Example 9. Solve the inequality

$$
\left|x^{3}-x\right| \geqslant 1-x
$$
Solution. The given inequality is equivalent to the system

$$
\left[\begin{array} { l } 
{ x ^ { 3 } - x \geqslant 1 - x , } \\
{ x ^ { 3 } - x \leqslant - ( 1 - x ) }
\end{array} \Leftrightarrow \left[\begin{array}{l}
x^{3}-1 \geqslant 0 \\
x^{3}-2 x+1 \leqslant 0
\end{array}\right.\right.
$$

The first inequality of the last system is satisfied only for $x \geqslant 1$ (since $x^{3}-1=(x-1)\left(x^{2}+x+1\right)$, and $x^{2}+x+1>0$ for any $\quad x$). Since $x^{3}-2 x+1=(x-1)\left(x+\frac{1+\sqrt{5}}{2}\right) \times$ $\times\left(x-\frac{\sqrt{\overline{5}}-1}{2}\right)$, the set of solutions to the inequality $x^{3}-2 x+$ $+1 \leqslant 0$ is the set $\left(-\infty ; \frac{-1-\sqrt{5}}{2}\right] \cup\left[\frac{\sqrt{5}-1}{2} ; 1\right]$.

By combining the solution sets of both inequalities in the system, we obtain the solution set of the original inequality:

$$
\left(-\infty ; \frac{-1-\sqrt{5}}{2}\right] \cup\left[\frac{\sqrt{5}-1}{2} ;+\infty\right)
$$

An inequality of the form

$$
|f(|x|)|-g(x)
\end{array}\right.
$$

The choice of solution method depends on the specific inequality and the complexity of the functions $f$ and $g$.