓓ -2 + (−3)

Try It 3.35

Model each addition.

  1. ⓐ 3 + 4
  2. ⓑ −1 + 4
  3. ⓒ 4 + (−6)
  4. ⓓ −2 + (−2)

Try It 3.36

  1. ⓐ 5 + 1
  2. ⓑ −3 + 7
  3. ⓒ 2 + (−8)
  4. ⓓ −3 + (−4)

Simplify Expressions with Integers

Now that you have modeled adding small positive and negative integers, you can visualize the model in your mind to simplify expressions with any integers.

For example, if you want to add 37+(−53),37+(−53), you don’t have to count out 3737 blue counters and 5353 red counters.

Picture 3737 blue counters with 5353 red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because 53−37=16,53−37=16, there are 1616 more negative counters.

37+(−53)=−1637+(−53)=−16

Let’s try another one. We’ll add −74+(−27).−74+(−27). Imagine 7474 red counters and 2727 more red counters, so we have 101101 red counters all together. This means the sum is −101.−101.

−74+(−27)=−101−74+(−27)=−101

Look again at the results of Example 3.14 - Example 3.17.

5+35+3 −5+(−3)−5+(−3)
both positive, sum positive both negative, sum negative
When the signs are the same, the counters would be all the same color, so add them.
−5+3−5+3 5+(−3)5+(−3)
different signs, more negatives different signs, more positives
Sum negative sum positive
When the signs are different, some counters would make neutral pairs; subtract to see how many are left.
Table 3.1 Addition of Positive and Negative Integers

Example 3.19

Simplify:

  1. ⓐ19+(−47)19+(−47)
  2. ⓑ−32+40−32+40

Try It 3.37

Simplify each expression:

  1. ⓐ15+(−32)15+(−32)
  2. ⓑ−19+76−19+76

Try It 3.38

Simplify each expression:

  1. ⓐ−55+9−55+9
  2. ⓑ43+(−17)43+(−17)

Example 3.20

Simplify: −14+(−36).−14+(−36).

Try It 3.39

Simplify the expression:

−31+(−19)−31+(−19)

Try It 3.40

Simplify the expression:

−42+(−28)−42+(−28)

The techniques we have used up to now extend to more complicated expressions. Remember to follow the order of operations.

Example 3.21

Simplify: −5+3(−2+7).−5+3(−2+7).

Try It 3.41

Simplify the expression:

−2+5(−4+7)−2+5(−4+7)

Try It 3.42

Simplify the expression:

−4+2(−3+5)−4+2(−3+5)

Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers when evaluating expressions.

Example 3.22

Evaluate x+7whenx+7when

  1. ⓐx=−2x=−2
  2. ⓑx=−11.x=−11.

Try It 3.43

Evaluate each expression for the given values:

x+5whenx+5when

  1. ⓐ x=−3andx=−3and
  2. ⓑ x=−17x=−17

Try It 3.44

Evaluate each expression for the given values: y+7y+7 when

  1. ⓐy=−5y=−5
  2. ⓑy=−8y=−8

Example 3.23

When n=−5,n=−5, evaluate

  1. ⓐn+1n+1
  2. ⓑ−n+1.−n+1.

Try It 3.45

When n=−8,n=−8, evaluate

  1. ⓐn+2n+2
  2. ⓑ−n+2−n+2

Try It 3.46

Wheny=−9,evaluateWheny=−9,evaluate

  1. ⓐy+8y+8
  2. ⓑ−y+8.−y+8.

Next we'll evaluate an expression with two variables.

Example 3.24

Evaluate 3a+b3a+b when a=12a=12 and b=−30.b=−30.

Try It 3.47

Evaluate the expression:

a+2bwhena=−19andb=14.a+2bwhena=−19andb=14.

Try It 3.48

Evaluate the expression:

5p+qwhenp=4andq=−7.5p+qwhenp=4andq=−7.

Example 3.25

Evaluate (x+y)2(x+y)2 when x=−18x=−18 and y=24.y=24.

Try It 3.49

Evaluate:

(x+y)2(x+y)2 when x=−15x=−15 and y=29.y=29.

Try It 3.50

Evaluate:

(x+y)3(x+y)3 when x=−8x=−8 and y=10.y=10.

Translate Word Phrases to Algebraic Expressions

All our earlier work translating word phrases to algebra also applies to expressions that include both positive and negative numbers. Remember that the phrase the sum indicates addition.

Example 3.26

Translate and simplify: the sum of −9−9 and 5.5.

