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 asks 'which x 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:

|A| = k means A = k or A = -k when k \ge 0; if k < 0 there is no solution.

Common stuck point: Students often find only one of the two solution branches and stop — always set up and solve both cases.

Sense of Study hint: Write both cases explicitly before solving.

Worked Examples

Example 1

easy
Solve |x - 3| = 7.

Solution

  1. 1
    |A| = k means A = k or A = -k.
  2. 2
    Case 1: x - 3 = 7 \Rightarrow x = 10.
  3. 3
    Case 2: x - 3 = -7 \Rightarrow x = -4.
  4. 4
    Check: |10-3| = 7 ✓ and |-4-3| = 7 ✓

Answer

x = 10 \text{ or } x = -4
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.

Practice Problems

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

Example 1

easy
Solve |x| = 4.

Example 2

medium
Solve |x + 2| = -3.
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Background Knowledge

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

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