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2.5: Subtract Integers (Part 1)

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Learning Objectives
  • 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!

Before you get started, take this readiness quiz.

  1. Simplify: 12(81). If you missed this problem, review Example 2.1.8.
  2. Translate the difference of \(20\) and 15 into an algebraic expression. If you missed this problem, review Example 1.3.11.
  3. Add: 18+7. If you missed this problem, review Example 3.2.6.

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 53 as five take away three. When we use counters, we can think of subtraction the same way.

We will model four subtraction facts using the numbers 5 and 3.

535(3)535(3)

Example 2.5.1: model

Model: 53.

Solution

Interpret the expression. 5 − 3 means 5 take away 3.
Model the first number. Start with 5 positives. CNX_BMath_Figure_03_03_027_img-01.png
Take away the second number. So take away 3 positives. CNX_BMath_Figure_03_03_027_img-02.png
Find the counters that are left. CNX_BMath_Figure_03_03_027_img-03.png

The difference between 5 and 3 is 2.

Exercise 2.5.1

Model the expression: 64

Answer

CNX_BMath_Figure_03_03_003_img.jpg

2

Exercise 2.5.2

Model the expression: 74

Answer

CNX_BMath_Figure_03_03_004_img.jpg

3

Example 2.5.2: model

Model: 5(3).

Solution

Interpret the expression. −5 − (−3) means −5 take away −3.
Model the first number. Start with 5 negatives. CNX_BMath_Figure_03_03_028_img-01.png
Take away the second number. So take away 3 negatives. CNX_BMath_Figure_03_03_028_img-02.png
Find the number of counters that are left. CNX_BMath_Figure_03_03_028_img-03.png

The difference between 5 and 3 is 2.

Exercise 2.5.3

Model the expression: 6(4)

Answer

CNX_BMath_Figure_03_03_008_img.jpg

2

Exercise 2.5.4

Model the expression: 7(4)

Answer

CNX_BMath_Figure_03_03_009_img.jpg

3

Notice that Example 2.5.1 and Example 2.5.2 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.

This figure has a row of 5 blue circles. The first three are circled. Above the row is 5 minus 3 equals 2. Next to this is a row of 5 red circles. The first three are circled. Above the row is negative 5 minus negative 3 equals negative 2.

Figure 2.5.1

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 2.5.3: model

Model: 53.

Solution

Interpret the expression. −5 − 3 means −5 take away 3.
Model the first number. Start with 5 negatives. CNX_BMath_Figure_03_03_029_img-01.png
Take away the second number. So we need to take away 3 positives.  
But there are no positives to take away. Add neutral pairs until you have 3 positives. CNX_BMath_Figure_03_03_029_img-02.png
Now take away 3 positives. CNX_BMath_Figure_03_03_029_img-03.png
Count the number of counters that are left. CNX_BMath_Figure_03_03_029_img-04.png

The difference of 5 and 3 is 8.

Exercise 2.5.5

Model the expression: 64

Answer

CNX_BMath_Figure_03_03_012_img.jpg

10

Exercise 2.5.6

Model the expression: 74

Answer

CNX_BMath_Figure_03_03_013_img.jpg

11

Example 2.5.4: model

Model: 5(3).

Solution

Interpret the expression. 5 − (−3) means 5 take away −3.
Model the first number. Start with 5 positives. CNX_BMath_Figure_03_03_030_img-01.png
Take away the second number, so take away 3 negatives.  
But there are no negatives to take away. Add neutral pairs until you have 3 negatives. CNX_BMath_Figure_03_03_030_img-02.png
Then take away 3 negatives. CNX_BMath_Figure_03_03_030_img-03.png
Count the number of counters that are left. CNX_BMath_Figure_03_03_030_img-04.png

The difference of 5 and 3 is 8.

Exercise 2.5.7

Model the expression: 6(4)

Answer

CNX_BMath_Figure_03_03_016_img.jpg

10

Exercise 2.5.8

Model the expression: 7(4)

Answer

CNX_BMath_Figure_03_03_017_img.jpg

11

Example 2.5.5: model

Model each subtraction.

  1. 82
  2. 54
  3. 6(6)
  4. 8(3)

Solution

  1. 82: This means 8 take away 2.
Start with 8 positives. CNX_BMath_Figure_03_03_041_img-01.png
Take away 2 positives. CNX_BMath_Figure_03_03_041_img-02.png
How many are left? 8 − 2 = 6
  1. 54: This means 5 take away 4.
Start with 5 negatives. CNX_BMath_Figure_03_03_042_img-01.png
You need to take away 4 positives. Add 4 neutral pairs to get 4 positives. CNX_BMath_Figure_03_03_042_img-02.pngCNX_BMath_Figure_03_03_042_img-03.png
Take away 4 positives. CNX_BMath_Figure_03_03_042_img-04.png
How many are left? −5 − 4 = −9
  1. 6(6): This means 6 take away 6.
Start with 6 positives. CNX_BMath_Figure_03_03_043_img-01.png
Add 6 neutrals to get 6 negatives to take away. CNX_BMath_Figure_03_03_043_img-02.png
Remove 6 negatives. CNX_BMath_Figure_03_03_043_img-03.png
How many are left? 6 − (−6) = 12
  1. 8(3): This means 8 take away 3.
Start with 8 negatives. CNX_BMath_Figure_03_03_044_img-01.png
Take away 3 negatives. CNX_BMath_Figure_03_03_044_img-02.png
How many are left? −8 − (−3) = −5
Exercise 2.5.9

Model each subtraction.

