|2x-5|<-3 inequation

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

1.
$$2 x - 5 \geq 0$$
or
$$\frac{5}{2} \leq x \wedge x < \infty$$
we get the equation
$$\left(2 x - 5\right) + 3 = 0$$
after simplifying we get
$$2 x - 2 = 0$$
the solution in this interval:
$$x_{1} = 1$$
but x1 not in the inequality interval

2.
$$2 x - 5 < 0$$
or
$$-\infty < x \wedge x < \frac{5}{2}$$
we get the equation
$$\left(5 - 2 x\right) + 3 = 0$$
after simplifying we get
$$8 - 2 x = 0$$
the solution in this interval:
$$x_{2} = 4$$
but x2 not in the inequality interval


This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example

x0 = 0

$$\left|{-5 + 0 \cdot 2}\right| < -3$$

5 < -3

but

5 > -3

so the inequality has no solutions