9.117. For what values of $m$ is the inequality $\frac{x^{2}+m x-1}{2 x^{2}-2 x+3}<1$ satisfied for all $x$?
## Solution.

From the condition we get

$$
\begin{aligned}
& \frac{x^{2}+m x-1}{2 x^{2}-2 x+3}-10
\end{aligned}
$$

Since $2 x^{2}-2 x+3>0$ for $x \in R$, then $x^{2}-(m+2) x+4>0$ for $x \in R \Leftrightarrow$ $\Leftrightarrow(m+2)^{2}-16<0,(m+2)^{2}<16,-4<m+2<4,-6<m<2$.

Answer: $m \in(-6 ; 2)$.