Absolute Value Inequalities Examples in Math

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

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 inequalities describe values within or outside a fixed distance from a center.

|x-a|<r means stay inside a radius; |x-a|>r means outside it.

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: Translate absolute-value inequalities into equivalent compound inequalities using the distance interpretation.

Common stuck point: Use AND for less-than (|x|<k gives -k<x<k); use OR for greater-than (|x|>k gives x<-k or x>k).

Sense of Study hint: Draw the center and radius on a number line before writing inequality cases.

Worked Examples

Example 1

easy
Solve |x| < 5.

Solution

  1. 1
    |x| < 5 means x is less than 5 units from zero.
  2. 2
    This translates to -5 < x < 5.
  3. 3
    Interval notation: (-5, 5).

Answer

-5 < x < 5
For |A| < k (less than), the solution is a compound inequality -k < A < k. Think of it as 'between.'

Example 2

medium
Solve |2x - 1| \geq 3.

Practice Problems

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

Example 1

easy
Solve |x - 4| \leq 2.

Example 2

hard
Solve |3x + 6| > 0.
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Background Knowledge

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

absolute valueinequalitiesgraphing inequalities