Example 4. Solve the inequality

$$
\sqrt{x+2}>\sqrt{8-x^{2}}
$$
Solution. Inequality (4) is equivalent to the system of inequalities

$$
\left\{\begin{aligned}
8-x^{2} & \geqslant 0 \\
x+2 & >8-x^{2}
\end{aligned}\right.
$$

The solutions to the first inequality of this system are all $x$ for which $|x| \leqslant 2 \sqrt{2}$, i.e., all numbers in the interval $-2 \sqrt{2} \leqslant x \leqslant 2 \sqrt{2}$.

The solutions to the inequality $x+2>8-x^{2}$, i.e., the inequality $x^{2} + x - 6 > 0$, are all numbers from two intervals: $x < -3$ and $x > 2$.

Thus, the solutions to the system (5), and therefore to inequality (4), are all numbers in the interval $2 < x \leq 2 \sqrt{2}$.