Answer
$4\lt x\lt5$
See graph
Work Step by Step
$$|9-2x|-2\lt-1$$
$$|9-2x|-2+2\lt-1+2$$
$$|9-2x|\lt1$$
Apply absolute rule:
If $|u|\lt a,~a\gt 0$, then $-a\lt u\lt a$.
Take $u=9-2x$ and $a=1$.
$$-1\lt9-2x\lt1$$
$$-1\lt9-2\text{ and }9-2x\lt1$$
$$-1-9\lt9-2x-9\text{ and }9-2x-9\lt1-9$$
$$-10\lt-2x\text{ and }-2x\lt-8$$
$$\frac{-10}{-2}\gt\frac{-2x}{-2}\text{ and }\frac{-2x}{-2}\lt\frac{-8}{-2}$$
$$5\gt x\text{ and }x\gt4$$
Combining:
$$4\lt x\lt5$$
The graph of the solution set is as shown.
