Example 8. Solve the inequality

$$
\sqrt{-x^{2}+6 x-5}>8-2 x
$$
Solution. The quadratic trinomial $-x^{2}+6 x-5$ has roots $x_{1}=1$ and $x_{2}=5$; therefore, inequality (10) is equivalent to the combination of two systems

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-148.jpg?height=142&width=884&top_left_y=2079&top_left_x=583)

From the second system of this combination, we find $4(8-2 x)^{2}$, i.e., the inequality $5 x^{2}-38 x+69<0$, which gives $3<x<23 / 5$. Among these $x$, the condition $8-2 x \geqslant 0$ is satisfied only by numbers in the interval $3<x \leqslant 4$.

The solution set of the original inequality is the union of the sets $3<x \leqslant 4$ and $4<x \leqslant 5$, i.e., the interval $3<x \leqslant 5$.