Absolute Value Examples: 44 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Absolute Value.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The distance of a number from zero on the number line, always non-negative; written $|x|$ . For any real number, absolute value strips away the sign and returns only the magnitude.

$-5$ and $5$ are both 5 units from zero, so $|-5| = |5| = 5$ .

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Absolute value measures how far a number is from zero on the number line.

Common stuck point: The procedure for absolute value is the easy part; the trap is keeping the negative sign inside absolute value. Asking "Is the sign direction important, or only the distance?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the sign direction important, or only the distance?

Worked Examples

Example 1

easy

Evaluate

|{-7}| + |3|

Answer

10

First step

The absolute value of

-7

is the distance from 0:

|{-7}| = 7

Full solution

2
The absolute value of $3$ is $|3| = 3$ .
3
Add: $7 + 3 = 10$ .

Absolute value measures distance from zero on the number line and is always non-negative. For any number

a

|a| = a

a \ge 0

and

|a| = -a

a < 0

Example 2

medium

Evaluate

|5 - 12| - |2 - 9|

Example 3

medium

Compute

\big||{-3}|-|{-8}|\big|

Example 4

hard

Simplify

|x-2|

when

x<2

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Evaluate

|{-15}| - |{-4}|

Example 2

medium

Evaluate

|3 - 8| + |{-2}| - |5|

Example 3

easy

Compute

|-8|

Example 4

easy

Compute

|7|

Example 5

easy

Compute

|0|

Example 6

easy

Compute

-|6|

Example 7

easy

Compute

|3 - 8|

Example 8

easy

Compute

|-2| + |5|

Example 9

easy

What is the distance between

-3

and

4

on the number line?

Example 10

easy

Solve

|x| = 5

Example 11

medium

Solve

|x - 3| = 7

Example 12

medium

Compute

|{-4}| \times |{-3}|

Example 13

medium

Solve

|2x| = 10

Example 14

medium

Compute

|{-2}|^3

Example 15

medium

Evaluate

|x - 5|

when

x = 2

Example 16

medium

Solve

|x| < 4

. Write the solution as an interval.

Example 17

medium

Solve

|x| > 3

. Describe the solution.

Example 18

medium

Simplify

|x|

if you know

x < 0

Example 19

medium

Compute

\sqrt{(-6)^2}

Example 20

challenge

Solve

|2x - 1| = |x + 4|

Example 21

challenge

Solve

|x - 2| + |x + 1| = 5

Example 22

challenge

What is the minimum value of

|x - 1| + |x - 5|

, and where is it achieved?

Example 23

easy

Compute

|-12|

Example 24

easy

Compute

|9|

Example 25

easy

Compute

-|-5|

Example 26

easy

Compute

|10-4|

Example 27

easy

Compute

|2-9|

Example 28

medium

Compute

|3-7|+|7-3|

Example 29

medium

Compute

|-4|\cdot|-3|+|2|

Example 30

medium

Evaluate

|x|

x=-\frac{3}{4}

Example 31

medium

Compute

\sqrt{(-9)^2}

Example 32

medium

Order from smallest to largest:

|-3|,\ -|7|,\ |-5|,\ 0

Example 33

medium

|x|=8

, list all possible values of

x

Example 34

hard

Compute

|3-\pi|+|\pi-4|

Example 35

hard

For what value of

x

|x-3|+|x-7|

minimized?

Example 36

hard

Find all real

x

with

|x|+x=0

Example 37

medium

Compute

|2|^3-|-3|^2

Example 38

medium

Compute

\frac{|-12|}{|-4|}

Example 39

challenge

Find the smallest possible value of

|x-1|+|x-4|+|x-9|

over real

x

, and where it occurs.

Example 40

challenge

For what real

x

does

|x-2|=|2-x|

fail?

← Back to Absolute Value Formula Explained Practice Mode

Related Concepts

Integers Distance Formula Inequalities

Background Knowledge

These ideas may be useful before you work through the harder examples.

integers