Question

Find all values of $a$ for which the inequality $\frac{x-2a-3}{x-a+2}<0$ is satisfied for all $x$ belonging to the interval $1⩽x⩽2$.

Answer 1

$\frac{x-2a-3}{x-a+2}<0$

$\frac{x-(2a+3)}{x-(a-2)}<0$

$2a+3>a-2$

$2a+3\ge 2$

$2a\ge -1$

$a\ge \frac{-1}{2}$

$a-2\le 1$

$a\le -3$

$a\epsilon (-5,-3]\bigcup [\frac{-1}{2},\infty )$

when $2a+3<a-2$

$a<-5$

$a-2\ge 2$

$a\ge 4$ (not possible)

$2a+3\le 1$

$a\le -1$

$a\epsilon (-5,3]\bigcup [\frac{-1}{2},\infty )$