Absolute value equations and inequalities

Absolute value means distance

The absolute value of a number is its distance from \(0\) on the number line. Distance is never negative, so \(|7| = 7\) and \(|-7| = 7\).

The second line can look strange: if \(a\) is negative, then \(-a\) is positive. For \(a = -7\), \(-a = 7\).

More generally, \(|x - c|\) is the distance between \(x\) and \(c\). Keep that picture in mind: it explains every rule below.

Absolute value equations

Absolute value inequalities

Think distance again. \(|x| < 3\) says "\(x\) is less than \(3\) units from \(0\)": it's between \(-3\) and \(3\). And \(|x| > 3\) says "more than \(3\) units away": it's outside, beyond \(-3\) or beyond \(3\).

Variables on both sides: check for extraneous solutions

When the other side contains \(x\), like \(|x - 3| = 2x\), you can still split into two equations. But the right side might be negative for some of the answers, and an absolute value can never equal a negative number. Those answers are extraneous: they come out of the algebra but don't satisfy the original equation.

Practice