cos(x)<-√3/3 inequation

Given the inequality:
$$\cos{\left(x \right)} < \frac{\left(-1\right) \sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
Solve:
Given the equation
$$\cos{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}$$
Or
$$x = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}$$
, where n - is a integer
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}$$
This roots
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}$$
substitute to the expression
$$\cos{\left(x \right)} < \frac{\left(-1\right) \sqrt{3}}{3}$$
$$\cos{\left(\pi n - \frac{1}{10} + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)} \right)} < \frac{\left(-1\right) \sqrt{3}}{3}$$

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

but

   /                  /   ___ \\      ___ 
   |  1               |-\/ 3  ||   -\/ 3  
cos|- -- + pi*n + acos|-------|| > -------
   \  10              \   3   //      3   
   

Then
$$x < \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}$$
no execute
one of the solutions of our inequality is:
$$x > \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)} \wedge x < \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{3} \right)}$$

         _____  
        /     \  
-------ο-------ο-------
       x1      x2