9.116. For what values of $x$ does the difference $\frac{11 x^{2}-5 x+6}{x^{2}+5 x+6}-x$ take only negative values?
## Solution.

We have

$$
\begin{aligned}
& \frac{11 x^{2}-5 x+6}{x^{2}+5 x+6}-x0 \Leftrightarrow \\
& \Leftrightarrow \frac{(x-1)(x-2)(x-3)}{(x+3)(x+2)}>0 \Leftrightarrow(x-1)(x-2)(x-3)(x+3)(x+2)>0
\end{aligned}
$$

Using the interval method, we get $x \in(-3 ;-2) \bigcup(1 ; 2) \cup(3 ; \infty)$.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0653.jpg?height=85&width=517&top_left_y=868&top_left_x=420)

Answer: $x \in(-3 ;-2) \cup(1 ; 2) \cup(3 ; \infty)$.