9.158. $\log _{\frac{1}{3}}(x-1)+\log _{\frac{1}{3}}(x+1)+\log _{\sqrt{3}}(5-x)<1$.
## Solution.

Domain of definition: $\left\{\begin{array}{l}x-1>0, \\ x+1>0, \\ 5-x>0\end{array} \Leftrightarrow 11 \Leftrightarrow \log _{3} \frac{(5-x)^{2}}{(x-1)(x+1)}>1 \Leftrightarrow \\
& \Leftrightarrow \frac{\cdot(5-x)^{2}}{(x-1)(x+1)}>3 \Leftrightarrow \frac{(5-x)^{2}}{(x-1)(x+1)}-3>0 \Leftrightarrow \frac{(5-x)^{2}-3\left(x^{2}-1\right)}{(x-1)(x+1)}>0 \Leftrightarrow \\
& \Leftrightarrow \frac{-2 x^{2}-10 x+28}{(x-1)(x+1)}>0, \frac{x^{2}+5 x-14}{(x-1)(x+1)}<0, \frac{(x+7)(x-2)}{(x-1)(x+1)}<0 \Leftrightarrow \\
& \Leftrightarrow(x+7)(x-2)(x-1)(x+1)<0
\end{aligned}
$$

Considering the domain of definition, using the interval method, we get $2<x<5$.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0676.jpg?height=85&width=444&top_left_y=1288&top_left_x=434)

Answer: $x \in(2 ; 5)$.