# 3.1 Introduction to integers  (Page 3/9)

 Page 3 / 9

Find the opposite of each number:

1. $\phantom{\rule{0.2em}{0ex}}4\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}-3$

1. −4
2. 3

Find the opposite of each number:

1. $\phantom{\rule{0.2em}{0ex}}8\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}-5$

1. −8
2. 5

## 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.

 $10-4$ Between two numbers, the symbol indicates the operation of subtraction. We read $10-4$ as 10 minus $4$ . $-8$ In front of a number, the symbol indicates a negative number. We read $-8$ as negative eight . $-x$ In front of a variable or a number, it indicates the opposite. We read $-x$ as the opposite of $x$ . $-\left(-2\right)$ Here we have two signs. The sign in the parentheses indicates that the number is negative 2. The sign outside the parentheses indicates the opposite. We read $-\left(-2\right)$ as the opposite of $-2.$

## Opposite notation

$-a$ means the opposite of the number $a$

The notation $-a$ is read the opposite of $a.$

Simplify: $-\left(-6\right).$

## Solution

 $-\left(-6\right)$ The opposite of $-6$ is $6.$ $6$

Simplify:

$-\left(-1\right)$

1

Simplify:

$-\left(-5\right)$

5

## Integers

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

## Integers

Integers are counting numbers, their opposites    , and zero.

$\text{…}\phantom{\rule{0.2em}{0ex}}-3,-2,-1,0,1,2,3\text{…}$

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

Evaluate $-x:$

1. when $x=8$
2. when $x=-8.$

 ⓐ To evaluate $-x$ when $x=8$ , substitute $8$ for $x$ . $-x$ Simplify. $-8$
 ⓑ To evaluate $-x$ when $x=-8$ , substitute $-8$ for $x$ . $-x$ Simplify. $8$

Evaluate $-n:\phantom{\rule{1em}{0ex}}$

1. $\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}n=4\phantom{\rule{0.8em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}n=-4$

1. −4
2. 4

Evaluate: $-m:\phantom{\rule{1em}{0ex}}$

1. $\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}m=11\phantom{\rule{0.8em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}m=-11$

1. −11
2. 11

## Simplify expressions with absolute value

We saw that numbers such as $5$ and $-5$ 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 $|5|,$ and the absolute value of $-5$ is written as $|-5|$ as shown in [link] .

## Absolute value

The absolute value    of a number is its distance from $0$ on the number line.

The absolute value of a number $n$ is written as $|n|.$

$|n|\ge 0\phantom{\rule{0.2em}{0ex}}\text{for all numbers}$

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}|3|$
2. $\phantom{\rule{0.2em}{0ex}}|-44|$
3. $\phantom{\rule{0.2em}{0ex}}|0|$

## Solution

1.  $|3|$ 3 is 3 units from zero. $3$
2.  $|-44|$ −44 is 44 units from zero. $44$
3.  $|0|$ 0 is already zero. $0$

Simplify:

1. $\phantom{\rule{0.8em}{0ex}}|12|\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.8em}{0ex}}-|-28|$

1. 12
2. −28

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}|9|\phantom{\rule{1em}{0ex}}$
2. $\text{(b)}\phantom{\rule{0.2em}{0ex}}-|37|$

1. 9
2. −37

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

Evaluate:

1. $\phantom{\rule{0.2em}{0ex}}|x|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-35$
2. $\phantom{\rule{0.2em}{0ex}}|\mathit{\text{−y}}|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-20$
3. $\phantom{\rule{0.2em}{0ex}}-|u|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}u=12$
4. $\phantom{\rule{0.2em}{0ex}}-|p|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=-14$

## Solution

 ⓐ To find $|x|$ when $x=-35:$ $|x|$ Take the absolute value. $35$
 ⓑ To find $|-y|$ when $y=-20:$ $|-y|$ Simplify. $|20|$ Take the absolute value. $20$
 ⓒ To find $-|u|$ when $u=12:$ $-|u|$ Take the absolute value. $-12$
 ⓓ To find $-|p|$ when $p=-14:$ $-|p|$ Take the absolute value. $-14$

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

Evaluate:

1. $\phantom{\rule{0.2em}{0ex}}|x|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-17\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}|\mathit{\text{−y}}|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-39\phantom{\rule{1em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}-|m|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}m=22\phantom{\rule{1em}{0ex}}$
4. $\phantom{\rule{0.2em}{0ex}}-|p|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=-11$

1. 17
2. 39
3. −22
4. −11
1. $\phantom{\rule{0.2em}{0ex}}|y|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-23\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}|-y|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-21$
3. $\phantom{\rule{0.2em}{0ex}}-|n|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}n=37\phantom{\rule{1em}{0ex}}$
4. $\phantom{\rule{0.2em}{0ex}}-|q|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}q=-49$
1. 23
2. 21
3. −37
4. −49

Fill in $\text{<},\text{>},\text{or}=$ for each of the following:

1. $\phantom{\rule{0.2em}{0ex}}|-5|___-|-5|\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}8___-|-8|\phantom{\rule{1em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}-9___-|-9|\phantom{\rule{1em}{0ex}}$
4. $\phantom{\rule{0.2em}{0ex}}-|-7|___-7$

## Solution

To compare two expressions, simplify each one first. Then compare.

1.  $|-5|___-|-5|$ Simplify. $5___-5$ Order. $5>-5$
2.  $-|-8|=-8$ Simplify. $8___-8$ Order. $8>-8$
3.  $-|-9|=-9$ Simplify. $9___-9$ Order. $9>-9$
4.  $-|-7|=-7$ Simplify. $7___-7$ Order. $7>-7$

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