<s>[INST] Determine all real $x$ that satisfies $\sqrt{3-x} - \sqrt{x+1} > \frac{1}{2}$ I have this equation that I have to solve:

Determine all real $x$ that satisfies $\sqrt{3-x} - \sqrt{x+1} > \frac{1}{2}$

Maybe it involves means? The square roots are leaning towards it... Could someone help me?
 [/INST] the inequality is given by
$$\sqrt{(3-x)}>1/2+\sqrt{x+1}$$ and we have $$-1\le x\le 3$$
after squaring we get
$$\frac{7}{4}-2x>\sqrt{x+1}$$
Now it must be $$\frac{7}{8}>x\geq -1$$
squaring again we have
$$x^2-2x+\frac{33}{64}>0$$
Can you proceed?
solving the last inequality we obtain $$-1\le x<\frac{1}{8}(8-\sqrt{31})$$
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