Prealgebra 2e

# 3.3Subtract Integers

Prealgebra 2e3.3 Subtract Integers

### Learning Objectives

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

• Model subtraction of integers
• Simplify expressions with integers
• Evaluate variable expressions with integers
• Translate words phrases to algebraic expressions
• Subtract integers in applications

### Be Prepared 3.4

Before you get started, take this readiness quiz.

Simplify: $12−(8−1).12−(8−1).$
If you missed this problem, review Example 2.8.

### Be Prepared 3.5

Translate the difference of $2020$ and $−15−15$ into an algebraic expression.
If you missed this problem, review Example 1.36.

### Be Prepared 3.6

Add: $−18+7.−18+7.$
If you missed this problem, review Example 3.20.

### Model Subtraction of Integers

Remember the story in the last section about the toddler and the cookies? Children learn how to subtract numbers through their everyday experiences. Real-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers.

We will continue to use counters to model subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.

Perhaps when you were younger, you read $5−35−3$ as five take away three. When we use counters, we can think of subtraction the same way.

### Manipulative Mathematics

Doing the Manipulative Mathematics activity "Subtraction of Signed Numbers" will help you develop a better understanding of subtracting integers.

We will model four subtraction facts using the numbers $55$ and $3.3.$

$5−3−5−(−3)−5−35−(−3)5−3−5−(−3)−5−35−(−3)$

### Example 3.30

Model: $5−3.5−3.$

### Try It 3.59

Model the expression:

$6−46−4$

### Try It 3.60

Model the expression:

$7−47−4$

### Example 3.31

Model: $−5−(−3).−5−(−3).$

### Try It 3.61

Model the expression:

$−6−(−4)−6−(−4)$

### Try It 3.62

Model the expression:

$−7−(−4)−7−(−4)$

Notice that Example 3.30 and Example 3.31 are very much alike.

• First, we subtracted $33$ positives from $55$ positives to get $22$ positives.
• Then we subtracted $33$ negatives from $55$ negatives to get $22$ negatives.

Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.

Now let’s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a neutral pair does not change the value.

### Example 3.32

Model: $−5−3.−5−3.$

### Try It 3.63

Model the expression:

$−6−4−6−4$

### Try It 3.64

Model the expression:

$−7−4−7−4$

### Example 3.33

Model: $5−(−3).5−(−3).$

### Try It 3.65

Model the expression:

$6−(−4)6−(−4)$

### Try It 3.66

Model the expression:

$7−(−4)7−(−4)$

### Example 3.34

Model each subtraction.

1. 8 − 2
2. −5 − 4
3. 6 − (−6)
4. −8 − (−3)

### Try It 3.67

Model each subtraction.

1. 7 - (-8)
2. -7 - (-2)
3. 4 - 1
4. -6 - 8

### Try It 3.68

Model each subtraction.

1. 4 - (-6)
2. -8 - (-1)
3. 7 - 3
4. -4 - 2

### Example 3.35

Model each subtraction expression:

1. $2−8 2−8$
2. $−3−(−8)−3−(−8)$

### Try It 3.69

Model each subtraction expression.

1. $7−97−9$
2. $−5−(−9)−5−(−9)$

### Try It 3.70

Model each subtraction expression.

1. $4−74−7$
2. $−7−(−10)−7−(−10)$

### Simplify Expressions with Integers

Do you see a pattern? Are you ready to subtract integers without counters? Let’s do two more subtractions. We’ll think about how we would model these with counters, but we won’t actually use the counters.

• Subtract $−23−7.−23−7.$
Think: We start with $2323$ negative counters.
We have to subtract $77$ positives, but there are no positives to take away.
So we add $77$ neutral pairs to get the $77$ positives. Now we take away the $77$ positives.
So what’s left? We have the original $2323$ negatives plus $77$ more negatives from the neutral pair. The result is $3030$ negatives.
$−23−7=−30−23−7=−30$

Notice, that to subtract $7,7,$ we added $77$ negatives.
• Subtract $30−(−12).30−(−12).$
Think: We start with $3030$ positives.
We have to subtract $1212$ negatives, but there are no negatives to take away.
So we add $1212$ neutral pairs to the $3030$ positives. Now we take away the $1212$ negatives.
What’s left? We have the original $3030$ positives plus $1212$ more positives from the neutral pairs. The result is $4242$ positives.
$30−(−12)=4230−(−12)=42$

Notice that to subtract $−12,−12,$ we added $12.12.$

While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters.

