(|5*x-2|)>3 inequation

Given the inequality:
$$\left|{5 x - 2}\right| > 3$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{5 x - 2}\right| = 3$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$5 x - 2 \geq 0$$
or
$$\frac{2}{5} \leq x \wedge x < \infty$$
we get the equation
$$\left(5 x - 2\right) - 3 = 0$$
after simplifying we get
$$5 x - 5 = 0$$
the solution in this interval:
$$x_{1} = 1$$

2.
$$5 x - 2 < 0$$
or
$$-\infty < x \wedge x < \frac{2}{5}$$
we get the equation
$$\left(2 - 5 x\right) - 3 = 0$$
after simplifying we get
$$- 5 x - 1 = 0$$
the solution in this interval:
$$x_{2} = - \frac{1}{5}$$


$$x_{1} = 1$$
$$x_{2} = - \frac{1}{5}$$
$$x_{1} = 1$$
$$x_{2} = - \frac{1}{5}$$
This roots
$$x_{2} = - \frac{1}{5}$$
$$x_{1} = 1$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{5} + - \frac{1}{10}$$
=
$$- \frac{3}{10}$$
substitute to the expression
$$\left|{5 x - 2}\right| > 3$$
$$\left|{-2 + \frac{\left(-3\right) 5}{10}}\right| > 3$$

7/2 > 3

one of the solutions of our inequality is:
$$x < - \frac{1}{5}$$

 _____           _____          
      \         /
-------ο-------ο-------
       x2      x1

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < - \frac{1}{5}$$
$$x > 1$$