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Learn Index · Patterns and Algebra Grade 9–10

Absolute-value equations and inequalities

Equations and inequalities involving distance from zero.

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Absolute value as distance

The absolute value of x, written |x|, is its distance from zero. Distances are never negative.

More generally, |x − a| is the distance between x and a on the number line. So |x − 3| = 5 asks: "Which numbers are 5 units from 3?" The answers are 8 and −2.

Absolute-value equations

|x| = 7 has two solutions: x = 7 or x = −7.

Method: isolate the absolute value, then split into two cases.

Example: |2x − 1| = 9

  • Case 1: 2x − 1 = 9 → x = 5
  • Case 2: 2x − 1 = −9 → x = −4

Solutions: x = 5 or x = −4

Special cases

Equation Solutions
|x| = 0 one: x = 0
|x| = −3 none: distance can’t be negative

Always isolate first: 3|x + 2| − 4 = 11 → 3|x + 2| = 15 → |x + 2| = 5 → x = 3 or x = −7.

Checking for extraneous solutions

When the variable appears outside the absolute value too, check each answer. A value that doesn’t work in the original is an extraneous solution.

Example: |x + 4| = 2x

  • Case 1: x + 4 = 2x → x = 4. Check: |8| = 8 ✓
  • Case 2: x + 4 = −2x → x = −4/3. Check: |8/3| = 8/3, but 2x = −8/3 ✗

Only x = 4 works.

Absolute-value inequalities

"Less than": an "and" (between) inequality

The absolute-value inequality |x| < 4 means x is within 4 of zero: −4 < x < 4.

Example: |x − 2| ≤ 5 → −5 ≤ x − 2 ≤ 5 → −3 ≤ x ≤ 7

"Greater than": an "or" (outside) inequality

The inequality |x| > 4 means x is more than 4 from zero: x < −4 or x > 4.

Example: |3x + 1| > 8

  • 3x + 1 > 8 → x > 7/3
  • 3x + 1 < −8 → x < −3

Solution: x < −3 or x > 7/3

Memory aid: "less thAND" (between), "greatOR" (outside).

Real-life use: tolerance

A bolt should be 50 mm long, with an allowed error of 0.2 mm: |L − 50| ≤ 0.2 → 49.8 ≤ L ≤ 50.2.

Worked examples

Example 1: |x + 3| = 10 → x = 7 or x = −13.

Example 2: |x| − 6 = −2 → |x| = 4 → x = ±4.

Example 3: |x − 1| < 3 → −2 < x < 4.

Common mistakes

  • Only finding the positive case.
  • Not isolating the absolute value first.
  • Writing an "and" when it should be "or" (and vice versa).
  • Forgetting that |x| = negative has no solution.

Try it yourself

  1. |x| = 12
  2. |x − 5| = 3
  3. |2x + 4| = 0
  4. |x + 1| ≤ 4
  5. |x − 3| > 2

Answers: 1) ±12 2) 8 or 2 3) −2 4) −5 ≤ x ≤ 3 5) x < 1 or x > 5

Quick recap

  • Absolute value is distance, so it’s never negative.
  • An absolute-value equation splits into two cases.
  • "<" gives a between solution; ">" gives an outside solution.