1.3 Add and subtract integers  (Page 3/10)

We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value    of the resulting number.

In the next example, we simplify the expressions inside absolute value bars first, just like we do with parentheses.

Evaluate: ⓐ $\left|x\right|\text{when}x=\mathrm{-35}$ ⓑ $\left|\text{−}y\right|\text{when}y=\mathrm{-20}$ ⓒ $\text{−}\left|u\right|\text{when}u=12$ ⓓ $\text{−}\left|p\right|\text{when}p=\mathrm{-14}.$

Solution

ⓐ $\left|x\right|\text{when}x=\mathrm{-35}$

	$\left|x\right|$
Take the absolute value.	35

ⓑ $\left|\text{−}y\right|\text{when}y=\mathrm{-20}$

	$|-y|$
Simplify.	$\left|20\right|$
Take the absolute value.	20

ⓒ $\text{−}\left|u\right|\text{when}u=12$

	$-\left|u\right|$
Take the absolute value.	$-12$

ⓓ $\text{−}\left|p\right|\text{when}p=\mathrm{-14}$

	$-\left|p\right|$
Take the absolute value.	$-14$

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Add integers

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.

We will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.

We let one color (blue) represent positive. The other color (red) will represent the negatives. If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.

We will use the counters to show how to add the four addition facts using the numbers $5,\mathrm{-5}$ and $3,\mathrm{-3}.$

To add $5+3,$ we realize that $5+3$ means the sum of 5 and 3.

We start with 5 positives.
And then we add 3 positives.
We now have 8 positives. The sum of 5 and 3 is 8.

Now we will add $\mathrm{-5}+\left(\mathrm{-3}\right).$ Watch for similarities to the last example $5+3=8.$

To add $\mathrm{-5}+\left(\mathrm{-3}\right),$ we realize this means the sum of $\mathrm{-5}\text{and}-3.$

We start with 5 negatives.
And then we add 3 negatives.
We now have 8 negatives. The sum of −5 and −3 is −8.

In what ways were these first two examples similar?

• The first example adds 5 positives and 3 positives—both positives.
• The second example adds 5 negatives and 3 negatives—both negatives.