2.2 Absolute Value and Distance on the Number Line

Narrated lesson: Absolute Value and Distance on the Number Line

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Find Opposites

On the number line, the negative numbers are a mirror image of the positive numbers with zero in the middle. Because the numbers $2$ and $−2$ are the same distance from zero, they are called opposites. The opposite of $2$ is $−2,$ and the opposite of $−2$ is $2$ as shown in Figure 2.3(a). Similarly, $3$ and $−3$ are opposites as shown in Figure 2.3(b).

This figure shows two number lines. The first has points negative 2 and positive 2 labeled. Below the first line the statement is the numbers negative 2 and 2 are opposites. The second number line has the points negative 3 and 3 labeled. Below the number line is the statement negative 3 and 3 are opposites. — **Figure 2.3**

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 $“−”,$ in three different ways.

$10 minus 4$	Between two numbers, the symbol indicates the operation of subtraction. We read $10 minus 4$ 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 $x$ .
$minus open parenthesis −2 close parenthesis$	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 $minus open parenthesis −2 close parenthesis$ as the opposite of $−2.$

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.

Example 3

Evaluate $- x :$

ⓐ when $x equals 8$
ⓑ when $x equals −8.$

Show solution

ⓐ To evaluate $- x$ when $x equals 8$ , substitute $8$ for $x$ .
	$- x$

Simplify.	$−8$

ⓑ To evaluate $- x$ when $x equals −8$ , substitute $−8$ for $x$ .
	$- x$

Simplify.	$8$

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 $vertical bar 5 vertical bar,$ and the absolute value of $−5$ is written as $vertical bar −5 vertical bar$ as shown in Figure 2.6.

This figure is a number line. The points negative 5 and 5 are labeled. Above the number line the distance from negative 5 to 0 is labeled as 5 units. Also above the number line the distance from 0 to 5 is labeled as 5 units. — **Figure 2.6**

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

Example 5

Evaluate:

ⓐ $vertical bar x vertical bar when x equals −35$
ⓑ $vertical bar −y vertical bar when y equals −20$
ⓒ $- | u | when u = 12$
ⓓ $- | p | when p = −14$

Show solution

ⓐ To find $\| x \|$ when $x equals −35:$
	$\| x \|$

Take the absolute value.	$35$

ⓑ To find $\| - y \|$ when $y equals −20:$
	$\| - y \|$

Simplify.	$vertical bar 20 vertical bar$
Take the absolute value.	$20$

ⓒ To find $- \| u \|$ when $u equals 12:$
	$- \| u \|$

Take the absolute value.	$−12$

ⓓ To find $- \| p \|$ when $p equals −14:$
	$- \| p \|$

Take the absolute value.	$−14$

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

Example 6

Fill in $<, >, or equals$ for each of the following:

ⓐ $vertical bar −5 vertical bar ___ minus vertical bar −5 vertical bar$
ⓑ $8 ___ minus vertical bar −8 vertical bar$
ⓒ $−9 ___ minus vertical bar −9 vertical bar$
ⓓ $minus vertical bar −7 vertical bar ___ −7$

Show solution

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

ⓐ
	$vertical bar −5 vertical bar ___ minus vertical bar −5 vertical bar$
Simplify.	$5 ___ −5$
Order.	$5 greater than −5$

ⓑ
	$8 ___ minus vertical bar −8 vertical bar$
Simplify.	$8 ___ minus 8$
Order.	$8 greater than minus 8$

ⓒ
	$−9 ___ minus vertical bar −9 vertical bar$
Simplify.	$−9 ___ minus 9$
Order.	$−9 equals −9$

ⓓ
	$minus vertical bar −7 vertical bar ___ −7$
Simplify.	$−7 ___ minus 7$
Order.	$−7 = - 7$

Absolute value bars act like grouping symbols. First simplify inside the absolute value bars as much as possible. Then take the absolute value of the resulting number, and continue with any operations outside the absolute value symbols.

Example 7

Simplify:

ⓐ $vertical bar 9 −3 vertical bar$
ⓑ $4 vertical bar −2 vertical bar$

Show solution

For each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses.

ⓐ
	\|9−3\|
Simplify inside the absolute value sign.	\|6\|
Take the absolute value.	6

ⓑ
	4\|−2\|
Take the absolute value.	4⋅2
Multiply.	8

Example 8

Simplify: $| 8 + 7 | - | 5 + 6 | .$

Show solution

For each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses.

	\|8+7\|−\|5+6\|
Simplify inside each absolute value sign.	\|15\|−\|11\|
Subtract.	4

Example 9

Simplify: $24 - | 19 - 3 (6 - 2) | .$

Show solution

We use the order of operations. Remember to simplify grouping symbols first, so parentheses inside absolute value symbols would be first.

