9.012. Find the natural values of $x$ that satisfy the system of inequalities

$$
\left\{\begin{array}{l}
\log _{\sqrt{2}}(x-1)<4 \\
\frac{x}{x-3}+\frac{x-5}{x}<\frac{2 x}{3-x}
\end{array}\right.
$$
Solution.

From the condition

$$
\begin{aligned}
& \left\{\begin{array} { l } 
{ 0 0 \text { for } x \in R, \\
x(x-3)<0 .
\end{array}\right.
\end{aligned}
$$

Using the number line, we find the solution to the system $x=2$.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-520.jpg?height=132&width=608&top_left_y=1014&top_left_x=398)

Answer: $x=2$.