Try It 3.51

Translate and simplify the expression:

the sum of −7−7 and 44

Try It 3.52

Translate and simplify the expression:

the sum of −8−8 and −6−6

Example 3.27

Translate and simplify: the sum of 88 and −12,−12, increased by 3.3.

Try It 3.53

Translate and simplify:

the sum of 99 and −16,−16, increased by 4.4.

Try It 3.54

Translate and simplify:

the sum of −8−8 and −12,−12, increased by 7.7.

Add Integers in Applications

Recall that we were introduced to some situations in everyday life that use positive and negative numbers, such as temperatures, banking, and sports. For example, a debt of $5$5 could be represented as −$5.−$5. Let’s practice translating and solving a few applications.

Solving applications is easy if we have a plan. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question.

Example 3.28

The temperature in Buffalo, NY, one morning started at 7degrees7degrees below zero Fahrenheit. By noon, it had warmed up 12degrees.12degrees. What was the temperature at noon?

Try It 3.55

The temperature in Chicago at 5 A.M. was 10degrees10degrees below zero Celsius. Six hours later, it had warmed up 14 degrees Celsius.14 degrees Celsius. What is the temperature at 11 A.M.?

Try It 3.56

A scuba diver was swimming 16 feet16 feet below the surface and then dove down another 17 feet.17 feet. What is her new depth?

Example 3.29

A football team took possession of the football on their 42-yard line.42-yard line. In the next three plays, they lost 6 yards,6 yards, gained 4 yards,4 yards, and then lost 8 yards.8 yards. On what yard line was the ball at the end of those three plays?

Try It 3.57

The Bears took possession of the football on their 20-yard line.20-yard line. In the next three plays, they lost 9 yards,9 yards, gained 7 yards,7 yards, then lost 4 yards.4 yards. On what yard line was the ball at the end of those three plays?

Try It 3.58

The Chargers began with the football on their 25-yard line.25-yard line. They gained 5 yards,5 yards, lost 8 yards8 yards and then gained 15 yards15 yards on the next three plays. Where was the ball at the end of these plays?

Section 3.2 Exercises

Practice Makes Perfect

Model Addition of Integers

In the following exercises, model the expression to simplify.

63.

7 + 4 7 + 4

64.

8 + 5 8 + 5

65.

−6 + ( −3 ) −6 + ( −3 )

66.

−5 + ( −5 ) −5 + ( −5 )

67.

−7 + 5 −7 + 5

68.

−9 + 6 −9 + 6

69.

8 + ( −7 ) 8 + ( −7 )

70.

9 + ( −4 ) 9 + ( −4 )

Simplify Expressions with Integers

In the following exercises, simplify each expression.

71.

−21 + ( −59 ) −21 + ( −59 )

72.

−35 + ( −47 ) −35 + ( −47 )

73.

48 + ( −16 ) 48 + ( −16 )

74.

34 + ( −19 ) 34 + ( −19 )

75.

−200 + 65 −200 + 65

76.

−150 + 45 −150 + 45

77.

2 + ( −8 ) + 6 2 + ( −8 ) + 6

78.

4 + ( −9 ) + 7 4 + ( −9 ) + 7

79.

−14 + ( −12 ) + 4 −14 + ( −12 ) + 4

80.

−17 + ( −18 ) + 6 −17 + ( −18 ) + 6

81.

135 + ( −110 ) + 83 135 + ( −110 ) + 83

82.

140 + ( −75 ) + 67 140 + ( −75 ) + 67

83.

−32 + 24 + ( −6 ) + 10 −32 + 24 + ( −6 ) + 10

84.

−38 + 27 + ( −8 ) + 12 −38 + 27 + ( −8 ) + 12

85.

19 + 2 ( −3 + 8 ) 19 + 2 ( −3 + 8 )

86.

24 + 3 ( −5 + 9 ) 24 + 3 ( −5 + 9 )

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

87.

x+8x+8 when

  1. ⓐx=−26x=−26
  2. ⓑx=−95x=−95
88.

y+9y+9 when

  1. ⓐy=−29y=−29
  2. ⓑy=−84y=−84
89.

y+(−14)y+(−14) when

  1. ⓐy=−33y=−33
  2. ⓑy=30y=30
90.

x+(−21)x+(−21) when

  1. ⓐx=−27x=−27
  2. ⓑx=44x=44
91.

When a=−7,a=−7, evaluate:

  1. ⓐa+3a+3
  2. ⓑ−a+3−a+3
92.

