3.3 Subtract integers (Page 3/6)

Prealgebra Page 3 / 6

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:

ⓐ $13 - 8 and 13 + (−8)$
ⓑ $17 - 9 and −17 + (−9)$

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:

ⓐ $21 - 13 and 21 + (−13)$
ⓑ $−11 - 7 and −11 + (−7)$

ⓐ 8, 8
ⓑ −18, −18

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Simplify each expression:

ⓐ $15 - 7 and 15 + (−7)$
ⓑ $−14 - 8 and −14 + (−8)$

ⓐ 8, 8
ⓑ −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:

ⓐ $9 - (−15) and 9 + 15$
ⓑ $−7 - (−4) and −7 + 4$

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:

ⓐ $6 - (−13) and 6 + 13$
ⓑ $−5 - (−1) and −5 + 1$

ⓐ 19, 19
ⓑ −4, −4

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Simplify each expression:

ⓐ $4 - (−19) and 4 + 19$
ⓐ $−4 - (−7) and −4 + 7$

ⓐ 23, 23
ⓑ 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:

$8 - (−3 - 1) - 9$

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Simplify the expression:

$12 - (−9 - 6) - 14$

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Simplify: $3 \cdot 7 - 4 \cdot 7 - 5 \cdot 8 .$

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:

$6 \cdot 2 - 9 \cdot 1 - 8 \cdot 9 .$

−69

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Simplify the expression:

$2 \cdot 5 - 3 \cdot 7 - 4 \cdot 9$

−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 $x - 4 when$

ⓐ $x = 3$
ⓑ $x = 6 .$

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:

$y - 7 when$

ⓐ $y = 5$
ⓑ $y = −8$

ⓐ −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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