ctg((x/5)-(pi/10))<=-(sqrt(3)/3) inequation

Given the inequality:
$$\cot{\left(\frac{x}{5} - \frac{\pi}{10} \right)} \leq - \frac{\sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(\frac{x}{5} - \frac{\pi}{10} \right)} = - \frac{\sqrt{3}}{3}$$
Solve:
$$x_{1} = - \frac{7 \pi}{6}$$
$$x_{1} = - \frac{7 \pi}{6}$$
This roots
$$x_{1} = - \frac{7 \pi}{6}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{7 \pi}{6} - \frac{1}{10}$$
=
$$- \frac{7 \pi}{6} - \frac{1}{10}$$
substitute to the expression
$$\cot{\left(\frac{x}{5} - \frac{\pi}{10} \right)} \leq - \frac{\sqrt{3}}{3}$$
$$\cot{\left(\frac{- \frac{7 \pi}{6} - \frac{1}{10}}{5} - \frac{\pi}{10} \right)} \leq - \frac{\sqrt{3}}{3}$$

                    ___ 
    /1    pi\    -\/ 3  
-cot|-- + --| <= -------
    \50   3 /       3   
                 

but

                    ___ 
    /1    pi\    -\/ 3  
-cot|-- + --| >= -------
    \50   3 /       3   
                 

Then
$$x \leq - \frac{7 \pi}{6}$$
no execute
the solution of our inequality is:
$$x \geq - \frac{7 \pi}{6}$$

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