When b=−11,b=−11, evaluate:

  1. ⓐb+6b+6
  2. ⓑ−b+6−b+6
93.

When c=−9,c=−9, evaluate:

  1. ⓐc+(−4)c+(−4)
  2. ⓑ−c+(−4)−c+(−4)
94.

When d=−8,d=−8, evaluate:

  1. ⓐd+(−9)d+(−9)
  2. ⓑ−d+(−9)−d+(−9)
95.

m+nm+n when, m=−15m=−15, n=7n=7

96.

p+qp+q when, p=−9p=−9, q=17q=17

97.

r−3sr−3s when, r=16r=16, s=2s=2

98.

2t+u2t+u when, t=−6t=−6, u=−5u=−5

99.

(a+b)2(a+b)2 when, a=−7a=−7, b=15b=15

100.

(c+d)2(c+d)2 when, c=−5c=−5, d=14d=14

101.

(x+y)2(x+y)2 when, x=−3x=−3, y=14y=14

102.

(y+z)2(y+z)2 when, y=−3y=−3, z=15z=15

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

103.

The sum of −14−14 and 55

104.

The sum of −22−22 and 99

105.

88 more than −2−2

106.

55 more than −1−1

107.

−10−10 added to −15−15

108.

−6−6 added to −20−20

109.

66 more than the sum of −1−1 and −12−12

110.

33 more than the sum of −2−2 and −8−8

111.

the sum of 1010 and −19,−19, increased by 44

112.

the sum of 1212 and −15,−15, increased by 11

Add Integers in Applications

In the following exercises, solve.

113.

Temperature The temperature in St. Paul, Minnesota was −19°F−19°F at sunrise. By noon the temperature had risen 26°F.26°F. What was the temperature at noon?

114.

Temperature The temperature in Chicago was −15°F−15°F at 6 am. By afternoon the temperature had risen 28°F.28°F. What was the afternoon temperature?

115.

Credit Cards Lupe owes $73$73 on her credit card. Then she charges $45$45 more. What is the new balance?

116.

Credit Cards Frank owes $212$212 on his credit card. Then he charges $105$105 more. What is the new balance?

117.

Weight Loss Angie lost 3 pounds3 pounds the first week of her diet. Over the next three weeks, she lost 2 pounds,2 pounds, gained 1 pound,1 pound, and then lost 4 pounds.4 pounds. What was the change in her weight over the four weeks?

118.

Weight Loss April lost 5 pounds5 pounds the first week of her diet. Over the next three weeks, she lost 3 pounds,3 pounds, gained 2 pounds,2 pounds, and then lost 1 pound.1 pound. What was the change in her weight over the four weeks?

119.

Football The Rams took possession of the football on their own 35-yard line.35-yard line. In the next three plays, they lost 12 yards,12 yards, gained 8 yards,8 yards, then lost 6 yards.6 yards. On what yard line was the ball at the end of those three plays?

120.

Football The Cowboys began with the ball on their own 20-yard line.20-yard line. They gained 15 yards,15 yards, lost 3 yards3 yards and then gained 6 yards6 yards on the next three plays. Where was the ball at the end of these plays?

121.

Calories Lisbeth walked from her house to get a frozen yogurt, and then she walked home. By walking for a total of 20 minutes,20 minutes, she burned 90 calories.90 calories. The frozen yogurt she ate was 110 calories.110 calories. What was her total calorie gain or loss?

122.

Calories Ozzie rode his bike for 30 minutes,30 minutes, burning 168 calories.168 calories. Then he had a 140-calorie140-calorie iced blended mocha. Represent the change in calories as an integer.

Everyday Math

123.

Stock Market The week of September 15, 2008, was one of the most volatile weeks ever for the U.S. stock market. The change in the Dow Jones Industrial Average each day was:

Monday −504 Tuesday + 142 Wednesday −449 Thursday + 410 Friday + 369 Monday −504 Tuesday + 142 Wednesday −449 Thursday + 410 Friday + 369

What was the overall change for the week?

124.

Stock Market During the week of June 22, 2009, the change in the Dow Jones Industrial Average each day was:

Monday −201 Tuesday −16 Wednesday −23 Thursday + 172 Friday −34 Monday −201 Tuesday −16 Wednesday −23 Thursday + 172 Friday −34

What was the overall change for the week?

Writing Exercises

125.

Explain why the sum of −8−8 and 22 is negative, but the sum of 88 and −2−2 and is positive.

126.

Give an example from your life experience of adding two negative numbers.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

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