9.010. Find the positive integer values of $x$ that satisfy the inequality $\frac{5 x+1}{x-1}>2 x+2$.
Solution.

We have

$$
\begin{aligned}
& \frac{5 x+1}{x-1}-2 x-2>0 \Leftrightarrow \frac{5 x+1-2(x+1)(x-1)}{x-1}>0 \Leftrightarrow \frac{-2 x^{2}+5 x+3}{x-1}>0 \Leftrightarrow \\
& \Leftrightarrow\left(2 x^{2}-5 x-3\right)(x-1)<0 \Leftrightarrow 2\left(x+\frac{1}{2}\right)(x-3)(x-1)<0
\end{aligned}
$$

Using the number line, we find that $x=2$.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-519.jpg?height=152&width=610&top_left_y=1116&top_left_x=399)

Answer: $\quad x=2$.