Math
Absolute value equations.
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 · 5 phases
Step by step.
- Isolate the absolute value expression on one side of the equation.
- If |expression| = k where k > 0, write two equations: expression = k and expression = −k.
- Solve each equation separately.
- Check both solutions in the original equation to eliminate extraneous solutions.
- If k < 0, there is no solution; if k = 0, there is exactly one solution.
Examples
Real-world.
- 1 |x − 3| = 5 gives x − 3 = 5 or x − 3 = −5, so x = 8 or x = −2
- 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)