For |u|=d, no solution exists if d<0; otherwise solve u=d or u=-d. When d=0, these coincide. For |ax+b|=cx+d, first require cx+d 0, then solve ax+b=cx+d and ax+b=-(cx+d), checking both candidates. For |ax+b|=|cx+d|, both sides are nonnegative, so solve ax+b=+-(cx+d). For |ax2+bx+c|=d, solve each quadratic ax2+bx+c=+- d. Keep all real roots, remove duplicates and check in the original equation.

For \(|u|=d\), no solution exists if \(d<0\); otherwise solve \(u=d\) or \(u=-d\). When \(d=0\), these coincide.

For \(|ax+b|=cx+d\), first require \(cx+d\ge0\), then solve \(ax+b=cx+d\) and \(ax+b=-(cx+d)\), checking both candidates. For \(|ax+b|=|cx+d|\), both sides are nonnegative, so solve \(ax+b=\pm(cx+d)\).

For \(|ax^2+bx+c|=d\), solve each quadratic \(ax^2+bx+c=\pm d\). Keep all real roots, remove duplicates and check in the original equation.

Solving absolute-value equations

Solving absolute-value equations