Prealgebra 2e

### Learning Objectives

By the end of this section, you will be able to:

• Simplify expressions with integers
• Evaluate variable expressions with integers
• Translate word phrases to algebraic expressions

### Be Prepared 3.3

Before you get started, take this readiness quiz.

Evaluate $x+8x+8$ when $x=6.x=6.$
If you missed this problem, review Example 2.13.

Simplify: $8+2(5+1).8+2(5+1).$
If you missed this problem, review Example 2.8.

Translate the sum of $33$ and negative $77$ into an algebraic expression.
If you missed this problem, review Table 2.7

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. 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.

Figure 3.17 A blue counter represents $+1.+1.$ A red counter represents $−1.−1.$ Together they add to zero.

### Manipulative Mathematics

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:

$(−2)+5(−2)+5$

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

1. 3 + 4
2. −1 + 4
3. 4 + (−6)
4. −2 + (−2)

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 $7.7.$

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 $55$ 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

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 $11$

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 $7373$ on her credit card. Then she charges $4545$ more. What is the new balance?

116.

Credit Cards Frank owes $212212$ on his credit card. Then he charges $105105$ more. What is the new balance?

117.

Football A team lost $3 yards3 yards$ the first play. Then they lost $2 yards,2 yards,$ gained $1 yard,1 yard,$ and then lost $4 yards.4 yards.$ What was the change in overall yardage over the four plays?

118.

Card Games April lost $5 cards5 cards$ the first turn. Over the next three turns, she lost $3 cards,3 cards,$ gained $2 cards,2 cards,$ and then lost $1 card.1 card.$ What was the change in cards over the four turns?

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.

Scuba Diving A scuba diver swimming $8 feet8 feet$ below the surface dove $17 feet17 feet$ deeper; the pressure got to them and they rose five feet. What is their new depth?

122.

Gas Consumption: Ozzie rode their motorcycle for $30 minutes,30 minutes,$ using $168 fluid ounces of gas.168 fluid ounces of gas.$ Then they stopped and got $140-fluid ounces of gas.140-fluid ounces of gas.$ Represent the change in gas amount 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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