9. 1 For which values of $x$ is the inequality
$$
\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9
$$

true.
[Solution] From the assumption, we know that $1+2 x \geqslant 0,1-\sqrt{1+2 x} \neq 0$, so $x \geqslant-\frac{1}{2}$ and $x \neq 0$.
Also, when $x \neq 0$,
$$
\frac{-2 x}{1-\sqrt{1+2 x}}=1+\sqrt{1+2 x} \text {, }
$$

thus the original inequality becomes
$$
2+2 x+2 \sqrt{1+2 x}<2 x+9,
$$

solving this yields $-\frac{1}{2} \leqslant x<\frac{45}{8}$.
From (1) and (2), we get $-\frac{1}{2} \leqslant x<0$ or $0<x<\frac{45}{8}$.