Example 8. Solve the inequality

$$
\left|\frac{x^{2}-5 x+4}{x^{2}-4}\right| \leqslant 1
$$
Solution. The given inequality is equivalent to the system

$$
\left\{\begin{array}{l}
\frac{x^{2}-5 x+4}{x^{2}-4} \leq 1 \\
\frac{x^{2}-5 x+4}{x^{2}-4} \geqslant-1
\end{array}\right.
$$

The first inequality of the system is equivalent to the inequality

$$
\frac{x-8 / 5}{(x-2)(x+2)} \geqslant 0
$$

Applying the method of intervals, we find its solutions - all numbers from the intervals $-2<x \leqslant 8 / 5$ and $2<x<+\infty$, i.e., the set $(-2 ; 8 / 5] \cup(2 ;+\infty)$.

The second inequality of the system is equivalent to the inequality

$$
\frac{x(x-5 / 2)}{(x+2)(x-2)} \geqslant 0
$$

and, applying the method of intervals again, we find the set of its solutions, consisting of the union of three intervals: $(-\infty ;-2) U$ $U[0 ; 2) \cup[5 / 2 ;+\infty)$.

The intersection of the two obtained sets constitutes the set of solutions to the original inequality, i.e., the set $[0 ; 8 / 5] U[5 / 2 ;+\infty)$.