9.118. For what values of $m$ is the inequality $\frac{x^{2}-m x-2}{x^{2}-3 x+4}>-1$ satisfied for all $x$?
Solution.

From the condition, we have $\frac{2 x^{2}-(m+3) x+2}{x^{2}-3 x+4}>0$. Since $x^{2}-3 x+4>0$ for $x \in R$, then $2 x^{2}-(m+3) x+2>0$ for any $x$, from which $D=(m+3)^{2}-16<0$, $(m+3)^{2}<16, -4<m+3<4, -7<m<1$.

Answer: $m \in(-7 ; 1)$.