	$24 - \| 19 - 3 (6 - 2) \|$
Simplify in the parentheses first.	$24 - \| 19 - 3 (4) \|$
Multiply $3 open parenthesis 4 close parenthesis$ .	$24 - \| 19 - 12 \|$
Subtract inside the absolute value sign.	$24 minus vertical bar 7 vertical bar$
Take the absolute value.	$24 minus 7$
Subtract.	$17$

Practice Makes Perfect

Find Opposites

In the following exercises, find the opposite of each number.

ⓐ $2$
ⓑ $−6$

Show solution

ⓐ −2
ⓑ 6

ⓐ $9$
ⓑ $−4$

ⓐ $−8$
ⓑ $1$

Show solution

ⓐ 8
ⓑ −1

ⓐ $−2$
ⓑ $6$

In the following exercises, simplify.

$minus open parenthesis −4 close parenthesis$

Show solution

$minus open parenthesis −8 close parenthesis$

$minus open parenthesis −15 close parenthesis$

Show solution

$minus open parenthesis −11 close parenthesis$

In the following exercises, evaluate.

$- m when$

ⓐ $m equals 3$
ⓑ $m equals −3$

Show solution

ⓐ −3
ⓑ 3

$- p when$

ⓐ $p equals 6$
ⓑ $p equals −6$

$- c when$

ⓐ $c equals 12$
ⓑ $c equals −12$

Show solution

ⓐ −12;
ⓑ 12

$- d when$

ⓐ $d equals 21$
ⓑ $d equals −21$

Simplify Expressions with Absolute Value

In the following exercises, simplify each absolute value expression.

ⓐ $vertical bar 7 vertical bar$
ⓑ $vertical bar −25 vertical bar$
ⓒ $vertical bar 0 vertical bar$

Show solution

ⓐ 7
ⓑ 25
ⓒ 0

ⓐ $vertical bar 5 vertical bar$
ⓑ $vertical bar 20 vertical bar$

ⓐ $vertical bar −32 vertical bar$
ⓑ $vertical bar −18 vertical bar$

Show solution

ⓐ 32
ⓑ 18
ⓒ 16

ⓐ $vertical bar −41 vertical bar$
ⓑ $vertical bar −40 vertical bar$

In the following exercises, evaluate each absolute value expression.

ⓐ $vertical bar x vertical bar when x equals −28$
ⓑ $| - u | when u = −15$

Show solution

ⓐ 28
ⓑ 15

ⓐ $vertical bar y vertical bar when y equals −37$
ⓑ $| - z | when z = −24$

ⓐ $- | p | when p = 19$
ⓑ $- | q | when q = −33$

Show solution

ⓐ −19
ⓑ −33

ⓐ $- | a | when a = 60$
ⓑ $- | b | when b = −12$

In the following exercises, fill in $<, >, or equals$ to compare each expression.

ⓐ $−6 __ vertical bar −6 vertical bar$
ⓑ $minus vertical bar −3 vertical bar __ −3$

Show solution

ⓐ <
ⓑ =

ⓐ $−8 __ vertical bar −8 vertical bar$
ⓑ $minus vertical bar −2 vertical bar __ −2$

ⓐ $vertical bar −3 vertical bar __ minus vertical bar −3 vertical bar$
ⓑ $4 __ minus vertical bar −4 vertical bar$

Show solution

ⓐ >
ⓑ >

ⓐ $vertical bar −5 vertical bar __ minus vertical bar −5 vertical bar$
ⓑ $9 __ minus vertical bar −9 vertical bar$

In the following exercises, simplify each expression.

$vertical bar 8 minus 4 vertical bar$

Show solution

$vertical bar 9 minus 6 vertical bar$

$8 vertical bar −7 vertical bar$

Show solution

$5 vertical bar −5 vertical bar$

$| 15 - 7 | - | 14 - 6 |$

Show solution

$| 17 - 8 | - | 13 - 4 |$

$18 - | 2 (8 - 3) |$

Show solution

$15 - | 3 (8 - 5) |$

$8 open parenthesis 14 minus 2 vertical bar −2 vertical bar close parenthesis$

Show solution

$6 open parenthesis 13 minus 4 vertical bar −2 vertical bar close parenthesis$

Corequisite Skills for Introductory Statistics is a custom course edition compiled for Introductory Statistics corequisite courses from OpenStax Contemporary Mathematics, Elementary Algebra 2e, Intermediate Algebra 2e, Prealgebra 2e. Content by OpenStax is reproduced as published, apart from any corrections recorded for this edition. Sections are renumbered to follow the course outline. The course outline is adapted from Larry Green's Support Course for Elementary Statistics (LibreTexts, https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Support_Course_for_Elementary_Statistics), used under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Lessons 2.4 and 6.7 are written for this edition by XYZ Homework. Every OpenStax lesson is abridged to the subsections and practice exercises this course uses. Lessons are headed with their course titles, not the source section titles. Access the complete originals free at openstax.org. License: CC-BY-NC-SA-4.0.

ⓑ
	$vertical bar −44 vertical bar$
−44 is 44 units from zero.	$44$