Example 4. Solve the inequality
$$
\frac{2 x^{2}}{1+x^{2}}-\frac{5 x}{\sqrt{1+x^{2}}}+2>0 .
$$
Let $x=\operatorname{tg} \theta\left(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\right)$,

i.e., $2 \sin ^{2} \theta-5 \sin \theta+2>0$,
or $(\sin \theta-2)(2 \sin \theta-1)>0$.
Since $\sin \theta-2<0$, it follows that $2 \sin \theta-1<0$, $\sin \theta<\frac{1}{2},-\frac{\pi}{2}<\theta<\frac{\pi}{6}$. Also, $\operatorname{tg} \theta$ is a monotonic function on $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, so $\cdots \infty<x=$ $\operatorname{tg} \theta<\operatorname{tg} \frac{\pi}{6}$, i.e., $\cdots \infty<x<\frac{\sqrt{3}}{3}$, which is the solution to the original inequality.