9. (12 points) Solve the inequality
$$
\sqrt{x+\frac{1}{x^{2}}}-\sqrt{x-\frac{1}{x^{2}}}<\frac{1}{x} .
$$
From the problem, we have
$$
x+\frac{1}{x^{2}}>x-\frac{1}{x^{2}} \geqslant 0 \Rightarrow \frac{x^{3}-1}{x^{2}} \geqslant 0 \Rightarrow x \geqslant 1 \text {. }
$$

Therefore, the original inequality is equivalent to
$$
\begin{array}{l}
\sqrt{x^{3}+1}-\sqrt{x^{3}-1} < \frac{1}{2} \Rightarrow x^{3}>\frac{5}{4} \Rightarrow x>\frac{\sqrt[3]{10}}{2}
$$