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

Model: −5 + 3 .

Solution

Interpret the expression. −5 + 3 means the sum of −5 and 3 .
Model the first number. Start with 5 negatives. .
Model the second number. Add 3 positives. .
Remove any neutral pairs. .
Count the result. .
The sum of −5 and 3 is −2. −5 + 3 = −2

Notice that there were more negatives than positives, so the result is negative.

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Model the expression, and then simplify:

2 + ( −4 )


This figure shows two rows of counter circles. The first row has 2 dark pink circles, representing negative counters. The second row has 4 light pink circles, representing positive counters.
2

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Model the expression, and then simplify:

2 + ( −5 )


This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 5 dark pink circles, representing negative counters.
−3

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Model: 5 + ( −3 ) .

Solution

Interpret the expression. 5 + ( −3 ) means the sum of 5 and −3 .
Model the first number. Start with 5 positives. .
Model the second number. Add 3 negatives. .
Remove any neutral pairs. .
Count the result. .
The sum of 5 and −3 is 2. 5 + ( −3 ) = 2
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Model the expression, and then simplify:

( −2 ) + 4


This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 4 dark pink circles, representing negative counters.
−2

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Model the expression:

( −2 ) + 5


This figure shows two rows of counters. The first row shows 2 dark pink circles, representing negative counters. The second row has 5 light pink circles, representing negative counters. The neutral pairs of one positive and one negative counter are circled leaving three positive counters.
3

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Modeling addition of positive and negative integers

Model each addition.

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

4 + 2
Start with 4 positives. .
Add two positives. .
How many do you have? 6 4 + 2 = 6
3 + 6
Start with 3 negatives. .
Add 6 positives. .
Remove neutral pairs. .
How many are left? .
3 −3 + 6 = 3
4 + ( −5 )
Start with 4 negatives. .
Add 5 negatives. .
Remove neutral pairs. .
How many are left? .
−1 4 + ( −5 ) = −1
−2 + ( −3 )
Start with 2 negatives. .
Add 3 negatives. .
How many do you have? −5 −2 + ( −3 ) = −5

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Model each addition.

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


  1. .

  2. .

  3. .

  4. .

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


  1. .

  2. .

  3. .

  4. .

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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 ) , you don’t have to count out 37 blue counters and 53 red counters.

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

37 + ( −53 ) = −16

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

−74 + ( −27 ) = −101
Addition of positive and negative integers
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 )
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.

Simplify:

  1. 19 + ( −47 )
  2. −32 + 40

Solution

ⓐ Since the signs are different, we subtract 19 from 47 . The answer will be negative because there are more negatives than positives.

19 + ( −47 ) −28

ⓑ The signs are different so we subtract 32 from 40 . The answer will be positive because there are more positives than negatives

−32 + 40 8
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Simplify each expression:

  1. 15 + ( −32 )
  2. −19 + 76
  1. ⓐ −17
  2. ⓑ 57
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Simplify each expression:

  1. −55 + 9
  2. 43 + ( −17 )

  1. ⓐ −46
  2. ⓑ 26

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Simplify: −14 + ( −36 ) .

Solution

Since the signs are the same, we add. The answer will be negative because there are only negatives.

−14 + ( −36 ) −50
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Simplify the expression:

−31 + ( −19 )

−50

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Simplify the expression:

−42 + ( −28 )

−70

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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