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Absolute Value Equations and Inequalities

A free College Algebra lesson from the “Linear Equations and Inequalities” unit, with a worked example and practice problems including step-by-step solutions.

An absolute value equation describes distance from zero or distance from a target value. Because distance can be reached in two directions, equations like |x - a| = b usually create two cases: x - a = b and x - a = -b. This matters because absolute value problems often represent tolerances, error bounds, and symmetric distances. When practicing, isolate the absolute value first, then split into two equations if the right side is positive. A common mistake is forgetting the negative case or trying to split before the absolute value expression is isolated.

What you'll learn

Why it matters: Manufacturing tolerances, measurement error, and target ranges all use absolute value because they measure distance from a center point. Splitting into two cases is not a trick; it represents the two directions a value can be from the target.

Worked example

Problem. Solve |x - 4| = 9.

  1. Write two cases: x - 4 = 9 or x - 4 = -9.
  2. Solve each equation.
  3. x = 13 or x = -5.

Answer: x = 13 or x = -5

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.

Practice problems

1. Solve |x| = 7.

Choices: x = 7 or x = -7 · x = 7 only · x = -7 only · No solution

Show solution
  1. Both numbers are 7 units from zero.

Answer: x = 7 or x = -7

2. Solve |x + 2| = 5.

Choices: x = 3 or x = -7 · x = 7 or x = -3 · x = 5 or x = -5 · No solution

Show solution
  1. x + 2 = 5 or -5.

Answer: x = 3 or x = -7

3. Solve |x - 1| = -4.

Choices: No solution · x = 5 or x = -3 · x = -4 · x = 4

Show solution
  1. Absolute value cannot equal a negative number.

Answer: No solution

4. Solve |x - 3| = 10.

Choices: x = 13 or x = -7 · x = 7 or x = -13 · x = 10 or x = 3 · No solution

Show solution
  1. Set x - 3 equal to 10 and -10.

Answer: x = 13 or x = -7

5. Solve |2x| = 14.

Choices: x = 7 or x = -7 · x = 14 or x = -14 · x = 2 or x = -2 · No solution

Show solution
  1. 2x can equal 14 or -14.

Answer: x = 7 or x = -7

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