What the modulus means
The modulus (or absolute value) of a number is its distance from zero, ignoring sign. We write |x|. So |5| = 5 and |−5| = 5. The modulus is never negative.
Key idea
|x| = x when x ≥ 0, and |x| = −x when x < 0. To solve |A| = b (with b ≥ 0), write A = b or A = −b.
The graph of y = |x|
The graph is a V-shape with its vertex at the origin. For x ≥ 0 it is the line y = x; for x < 0 it is the line y = −x. Adding a constant, y = |x| + 2, lifts the whole V up by 2, so the range becomes y ≥ 2.
Worked example
Solve |2x − 1| = 5. Either 2x − 1 = 5, giving 2x = 6 and x = 3; or 2x − 1 = −5, giving 2x = −4 and x = −2. So x = 3 or x = −2.
Remember
- |x| ≥ 0 for every x.
- |A| = b has two cases: A = b and A = −b.
- |A| = negative number has no solution.
- Always check both answers in the original equation.