Have you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the Subtraction Property, written as follows:

### Subtraction Property

$a−b=a+(−b)a−b=a+(−b)$

Look at these two examples.

We see that $6−46−4$ gives the same answer as $6+(−4).6+(−4).$

Of course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already knew how to subtract $6−46−4$ long ago. But knowing that $6−46−4$ gives the same answer as $6+(−4)6+(−4)$ helps when we are subtracting negative numbers.

### Example 3.36

Simplify:

1. $13−8and13+(−8)13−8and13+(−8)$
2. $−17−9and−17+(−9)−17−9and−17+(−9)$

### Try It 3.71

Simplify each expression:

1. $21−13and21+(−13)21−13and21+(−13)$
2. $−11−7and−11+(−7)−11−7and−11+(−7)$

### Try It 3.72

Simplify each expression:

1. $15−7and15+(−7)15−7and15+(−7)$
2. $−14−8and−14+(−8)−14−8and−14+(−8)$

Now look what happens when we subtract a negative.

We see that $8−(−5)8−(−5)$ gives the same result as $8+5.8+5.$ Subtracting a negative number is like adding a positive.

### Example 3.37

Simplify:

1. $9−(−15)and9+159−(−15)and9+15$
2. $−7−(−4)and−7+4−7−(−4)and−7+4$

### Try It 3.73

Simplify each expression:

1. $6−(−13)and6+136−(−13)and6+13$
2. $−5−(−1)and−5+1−5−(−1)and−5+1$

### Try It 3.74

Simplify each expression:

1. $4−(−19)and4+194−(−19)and4+19$
2. $−4−(−7)and−4+7−4−(−7)and−4+7$

Look again at the results of Example 3.30 - Example 3.33.

 $5–35–3$ $–5–(–3)–5–(–3)$ $22$ $–2–2$ 2 positives 2 negatives When there would be enough counters of the color to take away, subtract. $–5–3–5–3$ $5–(–3)5–(–3)$ $–8–8$ $88$ 5 negatives, want to subtract 3 positives 5 positives, want to subtract 3 negatives need neutral pairs need neutral pairs When there would not be enough of the counters to take away, add neutral pairs.
Table 3.4 Subtraction of Integers

### Example 3.38

Simplify: $−74−(−58).−74−(−58).$

### Try It 3.75

Simplify the expression:

$−67−(−38)−67−(−38)$

### Try It 3.76

Simplify the expression:

$−83−(−57)−83−(−57)$

### Example 3.39

Simplify: $7−(−4−3)−9.7−(−4−3)−9.$

### Try It 3.77

Simplify the expression:

$8−(−3−1)−98−(−3−1)−9$

### Try It 3.78

Simplify the expression:

$12−(−9−6)−1412−(−9−6)−14$

### Example 3.40

Simplify: $3·7−4·7−5·8.3·7−4·7−5·8.$

### Try It 3.79

Simplify the expression:

$6·2−9·1−8·9.6·2−9·1−8·9.$

### Try It 3.80

Simplify the expression:

$2·5−3·7−4·92·5−3·7−4·9$

### Evaluate Variable Expressions with Integers

Now we’ll practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.

### Example 3.41

Evaluate $x−4whenx−4when$

1. $x=3x=3$
2. $x=−6.x=−6.$

### Try It 3.81

Evaluate each expression:

$y−7wheny−7when$

1. $y=5y=5$
2. $y=−8y=−8$

### Try It 3.82

Evaluate each expression:

$m−3whenm−3when$

1. $m=1m=1$
2. $m=−4m=−4$

### Example 3.42

Evaluate $20−zwhen20−zwhen$

1. $z=12z=12$
2. $z=−12z=−12$

### Try It 3.83

Evaluate each expression:

$17−kwhen17−kwhen$

1. $k=19k=19$
2. $k=−19k=−19$

### Try It 3.84

Evaluate each expression:

$−5−bwhen−5−bwhen$

1. $b=14b=14$
2. $b=−14b=−14$

### Translate Word Phrases to Algebraic Expressions

When we first introduced the operation symbols, we saw that the expression $a−ba−b$ may be read in several ways as shown below.

