2. (HUN) Find all real numbers $x$ for which  $$ \sqrt{3-x}-\sqrt{x+1}>\frac{1}{2} . $$
2. We note that $f(x)=\sqrt{3-x}-\sqrt{x+1}$ is well-defined only for $-1 \leq x \leq 3$ and is decreasing (and obviously continuous) on this interval. We also note that $f(-1)=2>1 / 2$ and $f(1-\sqrt{31} / 8)=\sqrt{(1 / 4+\sqrt{31} / 4)^{2}}-$ $\sqrt{(1 / 4-\sqrt{31} / 4)^{2}}=1 / 2$. Hence the inequality is satisfied for $-1 \leq x<$ $1-\sqrt{31} / 8$.