Example 5. Solve the inequality

$$
\log _{3}\left(x^{2}-2\right)<\log _{3}\left(\frac{3}{2}|x|-1\right)
$$
Solution. Since the base of the logarithms is greater than one, the given inequality is equivalent to the system of inequalities

$$
\left\{\begin{array}{l}
x^{2}-20
\end{array}\right.
$$

which, taking into account that $x^{2}=|x|^{2}$, can be rewritten as

$$
\left\{\begin{array}{l}
|x|^{2}-20
\end{array}\right.
$$

By setting $t=|x|$, we obtain the system (with respect to $t$)

$$
\begin{aligned}
& \left\{\begin{array} { l } 
{ t \geqslant 0 , } \\
{ t ^ { 2 } - 2 > 0 , } \\
{ 2 t ^ { 2 } - 3 t - 2 0 \\
2\left(t+\frac{1}{2}\right)(t-2)0, \Leftrightarrow \sqrt{2}<t<2 \\
t-2<0
\end{array}\right.
\end{aligned}
$$

Thus, the original inequality is equivalent to the inequality

$$
\sqrt{\overline{2}}<|x|<2
$$

from which it follows that the set of all solutions to the original inequality consists of the intervals $-2<x<-\sqrt{ } \overline{2}$ and $\sqrt{ } \overline{2}<x<2$.