Figure 3.18

Be careful to get $aa$ and $bb$ in the right order!

### Example 3.43

Translate and then simplify:

1. the difference of $1313$ and $−21−21$
2. subtract $2424$ from $−19−19$

### Try It 3.85

Translate and simplify:

1. the difference of $1414$ and $−23−23$
2. subtract $2121$ from $−17−17$

### Try It 3.86

Translate and simplify:

1. the difference of $1111$ and $−19−19$
2. subtract $1818$ from $−11−11$

### Subtract Integers in Applications

It’s hard to find something if we don’t know what we’re looking for or what to call it. So when we solve an application problem, we first need to determine what we are asked to find. Then we can write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

### How To

#### Solve Application Problems.

1. Step 1. Identify what you are asked to find.
2. Step 2. Write a phrase that gives the information to find it.
3. Step 3. Translate the phrase to an expression.
4. Step 4. Simplify the expression.
5. Step 5. Answer the question with a complete sentence.

### Example 3.44

In the morning, the temperature in Urbana, Illinois was $1111$ degrees Fahrenheit. By mid-afternoon, the temperature had dropped to $−9−9$ degrees Fahrenheit. What was the difference between the morning and afternoon temperatures?

### Try It 3.87

In the morning, the temperature in Anchorage, Alaska was $15 degrees Fahrenheit.15 degrees Fahrenheit.$ By mid-afternoon the temperature had dropped to $30 degrees30 degrees$ below zero. What was the difference between the morning and afternoon temperatures?

### Try It 3.88

The temperature in Denver was $−6−6$ degrees Fahrenheit at lunchtime. By sunset the temperature had dropped to $−15 degree Fahrenheit.−15 degree Fahrenheit.$ What was the difference between the lunchtime and sunset temperatures?

Geography provides another application of negative numbers with the elevations of places below sea level.

### Example 3.45

Dinesh hiked from Mt. Whitney, the highest point in California, to Death Valley, the lowest point. The elevation of Mt. Whitney is $14,49714,497$ feet above sea level and the elevation of Death Valley is $282282$ feet below sea level. What is the difference in elevation between Mt. Whitney and Death Valley?

### Try It 3.89

One day, John hiked to the $10,023 foot10,023 foot$ summit of Haleakala volcano in Hawaii. The next day, while scuba diving, he dove to a cave $80 feet80 feet$ below sea level. What is the difference between the elevation of the summit of Haleakala and the depth of the cave?

### Try It 3.90

The submarine Nautilus is at $340 feet340 feet$ below the surface of the water and the submarine Explorer is $573 feet573 feet$ below the surface of the water. What is the difference in the position of the Nautilus and the Explorer?

Managing your money can involve both positive and negative numbers. You might have overdraft protection on your checking account. This means the bank lets you write checks for more money than you have in your account (as long as they know they can get it back from you!)

### Example 3.46

Leslie has $2525$ in her checking account and she writes a check for $8.8.$

1. What is the balance after she writes the check?
2. She writes a second check for $20.20.$ What is the new balance after this check?
3. Leslie’s friend told her that she had lost a check for $1010$ that Leslie had given her with her birthday card. What is the balance in Leslie’s checking account now?

### Try It 3.91

Araceli has $7575$ in her checking account and writes a check for $27.27.$

1. What is the balance after she writes the check?
2. She writes a second check for $50.50.$ What is the new balance?
3. The check for $2020$ that she sent a charity was never cashed. What is the balance in Araceli’s checking account now?

### Try It 3.92

Genevieve’s bank account was overdrawn and the balance is $−78.−78.$

1. She deposits a check for $2424$ that she earned babysitting. What is the new balance?
2. She deposits another check for $49.49.$ Is she out of debt yet? What is her new balance?

### Section 3.3 Exercises

#### Practice Makes Perfect

Model Subtraction of Integers

In the following exercises, model each expression and simplify.

127.

$8 − 2 8 − 2$

128.

$9 − 3 9 − 3$

129.

$−5 − ( −1 ) −5 − ( −1 )$

130.

$−6 − ( −4 ) −6 − ( −4 )$

131.

$−5 − 4 −5 − 4$

132.

$−7 − 2 −7 − 2$

133.

$8 − ( −4 ) 8 − ( −4 )$

134.

