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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

Practice problems

1. Solve |x| = 7.

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve |x| = 7.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Both numbers are 7 units from zero.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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. Core Practice: First identify exactly what the question is asking: Solve |x + 2| = 5.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. x + 2 = 5 or -5.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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. Challenge: First identify exactly what the question is asking: Solve |x - 1| = -4.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Absolute value cannot equal a negative number.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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. Core Practice: First identify exactly what the question is asking: Solve |x - 3| = 10.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Set x - 3 equal to 10 and -10.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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. Core Practice: First identify exactly what the question is asking: Solve |2x| = 14.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. 2x can equal 14 or -14.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x = 7 or x = -7

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