|2x+5|<4 inequation

Given the inequality:
$$\left|{2 x + 5}\right| < 4$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{2 x + 5}\right| = 4$$
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
$$\left(2 x + 5\right) - 4 = 0$$
after simplifying we get
$$2 x + 1 = 0$$
the solution in this interval:
$$x_{1} = - \frac{1}{2}$$

2.
$$2 x + 5 < 0$$
or
$$-\infty < x \wedge x < - \frac{5}{2}$$
we get the equation
$$\left(- 2 x - 5\right) - 4 = 0$$
after simplifying we get
$$- 2 x - 9 = 0$$
the solution in this interval:
$$x_{2} = - \frac{9}{2}$$


$$x_{1} = - \frac{1}{2}$$
$$x_{2} = - \frac{9}{2}$$
$$x_{1} = - \frac{1}{2}$$
$$x_{2} = - \frac{9}{2}$$
This roots
$$x_{2} = - \frac{9}{2}$$
$$x_{1} = - \frac{1}{2}$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{9}{2} + - \frac{1}{10}$$
=
$$- \frac{23}{5}$$
substitute to the expression
$$\left|{2 x + 5}\right| < 4$$
$$\left|{\frac{\left(-23\right) 2}{5} + 5}\right| < 4$$

21/5 < 4

but

21/5 > 4

Then
$$x < - \frac{9}{2}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{9}{2} \wedge x < - \frac{1}{2}$$

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       x2      x1