9.111. Find the non-negative integer values of $x$ that satisfy the inequality $\frac{x+3}{x^{2}-4}-\frac{1}{x+2}<\frac{2 x}{2 x-x^{2}}$.
Solution.

Rewrite the inequality as $\frac{x+3}{x^{2}-4}-\frac{1}{x+2}+\frac{2 x}{x(x-2)}<0 \Leftrightarrow$

$$
\begin{aligned}
& \Leftrightarrow \frac{x+3}{(x-2)(x+2)}-\frac{1}{x+2}+\frac{2}{x-2}<0 \text { for } x \neq 0: \Leftrightarrow\left\{\begin{array}{l}
\frac{2 x+9}{(x-2)(x+2)}<0, \\
x \neq 0
\end{array} \Leftrightarrow\right. \\
& \Leftrightarrow\left\{\begin{array}{l}
(2 x+9)(x-2)(x+2)<0, \\
x \neq 0 .
\end{array}\right.
\end{aligned}
$$

Using the number line, we get $x=1$.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0651.jpg?height=121&width=440&top_left_y=850&top_left_x=462)

Answer: $x=1$.