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|x-a|<r means stay inside a radius; ∣x−a∣>r|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: An absolute-value inequality describes all values within a distance from a center (<<, an interval) or outside it (>>, two rays).

Common stuck point: The procedure for absolute value inequalities is the easy part; the trap is treating >> like <<. Asking "Are absolute-value bars set << or >> a value, asking for a range within or outside a distance (not an exact distance)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are absolute-value bars set << or >> a value, asking for a range within or outside a distance (not an exact distance)?

Worked Examples

Example 1

easy
Solve ∣x∣<5|x| < 5.

Answer

−5<x<5-5 < x < 5

First step

1
∣x∣<5|x| < 5 means xx is less than 5 units from zero.

Full solution

  1. 2
    This translates to −5<x<5-5 < x < 5.
  2. 3
    Interval notation: (−5,5)(-5, 5).
For ∣A∣<k|A| < k (less than), the solution is a compound inequality −k<A<k-k < A < k. Think of it as 'between.'

Example 2

medium
Solve ∣2x−1∣≥3|2x - 1| \geq 3.

Example 3

easy
Solve ∣x−1∣<3|x-1|<3 and graph on a number line.

Example 4

medium
A bolt's diameter must be within 0.020.02 mm of 55 mm. Express this as an absolute value inequality and find the acceptable range.

Example 5

hard
For what kk does ∣x−1∣≤k|x-1|\le k contain exactly 55 integers?

Practice Problems

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

Example 1

easy
Solve ∣x−4∣≤2|x - 4| \leq 2.

Example 2

hard
Solve ∣3x+6∣>0|3x + 6| > 0.

Example 3

easy
Solve ∣x∣<3|x|<3.

Example 4

easy
Solve ∣x∣>2|x|>2.

Example 5

easy
Solve ∣x∣≤4|x|\le4.

Example 6

easy
Solve ∣x∣≥5|x|\ge5.

Example 7

easy
Solve ∣x−1∣<2|x-1|<2.

Example 8

easy
Solve ∣x+2∣>3|x+2|>3.

Example 9

easy
Solve ∣x∣<0|x|<0.

Example 10

easy
Solve ∣x∣≥0|x|\ge0.

Example 11

medium
Solve 2∣x∣<102|x|<10.

Example 12

medium
Solve ∣2x−1∣≤5|2x-1|\le5.

Example 13

medium
Solve ∣3x+2∣>7|3x+2|>7.

Example 14

medium
Solve ∣x−3∣+2<6|x-3|+2<6.

Example 15

medium
Solve ∣x+4∣≥6|x+4|\ge6 and write it in interval notation.

Example 16

medium
Solve −∣x−2∣>−5-|x-2|>-5.

Example 17

medium
Solve ∣2x∣≤0|2x|\le0.

Example 18

medium
Solve 3∣x+1∣−2≤73|x+1|-2\le7.

Example 19

medium
Solve ∣x+3∣≥1|x+3|\ge1 and write it in interval notation.

Example 20

challenge
Solve ∣x−2∣<∣x+4∣|x-2|<|x+4|.

Example 21

challenge
For what values of kk does ∣x−3∣<k|x-3|<k have no solution?

Example 22

challenge
Solve ∣x∣+∣x−4∣≤6|x|+|x-4|\le6.

Example 23

easy
Solve ∣x∣<6|x|<6.

Example 24

easy
Solve ∣x∣≥7|x|\ge 7.

Example 25

easy
Solve ∣x−2∣≤5|x-2|\le 5.

Example 26

easy
Solve ∣x+3∣>1|x+3|>1.

Example 27

easy
Solve ∣x∣>0|x|>0.

Example 28

medium
Solve ∣2x+3∣<9|2x+3|<9.

Example 29

medium
Solve ∣3x−5∣≥4|3x-5|\ge 4.

Example 30

medium
Solve ∣x−4∣+2≤7|x-4|+2\le 7.

Example 31

medium
Solve 3∣x+1∣>123|x+1|>12.

Example 32

medium
Solve ∣2−x∣<4|2-x|<4.

Example 33

medium
Solve 5−∣x∣≥15-|x|\ge 1.

Example 34

hard
Solve ∣x−1∣<2x|x-1|<2x.

Example 35

hard
Solve ∣x+1∣>2x−3|x+1|>2x-3.

Example 36

hard
Solve ∣x−2∣+∣x+3∣≥10|x-2|+|x+3|\ge 10.

Example 37

medium
Solve −2∣x−1∣+10≥4-2|x-1|+10\ge 4.

Example 38

medium
Solve ∣x−3∣≥0|x-3|\ge 0.

Example 39

challenge
Solve ∣x∣+∣x−4∣≤10|x|+|x-4|\le 10.

Example 40

challenge
For what real aa does ∣x−1∣+∣x−a∣≥4|x-1|+|x-a|\ge 4 hold for ALL real xx?
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Background Knowledge

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

absolute valueinequalitiesgraphing inequalities