|2x+5|=>1 inequation

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

2.
$$2 x + 5 < 0$$
or
$$-\infty < x \wedge x < - \frac{5}{2}$$
we get the equation
$$\left(- 2 x - 5\right) - 1 = 0$$
after simplifying we get
$$- 2 x - 6 = 0$$
the solution in this interval:
$$x_{2} = -3$$


$$x_{1} = -2$$
$$x_{2} = -3$$
$$x_{1} = -2$$
$$x_{2} = -3$$
This roots
$$x_{2} = -3$$
$$x_{1} = -2$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-3 + - \frac{1}{10}$$
=
$$- \frac{31}{10}$$
substitute to the expression
$$\left|{2 x + 5}\right| \geq 1$$
$$\left|{\frac{\left(-31\right) 2}{10} + 5}\right| \geq 1$$

6/5 >= 1

one of the solutions of our inequality is:
$$x \leq -3$$

 _____           _____          
      \         /
-------•-------•-------
       x2      x1

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq -3$$
$$x \geq -2$$