$7 − ( −3 ) 7 − ( −3 )$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

135.
1. $15−615−6$
2. $15+(−6)15+(−6)$
136.
1. $12−912−9$
2. $12+(−9)12+(−9)$
137.
1. $44−2844−28$
2. $44+(−28)44+(−28)$
138.
1. $35−1635−16$
2. $35+(−16)35+(−16)$
139.
1. $8−(−9)8−(−9)$
2. $8+98+9$
140.
1. $4−(−4)4−(−4)$
2. $4+44+4$
141.
1. $27−(−18)27−(−18)$
2. $27+1827+18$
142.
1. $46−(−37)46−(−37)$
2. $46+3746+37$

In the following exercises, simplify each expression.

143.

$15 − ( −12 ) 15 − ( −12 )$

144.

$14 − ( −11 ) 14 − ( −11 )$

145.

$10 − ( −19 ) 10 − ( −19 )$

146.

$11 − ( −18 ) 11 − ( −18 )$

147.

$48 − 87 48 − 87$

148.

$45 − 69 45 − 69$

149.

$31 − 79 31 − 79$

150.

$39 − 81 39 − 81$

151.

$−31 − 11 −31 − 11$

152.

$−32 − 18 −32 − 18$

153.

$−17 − 42 −17 − 42$

154.

$−19 − 46 −19 − 46$

155.

$−103 − ( −52 ) −103 − ( −52 )$

156.

$−105 − ( −68 ) −105 − ( −68 )$

157.

$−45 − ( −54 ) −45 − ( −54 )$

158.

$−58 − ( −67 ) −58 − ( −67 )$

159.

$8 − 3 − 7 8 − 3 − 7$

160.

$9 − 6 − 5 9 − 6 − 5$

161.

$−5 − 4 + 7 −5 − 4 + 7$

162.

$−3 − 8 + 4 −3 − 8 + 4$

163.

$−14 − ( −27 ) + 9 −14 − ( −27 ) + 9$

164.

$−15 − ( −28 ) + 5 −15 − ( −28 ) + 5$

165.

$71 + ( −10 ) − 8 71 + ( −10 ) − 8$

166.

$64 + ( −17 ) − 9 64 + ( −17 ) − 9$

167.

$−16 − ( −4 + 1 ) − 7 −16 − ( −4 + 1 ) − 7$

168.

$−15 − ( −6 + 4 ) − 3 −15 − ( −6 + 4 ) − 3$

169.

$( 2 − 7 ) − ( 3 − 8 ) ( 2 − 7 ) − ( 3 − 8 )$

170.

$( 1 − 8 ) − ( 2 − 9 ) ( 1 − 8 ) − ( 2 − 9 )$

171.

$− ( 6 − 8 ) − ( 2 − 4 ) − ( 6 − 8 ) − ( 2 − 4 )$

172.

$− ( 4 − 5 ) − ( 7 − 8 ) − ( 4 − 5 ) − ( 7 − 8 )$

173.

$25 − [ 10 − ( 3 − 12 ) ] 25 − [ 10 − ( 3 − 12 ) ]$

174.

$32 − [ 5 − ( 15 − 20 ) ] 32 − [ 5 − ( 15 − 20 ) ]$

175.

$6 ⋅ 3 − 4 ⋅ 3 − 7 ⋅ 2 6 ⋅ 3 − 4 ⋅ 3 − 7 ⋅ 2$

176.

$5 ⋅ 7 − 8 ⋅ 2 − 4 ⋅ 9 5 ⋅ 7 − 8 ⋅ 2 − 4 ⋅ 9$

177.

$5 2 − 6 2 5 2 − 6 2$

178.

$6 2 − 7 2 6 2 − 7 2$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression for the given values.

179.

$x−6whenx−6when$

1. $x=3x=3$
2. $x=−3x=−3$
180.

$x−4whenx−4when$

1. $x=5x=5$
2. $x=−5x=−5$
181.

$5−ywhen5−ywhen$

1. $y=2y=2$
2. $y=−2y=−2$
182.

$8−ywhen8−ywhen$

1. $y=3y=3$
2. $y=−3y=−3$
183.

$4 x 2 − 15 x + 1 when x = 3 4 x 2 − 15 x + 1 when x = 3$

184.

