cosx≤-0,2 inequation

Given the inequality:
$$\cos{\left(x \right)} \leq - \frac{1}{5}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(x \right)} = - \frac{1}{5}$$
Solve:
Given the equation
$$\cos{\left(x \right)} = - \frac{1}{5}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
Or
$$x = \pi n + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
, where n - is a integer
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
This roots
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \operatorname{acos}{\left(- \frac{1}{5} \right)}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
substitute to the expression
$$\cos{\left(x \right)} \leq - \frac{1}{5}$$
$$\cos{\left(\pi n - \frac{1}{10} + \operatorname{acos}{\left(- \frac{1}{5} \right)} \right)} \leq - \frac{1}{5}$$

cos(-1/10 + pi*n + acos(-1/5)) <= -1/5

but

cos(-1/10 + pi*n + acos(-1/5)) >= -1/5

Then
$$x \leq \pi n + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
no execute
one of the solutions of our inequality is:
$$x \geq \pi n + \operatorname{acos}{\left(- \frac{1}{5} \right)} \wedge x \leq \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$

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