3.1 Introduction to integers  (Page 3/9)

Opposite notation

Just as the same word in English can have different meanings, the same symbol in algebra can have different meanings. The specific meaning becomes clear by looking at how it is used. You have seen the symbol $\text{“−”,}$ in three different ways.

Integers

The set of counting numbers, their opposites, and $0$ is the set of integers    .

We must be very careful with the signs when evaluating the opposite of a variable.

Evaluate $-x:$

1. ⓐ when $x=8$
2. ⓑ when $x=\mathrm{-8}.$

ⓐ To evaluate $-x$ when $x=8$ , substitute $8$ for $x$ .
	$-x$
Simplify.	$\mathrm{-8}$

ⓑ To evaluate $-x$ when $x=\mathrm{-8}$ , substitute $\mathrm{-8}$ for $x$ .
	$-x$
Simplify.	$8$

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Simplify expressions with absolute value

We saw that numbers such as $5$ and $\mathrm{-5}$ are opposites    because they are the same distance from $0$ on the number line. They are both five units from $0.$ The distance between $0$ and any number on the number line is called the absolute value    of that number. Because distance is never negative, the absolute value    of any number is never negative.

The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of $5$ is written as $\left|5\right|,$ and the absolute value of $\mathrm{-5}$ is written as $\left|\mathrm{-5}\right|$ as shown in [link] .

We treat absolute value bars just like we treat parentheses in the order of operations. We simplify the expression inside first.

Evaluate:

1. ⓐ $\left|x\right|\text{when}x=\mathrm{-35}$
2. ⓑ $\left|\mathit{\text{−y}}\right|\text{when}y=\mathrm{-20}$
3. ⓒ $-\left|u\right|\text{when}u=12$
4. ⓓ $-\left|p\right|\text{when}p=\mathrm{-14}$

Solution

ⓐ To find $\left|x\right|$ when $x=\mathrm{-35}:$
	$\left|x\right|$
Take the absolute value.	$35$

ⓑ To find $|-y|$ when $y=\mathrm{-20}:$
	$|-y|$
Simplify.	$\left|20\right|$
Take the absolute value.	$20$

ⓒ To find $-\left|u\right|$ when $u=12:$
	$-\left|u\right|$
Take the absolute value.	$\mathrm{-12}$

ⓓ To find $-\left|p\right|$ when $p=\mathrm{-14}:$
	$-\left|p\right|$
Take the absolute value.	$\mathrm{-14}$

Notice that the result is negative only when there is a negative sign outside the absolute value symbol.

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