$5 x 2 − 14 x + 7 when x = 2 5 x 2 − 14 x + 7 when x = 2$

185.

$−12 − 5 x 2 when x = 6 −12 − 5 x 2 when x = 6$

186.

$−19 − 4 x 2 when x = 5 −19 − 4 x 2 when x = 5$

Translate Word Phrases to Algebraic Expressions

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

187.
1. The difference of $33$ and $−10−10$
2. Subtract $−20−20$ from $4545$
188.
1. The difference of $88$ and $−12−12$
2. Subtract $−13−13$ from $5050$
189.
1. The difference of $−6−6$ and $99$
2. Subtract $−12−12$ from $−16−16$
190.
1. The difference of $−8−8$ and $99$
2. Subtract $−15−15$ from $−19−19$
191.
1. $88$ less than $−17−17$
2. $−24−24$ minus $3737$
192.
1. $55$ less than $−14−14$
2. $−13−13$ minus $4242$
193.
1. $2121$ less than$66$
2. $3131$ subtracted from $−19−19$
194.
1. $3434$ less than$77$
2. $2929$ subtracted from $−50−50$

Subtract Integers in Applications

In the following exercises, solve the following applications.

195.

Temperature One morning, the temperature in Urbana, Illinois, was $28° Fahrenheit.28° Fahrenheit.$ By evening, the temperature had dropped $38° Fahrenheit.38° Fahrenheit.$ What was the temperature that evening?

196.

Temperature On Thursday, the temperature in Spincich Lake, Michigan, was $22° Fahrenheit.22° Fahrenheit.$ By Friday, the temperature had dropped $35° Fahrenheit.35° Fahrenheit.$ What was the temperature on Friday?

197.

Temperature On January 15, the high temperature in Anaheim, California, was $84° Fahrenheit.84° Fahrenheit.$ That same day, the high temperature in Embarrass, Minnesota was $−12° Fahrenheit.−12° Fahrenheit.$ What was the difference between the temperature in Anaheim and the temperature in Embarrass?

198.

Temperature On January 21, the high temperature in Palm Springs, California, was $89°,89°,$ and the high temperature in Whitefield, New Hampshire was $−31°.−31°.$ What was the difference between the temperature in Palm Springs and the temperature in Whitefield?

199.

Football At the first down, the Warriors football team had the ball on their $30-yard line.30-yard line.$ On the next three downs, they gained $2 yards,2 yards,$ lost $7 yards,7 yards,$ and lost $4 yards.4 yards.$ What was the yard line at the end of the third down?

200.

Football At the first down, the Barons football team had the ball on their $20-yard line.20-yard line.$ On the next three downs, they lost $8 yards,8 yards,$ gained $5 yards,5 yards,$ and lost $6 yards.6 yards.$ What was the yard line at the end of the third down?

201.

Checking Account John has $148148$ in his checking account. He writes a check for $83.83.$ What is the new balance in his checking account?

202.

Checking Account Ellie has $426426$ in her checking account. She writes a check for $152.152.$ What is the new balance in her checking account?

203.

Checking Account Gina has $210210$ in her checking account. She writes a check for $250.250.$ What is the new balance in her checking account?

204.

Checking Account Frank has $9494$ in his checking account. He writes a check for $110.110.$ What is the new balance in his checking account?

205.

Checking Account Bill has a balance of $−14−14$ in his checking account. He deposits $4040$ to the account. What is the new balance?

206.

Checking Account Patty has a balance of $−23−23$ in her checking account. She deposits $8080$ to the account. What is the new balance?

#### Everyday Math

207.

Camping Rene is on an Alpine hike. The temperature is$−7°.−7°.$ Rene’s sleeping bag is rated “comfortable to $−20°”.−20°”.$ How much can the temperature change before it is too cold for Rene’s sleeping bag?

208.

Scuba Diving Shelly’s scuba watch is guaranteed to be watertight to $−100feet.−100feet.$ She is diving at $−45feet−45feet$ on the face of an underwater canyon. By how many feet can she change her depth before her watch is no longer guaranteed?

#### Writing Exercises

209.

Explain why the difference of $99$ and $−6−6$ is $15.15.$

210.

Why is the result of subtracting $3−(−4)3−(−4)$ the same as the result of adding $3+4?3+4?$

#### Self Check

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

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Do you know how you learn best?
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