Disequazione $|2x-1|<|x+3|$

Text

Solve the inequality $|2x-1|<|x+3|.$

Solution

Since both sides are non-negative, we may square while keeping the direction: $(2x-1)^2<(x+3)^2\implies 4x^2-4x+1<x^2+6x+9\implies 3x^2-10x-8<0.$ The roots of $3x^2-10x-8$ are $x=\dfrac{10\pm\sqrt{100+96}}{6}=\dfrac{10\pm 14}{6}$, i.e. $x=-\tfrac23$ and $x=4$. The trinomial is negative between the roots: $\boxed{-\tfrac23<x<4.}$