1.3 Add and subtract integers  (Page 2/10)

Find: ⓐ the opposite of 7 ⓑ the opposite of $\mathrm{-10}$ ⓒ $\text{−}\left(\mathrm{-6}\right).$

Solution

ⓐ −7 is the same distance from 0 as 7, but on the opposite side of 0.	The opposite of 7 is −7.
ⓑ 10 is the same distance from 0 as −10, but on the opposite side of 0.	The opposite of −10 is 10.
ⓒ −(−6)	The opposite of −(−6) is −6.

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Our work with opposites gives us a way to define the integers.The whole numbers and their opposites are called the integers    . The integers are the numbers $\text{…}-3,\mathrm{-2},\mathrm{-1},0,1,2,3\text{…}$

When evaluating the opposite of a variable    , we must be very careful. Without knowing whether the variable represents a positive or negative number, we don’t know whether $\text{−}x$ is positive or negative. We can see this in [link] .

Simplify: expressions with absolute value

We saw that numbers such as $2\text{and}\mathrm{-2}$ are opposites because they are the same distance from 0 on the number line. They are both two units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.

For example,

• $\mathrm{-5}\text{is}5$ units away from $0,$ so $\left|\mathrm{-5}\right|=5.$
• $5\text{is}5$ units away from $0,$ so $\left|5\right|=5.$

[link] illustrates this idea.

The absolute value of a number is never negative (because distance cannot be negative). The only number with absolute value equal to zero is the number zero itself, because the distance from $0\text{to}0$ on the number line is zero units.

Mathematicians say it more precisely, “absolute values are always non-negative.” Non-negative means greater than or equal to zero.

In the next example, we’ll order expressions with absolute values. Remember, positive numbers are always greater than negative numbers!