Example 20. Solve the inequality

$$
\frac{\log _{5}\left(x^{2}-4 x-11\right)^{2}-\log _{11}\left(x^{2}-4 x-11\right)^{3}}{2-5 x-3 x^{2}} \geqslant 0
$$
Solution. The domain of definition (ODZ) of the given inequality consists of all $x$ satisfying the system

$$
\left\{\begin{array}{l}
x^{2}-4 x-11>0 \\
-3 x^{2}-5 x+2 \neq 0
\end{array}\right.
$$

i.e., it is the union of three intervals: $-\infty<x<-2$, $-2<x<2-\sqrt{15}, 2+\sqrt{15}<x<+\infty$. Since for such values of $x$ we have

$$
\log _{5}\left(x^{2}-4 x-11\right)^{2}=2 \log _{5}\left(x^{2}-4 x-11\right)
$$

and

$$
\log _{i \overline{1}}\left(x^{2}-4 x-11\right)^{3}=\frac{3 \log _{5}\left(x^{2}-4 x-11\right)}{\log _{5} 11}
$$

the original inequality is equivalent on its ODZ to the inequality

$$
\left(2-\frac{3}{\log _{5} 11}\right) \frac{\log _{5}\left(x^{2}-4 x-11\right)}{2-5 x-3 x^{2}} \geqslant 0
$$

From the valid numerical inequality $11<5^{3 / 2}$ it follows that $\log _{5} 11<3 / 2$ and $2-3 / \log _{5} 11<0$. Therefore, inequality (15) is equivalent to the inequality

$$
\frac{\log _{5}\left(x^{2}-4 x-11\right)}{3 x^{2}+5 x-2} \geqslant 0
$$

This inequality on the ODZ of the original inequality is equivalent to the union of two systems

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-196.jpg?height=139&width=1080&top_left_y=727&top_left_x=488)

i.e., the union of

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-196.jpg?height=139&width=842&top_left_y=930&top_left_x=610)

The set of solutions of the first system in the union (17) consists of the intervals $-\infty<x<-2,6 \leqslant x<+\infty$, which belong to the ODZ of the original inequality. The set of solutions of the second system in the union (17) is the interval $-2<x<1 / 3$. Among these numbers, only those from the interval $-2<x<2-\sqrt{15}$ belong to the ODZ of the original inequality.

Thus, the set of all solutions of the original inequality consists of three intervals: $-\infty<x<-2,6 \leqslant x<+\infty,-2<x<$ $<2-\sqrt{15}$.