Problem 11.2. Solve the inequality

$$
\log _{a}\left(x^{2}-x-2\right)>\log _{a}\left(3+2 x-x^{2}\right)
$$

if it is known that $x=a+1$ is a solution.

Emil Kolev
11.2. The inequality is defined for $x^{2}-x-2>0$ and $3+2 x-x^{2}>0$, whence $x \in(2,3)$. Since $x=a+1$ is a solution, we have $a \in(1,2)$. Then the inequality is equivalent to $x^{2}-x-2>3+2 x-x^{2} \Longleftrightarrow(x+1)(2 x-5)>0$ and therefore $x \in\left(\frac{5}{2}, 3\right)$.