Absolute Value Equations Examples in Math

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

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

Concept Recap

Absolute value equations solve for values whose distance from zero or another number matches a target amount.

An absolute-value equation is a distance problem — ∣x−2∣=5|x-2|=5 asks 'which xx is distance 5 from 2?' — two answers.

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: An absolute-value equation asks which values sit a fixed distance from a center, which usually splits into two cases.

Common stuck point: The procedure for absolute value equations is the easy part; the trap is keeping only the positive case. Asking "Is an expression inside absolute-value bars set equal to a constant, asking which values are that distance away?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is an expression inside absolute-value bars set equal to a constant, asking which values are that distance away?

Worked Examples

Example 1

easy
Solve ∣x−3∣=7|x - 3| = 7.

Answer

x=10 or x=−4x = 10 \text{ or } x = -4

First step

1
∣A∣=k|A| = k means A=kA = k or A=−kA = -k.

Full solution

  1. 2
    Case 1: x−3=7⇒x=10x - 3 = 7 \Rightarrow x = 10.
  2. 3
    Case 2: x−3=−7⇒x=−4x - 3 = -7 \Rightarrow x = -4.
  3. 4
    Check: ∣10−3∣=7|10-3| = 7 ✓ and ∣−4−3∣=7|-4-3| = 7 ✓
Absolute value equations always split into two cases because the expression inside can be either positive or negative. Both cases must be checked.

Example 2

medium
Solve ∣2x+1∣=5|2x + 1| = 5.

Example 3

easy
Solve ∣x−8∣=3|x-8|=3. Show both branches.

Example 4

medium
Solve ∣2x−3∣=∣x+4∣|2x-3|=|x+4|.

Example 5

medium
A thermostat keeps room temperature within 22 degrees of 70∘70^\circF. Write and solve the equation for the extreme temperatures.

Example 6

hard
Solve ∣3x−1∣=∣x+5∣|3x-1|=|x+5|.

Practice Problems

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

Example 1

easy
Solve ∣x∣=4|x| = 4.

Example 2

medium
Solve ∣x+2∣=−3|x + 2| = -3.

Example 3

easy
Solve ∣x∣=5|x|=5.

Example 4

easy
Solve ∣x∣=0|x|=0.

Example 5

easy
Solve ∣x∣=−4|x|=-4.

Example 6

easy
Solve ∣x−3∣=2|x-3|=2.

Example 7

easy
Solve ∣x+1∣=4|x+1|=4.

Example 8

easy
Solve ∣2x∣=8|2x|=8.

Example 9

easy
Solve ∣x∣=7|x|=7.

Example 10

easy
Solve ∣x−10∣=0|x-10|=0.

Example 11

medium
Solve 2∣x−1∣=102|x-1|=10.

Example 12

medium
Solve ∣3x−2∣=7|3x-2|=7.

Example 13

medium
Solve ∣x−4∣+3=8|x-4|+3=8.

Example 14

medium
Solve ∣2x+1∣=∣x−2∣|2x+1|=|x-2|.

Example 15

medium
Solve ∣x+5∣=3x−1|x+5|=3x-1 (check for extraneous roots).

Example 16

medium
Solve −3∣x∣=−12-3|x|=-12.

Example 17

medium
Solve ∣5−x∣=2|5-x|=2.

Example 18

medium
Solve ∣x∣2=3\frac{|x|}{2}=3.

Example 19

medium
Solve 4∣x∣−1=114|x|-1=11.

Example 20

challenge
For what values of kk does ∣x−2∣=k|x-2|=k have exactly two solutions, exactly one, and none?

Example 21

challenge
Solve ∣x−1∣=2x−4|x-1|=2x-4 and identify any extraneous root.

Example 22

challenge
Solve ∣x2−4∣=3|x^2-4|=3.

Example 23

easy
Solve ∣x∣=9|x|=9.

Example 24

easy
Solve ∣x−7∣=2|x-7|=2.

Example 25

easy
Solve ∣x+6∣=0|x+6|=0.

Example 26

easy
Solve ∣4x∣=20|4x|=20.

Example 27

medium
Solve ∣3x+1∣=10|3x+1|=10.

Example 28

medium
Solve ∣x∣+4=13|x|+4=13.

Example 29

medium
Solve 3∣x−2∣=153|x-2|=15.

Example 30

medium
Solve ∣x+1∣+5=12|x+1|+5=12.

Example 31

medium
Solve ∣4−2x∣=6|4-2x|=6.

Example 32

medium
Solve ∣x+2∣3=4\frac{|x+2|}{3}=4.

Example 33

hard
Solve ∣x−3∣=2x+1|x-3|=2x+1 and check for extraneous roots.

Example 34

hard
Solve ∣x+2∣=3x−4|x+2|=3x-4 and check for extraneous roots.

Example 35

hard
Solve ∣x2−5∣=4|x^2-5|=4.

Example 36

hard
Solve ∣x−1∣+∣x+2∣=5|x-1|+|x+2|=5.

Example 37

medium
Solve 5−∣x+2∣=15-|x+2|=1.

Example 38

challenge
Find all xx satisfying ∣∣x−1∣−3∣=2||x-1|-3|=2.

Example 39

challenge
For what values of kk does ∣x−2∣+∣x+1∣=k|x-2|+|x+1|=k have infinitely many solutions? exactly two? exactly one?
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Background Knowledge

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

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