IMO 1962 - Problem 2

Problem: Find all real number $x$ for which $\sqrt{3-x}-\sqrt{x+1}>\frac{1}{2}$.
Solution: We note that $f(x)=\sqrt{3-x}-\sqrt{x+1}$ is well-definded only for $-1\le x\le 3$ and is dereasing (and obviously continous) on this interval. We also note that $f(-1)=2>\frac{1}{2}$ and $f(1-\frac{\sqrt{31}}{8})=\sqrt{(\frac{1}{4}+\frac{\sqrt{31}}{4})^2}-\sqrt{(\frac{1}{4}-\frac{\sqrt{31}}{4})^2}=\frac{1}{2}$. Hence the inequality is satisfied for $-1\le x<1-\frac{\sqrt{31}}{8}$.