Possible real values

Question

Set of all possible real values of a such that the inequality $(x-(a-1))(x-(a^2+2))<0$ holds for all $x$ belongs to $(1,3)$ needs to be found. I tried by putting $x=2$ in the inequality, but nothing good resulted. I thought I would get the idea but it was not that easy. Help me with it with any new idea and explain the mistake in mine.

Answer 1

Guide:

• Notice that $a^2+2 \ge a-1$ and $a^2+2$ and $a-1$ are the roots. We want $(1,3) \subseteq (a-1, a^2+2)$.

• Hence we want $ a-1 \le 1 $ and $a^2+2 \ge 3$.