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Problem 830

AMC 12 late, AIME early

Algebra Difficulty 4.5 Prove it Bulgarian Mathematical Competitions · Bulgaria

Solve the inequality
$\log_{a}\left(x^{2}-x-2\right)>\log_{a}\left(3+2x-x^{2}\right)$
if it is known that $x=a+1$ is a solution.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:
The inequality is defined for $x^{2}-x-2>0$ and $3+2x-x^{2}>0$ , whence $x \in (2,3)$ .

Since $x=a+1$ is a solution, we have $a \in (1,2)$ .

Then the inequality is equivalent to
$x^{2}-x-2 > 3+2x-x^{2} \Longleftrightarrow (x+1)(2x-5)>0$
and therefore $x \in \left(\frac{5}{2}, 3\right)$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.