  1. 7(8)
  2. 7(2)
  3. 41
  4. 68
Answer a

CNX_BMath_Figure_03_03_045_img.jpg

Answer b

CNX_BMath_Figure_03_03_046_img.jpg

Answer c

CNX_BMath_Figure_03_03_047_img.jpg

Answer d

CNX_BMath_Figure_03_03_048_img.jpg

Exercise 2.5.10

Model each subtraction.

  1. 4(6)
  2. 8(1)
  3. 73
  4. 42
Answer a

CNX_BMath_Figure_03_03_049_img.jpg

Answer b

CNX_BMath_Figure_03_03_050_img.jpg

Answer c

CNX_BMath_Figure_03_03_051_img.jpg

Answer d

CNX_BMath_Figure_03_03_052_img.jpg

Example 2.5.6: model

Model each subtraction expression:

  1. 28
  2. 3(8)

Solution

We start with 2 positives. CNX_BMath_Figure_03_03_031_img-01.png
We need to take away 8 positives, but we have only 2.  
Add neutral pairs until there are 8 positives to take away. CNX_BMath_Figure_03_03_031_img-02.png
Then take away eight positives. CNX_BMath_Figure_03_03_031_img-03.png
Find the number of counters that are left. There are 6 negatives. CNX_BMath_Figure_03_03_031_img-04.png

28=6

We start with 3 negatives. CNX_BMath_Figure_03_03_032_img-01.png
We need to take away 8 negatives, but we have only 3.  
Add neutral pairs until there are 8 negatives to take away CNX_BMath_Figure_03_03_032_img-02.png
Then take away the 8 negatives. CNX_BMath_Figure_03_03_032_img-03.png
Find the number of counters that are left. There are 5 positives. CNX_BMath_Figure_03_03_032_img-04.png

3(8)=5

Exercise 2.5.11

Model each subtraction expression.

  1. 79
  2. 5(9)
Answer a

CNX_BMath_Figure_03_03_020_img.jpg

2

Answer b

CNX_BMath_Figure_03_03_021_img.jpg

4

Exercise 2.5.12

Model each subtraction expression.

  1. 47
  2. 7(10)
Answer a

CNX_BMath_Figure_03_03_022_img.jpg

3

Answer b

CNX_BMath_Figure_03_03_023_img.jpg

3

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 237. 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. 237=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:

Definition: Subtraction Property

ab=a+(b)

Look at these two examples.

This figure has two columns. The first column has 6 minus 4. Underneath, there is a row of 6 blue circles, with the first 4 separated from the last 2. The first 4 are circled. Under this row there is 2. The second column has 6 plus negative 4. Underneath there is a row of 6 blue circles with the first 4 separated from the last 2. The first 4 are circled. Under the first four is a row of 4 red circles. Under this there is 2.

Figure 2.5.2

We see that 64 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 64 long ago. But knowing that 64 gives the same answer as 6+(4) helps when we are subtracting negative numbers.

Example 2.5.7: simplify

Simplify:

  1. 138 and 13+(8)
  2. 179 and 17+(9)

Solution

Subtract to simplify. 13 − 8 = 5
Add to simplify. 13 + (−8) = 5
Subtracting 8 from 13 is the same as adding −8 to 13.  
Subtract to simplify. −17 − 9 = −26
Add to simplify. −17 + (−9) = −26
Subtracting 9 from −17 is the same as adding −9 to −17.  
Exercise 2.5.13

Simplify each expression:

  1. 2113 and 21+(13)
  2. 117 and 11+(7)
Answer a

8, 8

Answer b

18, 18

Exercise 2.5.14

Simplify each expression:

  1. 157 and 15+(7)
  2. 148 and 14+(8)
Answer a

8, 8

Answer b

22, 22

Now look what happens when we subtract a negative.

This figure has two columns. The first column has 8 minus negative 5. Underneath, there is a row of 13 blue  circles. The first 8 are separated from the next 5. Under the last 5 blue circles there is a row of 5 red circles. They are circled. Under this there is 13. The second column has 8 plus 5. Underneath there is a row of 13 blue circles. The first 8 are separated from the last 5. Under this there is 13.

Figure 2.5.3

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

Example 2.5.8: simplify

Simplify:

  1. 9(15) and 9+15
  2. 7(4) and 7+4

Solution

  1. 9(15) and 9+15
Subtract to simplify. 9 − (−15) = 24
Add to simplify. 9 + 15 = 24

Subtracting 15 from 9 is the same as adding 15 to 9.

  1. 7(4) and 7+4
Subtract to simplify. −7 − (−4) = −3
Add to simplify. −7 + 4 = −3

Subtracting 4 from 7 is the same as adding 4 to 7.

Exercise 2.5.15

Simplify each expression:

  1. 6(13) and 6+13
  2. 5(1) and 5+1
Answer a

19, 19

Answer b

4, 4

Exercise 2.5.16

Simplify each expression:

  1. 4(19) and 4+19
  2. 4(7) and 4+7
Answer a

23, 23

Answer b

3, 3

Look again at the results of Example 2.5.1 - Example 2.5.4.

Table 2.5.1: Subtraction of Integers
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.

Contributors and Attributions

  • Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Creative Commons Attribution License 4.0 license.

This page titled 2.5: Subtract Integers (Part 1) is shared under a not declared license and was authored, remixed, and/or curated by OpenStax.

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