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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 )

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.

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.

Simplify:

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

Solution

13 8 and 13 + ( −8 )
Subtract to simplify. 13 8 = 5
Add to simplify. 13 + ( −8 ) = 5
Subtracting 8 from 13 is the same as adding −8 to 13.
−17 9 and −17 + ( −9 )
Subtract to simplify. −17 9 = −26
Add to simplify. −17 + ( −9 ) = −26
Subtracting 9 from −17 is the same as adding −9 to −17.
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Simplify each expression:

  1. 21 13 and 21 + ( −13 )
  2. −11 7 and −11 + ( −7 )
  1. 8, 8
  2. −18, −18
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Simplify each expression:

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

  1. 8, 8
  2. −22, −22

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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.

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

Simplify:

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

Solution

9 ( −15 ) and 9 + 15
Subtract to simplify. 9 ( −15 ) = −24
Add to simplify. 9 + 15 = 24
9−(−15) Subtracting −15 from 9 is the same as adding 15 to 9.
−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
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Simplify each expression:

  1. 6 ( −13 ) and 6 + 13
  2. −5 ( −1 ) and −5 + 1

  1. 19, 19
  2. −4, −4

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Simplify each expression:

  1. 4 ( −19 ) and 4 + 19
  2. −4 ( −7 ) and −4 + 7

  1. 23, 23
  2. 3, 3

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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.

Simplify: −74 ( −58 ) .

Solution

We are taking 58 negatives away from 74 negatives. .
Subtract. .
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Simplify the expression:

−67 ( −38 )

−29

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Simplify the expression:

−83 ( −57 )

−26

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Simplify: 7 ( −4 3 ) 9 .

Solution

We use the order of operations to simplify this expression, performing operations inside the parentheses first. Then we subtract from left to right.

Simplify inside the parentheses first. .
Subtract from left to right. .
Subtract. .
.
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Simplify the expression:

8 ( −3 1 ) 9

3

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Simplify the expression:

12 ( −9 6 ) 14

13

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Simplify: 3 · 7 4 · 7 5 · 8 .

Solution

We use the order of operations to simplify this expression. First we multiply, and then subtract from left to right.

Multiply first. .
Subtract from left to right. .
Subtract. .
.
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Simplify the expression:

6 · 2 9 · 1 8 · 9 .

−69

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Simplify the expression:

2 · 5 3 · 7 4 · 9

−47

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Evaluate variable expressions with integers

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

Evaluate x 4 when

  1. x = 3
  2. x = 6 .

Solution

To evaluate x 4 when x = 3 , substitute 3 for x in the expression.

.
. .
Subtract. .

To evaluate x 4 when x = −6 , substitute −6 for x in the expression.

.
. .
Subtract. .

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Evaluate each expression:

y 7 when

  1. y = 5
  2. y = −8

  1. −2
  2. −15

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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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