Inequality with two absolute values by squaring (|x| < |x-1|)

Statement

Solve the inequality $|x|<|x-1|$.

Solution

Both sides are absolute values, so we can square (the direction is preserved): $x^2<(x-1)^2 \iff x^2<x^2-2x+1 \iff 0<-2x+1 \iff x<\frac12.$ Solution: $\boxed{x<\dfrac12}$.

Links

Topics: Parabola
Concepts: Absolute value inequalities · Absolute value
Methods: Absolute value mixed cases
Skills: Solving inequalities
Exercise type: Solving an inequality