3.3 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
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.
Notice that Example 3.30 and Example 3.31 are very much alike.
- First, we subtracted 3 positives from 5 positives to get 2 positives.
- Then we subtracted 3 negatives from 5 negatives to get 2 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.
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.
Think: We start with 23 negative counters.
We have to subtract 7 positives, but there are no positives to take away.
So we add 7 neutral pairs to get the 7 positives. Now we take away the 7 positives.
So what’s left? We have the original 23 negatives plus 7 more negatives from the neutral pair. The result is 30 negatives.
−23−7=−30
Notice, that to subtract 7, we added 7 negatives.
Subtract 30−(−12).
Think: We start with 30 positives.
We have to subtract 12 negatives, but there are no negatives to take away.
So we add 12 neutral pairs to the 30 positives. Now we take away the 12 negatives.
What’s left? We have the original 30 positives plus 12 more positives from the neutral pairs. The result is 42 positives.
30−(−12)=42
Notice that to subtract −12, we added 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:
Look at these two examples.
We see that 6−4 gives the same answer as 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 6−4 long ago. But knowing that 6−4 gives the same answer as 6+(−4) helps when we are subtracting negative numbers.
Now look what happens when we subtract a negative.
We see that 8−(−5) gives the same result as 8+5. Subtracting a negative number is like adding a positive.
Look again at the results of Example 3.30 – Example 3.33.
| 5–3 | –5–(–3) |
| 2 | –2 |
| 2 positives | 2 negatives |
| When there would be enough counters of the color to take away, subtract. | |
| –5–3 | 5–(–3) |
| –8 | 8 |
| 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. |
Table3.4 Subtraction of Integers
Evaluate Variable Expressions with Integers
Now we’ll practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.
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!
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.
Geography provides another application of negative numbers with the elevations of places below sea level.
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!)
Section 3.3 Exercises
Practice Makes Perfect
Model Subtraction of Integers
In the following exercises, model each expression and simplify.
- 8−2
- 9−3
- −5−(−1)
- −6−(−4)
- −5−4
- −7−2
- 8−(−4)
- 7−(−3)
Simplify Expressions with Integers
In the following exercises, simplify each expression.
135.
ⓐ15−6
ⓑ15+(−6)
136.
ⓐ12−9
ⓑ12+(−9)
137.
ⓐ44−28
ⓑ44+(−28)
138.
ⓐ35−16
ⓑ35+(−16)
139.
ⓐ8−(−9)
ⓑ8+9
140.
ⓐ4−(−4)
ⓑ4+4
141.
ⓐ27−(−18)
ⓑ27+18
142.
ⓐ46−(−37)
ⓑ46+37
In the following exercises, simplify each expression.
- 15−(−12)
- 14−(−11)
- 10−(−19)
- 11−(−18)
- 48−87
- 45−69
- 31−79
- 39−81
- −31−11
- −32−18
- −17−42
- −19−46
- −103−(−52)
- −105−(−68)
- −45−(−54)
- −58−(−67)
- 8−3−7
- 9−6−5
- −5−4+7
- −3−8+4
- −14−(−27)+9
- −15−(−28)+5
- 71+(−10)−8
- 64+(−17)−9
- −16−(−4+1)−7
- −15−(−6+4)−3
- (2−7)−(3−8)
- (1−8)−(2−9)
- −(6−8)−(2−4)
- −(4−5)−(7−8)
- 25−[10−(3−12)]
- 32−[5−(15−20)]
- 6⋅3−4⋅3−7⋅2
- 5⋅7−8⋅2−4⋅9
- 52−62
- 62−72
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression for the given values.
- x−6when
ⓐx=3
ⓑx=−3 - x−4when
ⓐx=5
ⓑx=−5 - 5−ywhen
ⓐy=2
ⓑy=−2 - 8−ywhen
ⓐy=3
ⓑy=−3 - 4×2−15x+1whenx=3
- 5×2−14x+7whenx=2
- −12−5x2whenx=6
- −19−4x2whenx=5
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate each phrase into an algebraic expression and then simplify.
187.
ⓐ The difference of 3 and −10
ⓑ Subtract −20 from 45
188.
ⓐ The difference of 8 and −12
ⓑ Subtract −13 from 50
189.
ⓐ The difference of −6 and 9
ⓑ Subtract −12 from −16
190.
ⓐ The difference of −8 and 9
ⓑ Subtract −15 from −19
191.
ⓐ8 less than −17
ⓑ−24 minus 37
192.
ⓐ5 less than −14
ⓑ−13 minus 42
193.
ⓐ21 less than6
ⓑ31 subtracted from −19
194.
ⓐ34 less than7
ⓑ29 subtracted from −50
Subtract Integers in Applications
In the following exercises, solve the following applications.
- Temperature One morning, the temperature in Urbana, Illinois, was 28° Fahrenheit. By evening, the temperature had dropped 38° Fahrenheit. What was the temperature that evening?
- Temperature On Thursday, the temperature in Spincich Lake, Michigan, was 22° Fahrenheit. By Friday, the temperature had dropped 35° Fahrenheit. What was the temperature on Friday?
- Temperature On January 15, the high temperature in Anaheim, California, was 84° Fahrenheit. That same day, the high temperature in Embarrass, Minnesota was −12° Fahrenheit. What was the difference between the temperature in Anaheim and the temperature in Embarrass?
- Temperature On January 21, the high temperature in Palm Springs, California, was 89°, and the high temperature in Whitefield, New Hampshire was −31°. What was the difference between the temperature in Palm Springs and the temperature in Whitefield?
- Football At the first down, the Warriors football team had the ball on their 30-yard line. On the next three downs, they gained 2 yards, lost 7 yards, and lost 4 yards. What was the yard line at the end of the third down?
- Football At the first down, the Barons football team had the ball on their 20-yard line. On the next three downs, they lost 8 yards, gained 5 yards, and lost 6 yards. What was the yard line at the end of the third down?
- Checking Account John has $148 in his checking account. He writes a check for $83. What is the new balance in his checking account?
- Checking Account Ellie has $426 in her checking account. She writes a check for $152. What is the new balance in her checking account?
- Checking Account Gina has $210 in her checking account. She writes a check for $250. What is the new balance in her checking account?
- Checking Account Frank has $94 in his checking account. He writes a check for $110. What is the new balance in his checking account?
- Checking Account Bill has a balance of −$14 in his checking account. He deposits $40 to the account. What is the new balance?
- Checking Account Patty has a balance of −$23 in her checking account. She deposits $80 to the account. What is the new balance?
Everyday Math
- Camping Rene is on an Alpine hike. The temperature is−7°. Rene’s sleeping bag is rated “comfortable to −20°”. How much can the temperature change before it is too cold for Rene’s sleeping bag?
- Scuba Diving Shelly’s scuba watch is guaranteed to be watertight to −100feet. She is diving at −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
- Explain why the difference of 9 and −6 is 15.
- Why is the result of subtracting 3−(−4) the same as the result of adding 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?