Absolute value equations

Math

Definition

Equations that contain an absolute value expression. To solve them, you set up two cases: one where the expression inside the absolute value is positive and one where it is negative, then solve each case separately.

How It Works

  1. Isolate the absolute value expression on one side of the equation.
  2. If |expression| = k where k > 0, write two equations: expression = k and expression = −k.
  3. Solve each equation separately.
  4. Check both solutions in the original equation to eliminate extraneous solutions.
  5. If k < 0, there is no solution; if k = 0, there is exactly one solution.

Examples

  • |x − 3| = 5 gives x − 3 = 5 or x − 3 = −5, so x = 8 or x = −2
  • |2x + 1| = −4 has no solution because absolute value cannot be negative
Key Fact

|expression| = k → expression = k or expression = −k (when k ≥ 0)

Related Terms

Absolute value Multi-step equations Compound inequalities Solving inequalities

Study This Concept

Practice absolute value equations with free review games in these units:

Algebra 2 Unit 1: Equations and Inequalities