cos(x)>=0.5 inequation

Given the inequality:
$$\cos{\left(x \right)} \geq \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(x \right)} = \frac{1}{2}$$
Solve:
Given the equation
$$\cos{\left(x \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$x = \pi n + \frac{\pi}{3}$$
$$x = \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
$$x_{1} = \pi n + \frac{\pi}{3}$$
$$x_{2} = \pi n - \frac{2 \pi}{3}$$
$$x_{1} = \pi n + \frac{\pi}{3}$$
$$x_{2} = \pi n - \frac{2 \pi}{3}$$
This roots
$$x_{1} = \pi n + \frac{\pi}{3}$$
$$x_{2} = \pi n - \frac{2 \pi}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{3}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{3}$$
substitute to the expression
$$\cos{\left(x \right)} \geq \frac{1}{2}$$
$$\cos{\left(\pi n - \frac{1}{10} + \frac{\pi}{3} \right)} \geq \frac{1}{2}$$

   /  1    pi       \       
cos|- -- + -- + pi*n| >= 1/2
   \  10   3        /       

but

   /  1    pi       \      
cos|- -- + -- + pi*n| < 1/2
   \  10   3        /      

Then
$$x \leq \pi n + \frac{\pi}{3}$$
no execute
one of the solutions of our inequality is:
$$x \geq \pi n + \frac{\pi}{3} \wedge x \leq \pi n - \frac{2 \pi}{3}$$

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