Modulus equations and inequalities

For \(k>0\), \(|u|=k\) means \(u=k\) or \(u=-k\); \(|u|<k\) means \(-k<u<k\); \(|u|>k\) means \(u<-k\) or \(u>k\). Replace strict signs by inclusive signs as appropriate.

Example

\(|2x-3|=5\) gives \(x=-1\) or \(x=4\). For \(|2x-3|<5\), \(-5<2x-3<5\), so \(-1<x<4\).

If the right-hand side is variable, check its sign before squaring. For \(|x|=x+2\), require \(x+2\ge0\); squaring gives \(x^2=(x+2)^2\), so \(x=-1\), which satisfies the original equation.

For more complicated modulus expressions use their piecewise definitions or graphing technology. Reject values outside the original domains.