Example 1. Solve the inequality

$$
\left|x^{2}-2 x\right|<x .
$$
Solution. From the properties of a quadratic trinomial, it follows that $x^{2}-2 x<0$, which for the specified values of $x$ is satisfied only when $1<x<2$. Therefore, the interval $(1 ; 2)$ is part of the solution set of inequality (1).

For $x \geqslant 2$, we have the inequality $x^{2}-2 x<x$, i.e., the inequality $x^{2}-3 x<0$, which for the specified values of $x$ is satisfied only for $2 \leqslant x<3$.

Combining the solutions obtained on each of the three intervals, we find the solution set of inequality (1) to be the interval $(1 ; 3)$.

An inequality of the form

$$
f(|x|)<g(x),
$$

where $f(x)$ and $g(x)$ are some functions, is equivalent to the combination of two systems

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-129.jpg?height=128&width=742&top_left_y=2223&top_left_x=657)