cos(2x)>=-2 inequation

Given the inequality:
$$\cos{\left(2 x \right)} \geq -2$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(2 x \right)} = -2$$
Solve:
Given the equation
$$\cos{\left(2 x \right)} = -2$$
- this is the simplest trigonometric equation
As right part of the equation
modulo =

True

but cos
can no be more than 1 or less than -1
so the solution of the equation d'not exist.
$$x_{1} = \pi - \frac{\operatorname{acos}{\left(-2 \right)}}{2}$$
$$x_{2} = \frac{\operatorname{acos}{\left(-2 \right)}}{2}$$
Exclude the complex solutions:
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

$$\cos{\left(0 \cdot 2 \right)} \geq -2$$

1 >= -2

so the inequality is always executed