|2*x-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
$$2 x - 8 = 0$$
the solution in this interval:
$$x_{1} = 4$$

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
$$2 - 2 x = 0$$
the solution in this interval:
$$x_{2} = 1$$


$$x_{1} = 4$$
$$x_{2} = 1$$
$$x_{1} = 4$$
$$x_{2} = 1$$
This roots
$$x_{2} = 1$$
$$x_{1} = 4$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1$$
=
$$\frac{9}{10}$$
substitute to the expression
$$\left|{2 x - 5}\right| < 3$$
$$\left|{-5 + \frac{2 \cdot 9}{10}}\right| < 3$$

16/5 < 3

but

16/5 > 3

Then
$$x < 1$$
no execute
one of the solutions of our inequality is:
$$x > 1 \wedge x < 4$$

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        /     \  
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       x2      x1