Example 3 (2nd IM() problem) For which values of $x$ is the inequality $\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9$ satisfied?
From the problem, we know that $1+2 x \geqslant 0,1-\sqrt{1+2 x} \neq 0$, which means $x \geqslant-\frac{1}{2}$ and $x \neq 0$. Also, when $x \neq 0$, $\frac{-2 x}{1-\sqrt{1+2 x}}=1+\sqrt{1+2 x}$, so the original inequality can be transformed into
$$
(1+\sqrt{1+2 x})^{2}<2 x+9 \Leftrightarrow \sqrt{1+2 x}<\frac{7}{2} \Leftrightarrow\left\{\begin{array}{l}
1+2 x<\frac{49}{4} \\
1+2 x \geqslant 0
\end{array} \Leftrightarrow-\frac{1}{2} \leqslant x<\frac{45}{8} .\right.
$$

Combining the conditions $x \geqslant-\frac{1}{2}$ and $x \neq 0$, the solution to the original inequality is
$$
-\frac{1}{2} \leqslant x<0 \text {, and } 0<x<\frac{45}{8} \text {. }
$$