Absolute value equation solver
An absolute value is a distance from zero, so there are usually two cases to solve.
Try:
What counts as an absolute value equation?
|u| = k means “u is k units from zero”, so u = k or u = −k. That’s why most absolute value equations have two solutions.
If the absolute value is set equal to a negative number there is no solution, because a distance can’t be negative. If it equals zero, there is exactly one.
How to solve |u| = k
- Isolate the absolute value, so it’s alone on one side.
- If k is negative, stop: there is no solution.
- Otherwise write two equations: u = k and u = −k.
- Solve each one.
- Check both answers in the original equation.
Frequently asked questions
Why are there two answers?
Two numbers are the same distance from zero: k and −k. So the expression inside the bars can equal either one.
Can I split into cases before isolating the absolute value?
No. In 3|x − 2| + 1 = 13 you must first get |x − 2| = 4. Splitting too early gives wrong answers.
What about |x| < 3?
That’s an absolute value inequality, and it becomes a compound inequality: −3 < x < 3. Try it on the inequality solver.