Com10atorics wrote: Solve the inequation $\sqrt {3-x}-\sqrt {x+1}>\frac {1}{2}$. $\iff$ $x\in[-1,3]$ and $\sqrt{3-x}>\frac 12+\sqrt{x+1}>0$ $\iff$ (squaring) $x\in[-1,3]$ and $\frac 74-2x>\sqrt{x+1}$ $\iff$ (squaring again) $x\in[-1,\frac 78]$ and $4x^2-8x+\frac{33}{16}>0$ And so $\boxed{x\in\left[-1,\frac{8-\sqrt{31}}8\right)}$