cos(x/5)>(-sqrt(2)/2) inequation

Given the inequality:
$$\cos{\left(\frac{x}{5} \right)} > \frac{\left(-1\right) \sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(\frac{x}{5} \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
Solve:
Given the equation
$$\cos{\left(\frac{x}{5} \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{5} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$\frac{x}{5} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{2}}{2} \right)}$$
Or
$$\frac{x}{5} = \pi n + \frac{3 \pi}{4}$$
$$\frac{x}{5} = \pi n - \frac{\pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{5}$$
$$x_{1} = 5 \pi n + \frac{15 \pi}{4}$$
$$x_{2} = 5 \pi n - \frac{5 \pi}{4}$$
$$x_{1} = 5 \pi n + \frac{15 \pi}{4}$$
$$x_{2} = 5 \pi n - \frac{5 \pi}{4}$$
This roots
$$x_{1} = 5 \pi n + \frac{15 \pi}{4}$$
$$x_{2} = 5 \pi n - \frac{5 \pi}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(5 \pi n + \frac{15 \pi}{4}\right) - \frac{1}{10}$$
=
$$5 \pi n - \frac{1}{10} + \frac{15 \pi}{4}$$
substitute to the expression
$$\cos{\left(\frac{x}{5} \right)} > \frac{\left(-1\right) \sqrt{2}}{2}$$
$$\cos{\left(\frac{5 \pi n - \frac{1}{10} + \frac{15 \pi}{4}}{5} \right)} > \frac{\left(-1\right) \sqrt{2}}{2}$$

                         ___ 
     n    /1    pi\   -\/ 2  
-(-1) *cos|-- + --| > -------
          \50   4 /      2   
                      

one of the solutions of our inequality is:
$$x < 5 \pi n + \frac{15 \pi}{4}$$

 _____           _____          
      \         /
-------ο-------ο-------
       x_1      x_2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 5 \pi n + \frac{15 \pi}{4}$$
$$x > 5 \pi n - \frac{5 \pi}{4}$$