Example 21. Solve the inequality

$$
\log _{2}\left(\sqrt{x^{2}-4 x}+3\right)>\log _{1 / 2} \frac{2}{\sqrt{x^{2}-4 x}+\sqrt{x+1}+1}+1
$$
Solution. The domain of admissible values of the inequality consists of all $x$ satisfying the system

$$
\left\{\begin{array}{l}
x^{2}-4 x \geqslant 0 \\
x+1 \geqslant 0
\end{array}\right.
$$

i.e., consists of the intervals $-1 \leqslant x \leqslant 0$ and $4 \leqslant x\sqrt{x^{2}-4 x}+\sqrt{x+1}+1
$$

i.e., the inequality

$$
\sqrt{x+1}<2
$$

The last inequality is equivalent to the inequality

$$
0 \leqslant x+1<4
$$

i.e., the inequality $-1 \leq x<3$. Of these values of $x$, only those from the interval $-1 \leq x \leq 0$ belong to the domain of admissible values of the original inequality.

Thus, the solution to the original inequality is the interval $-1 \leqslant x \leqslant 0$.