tan(5*x-pi/3)>=(-1)/sqrt(3) inequation

Given the inequality:
$$\tan{\left(5 x - \frac{\pi}{3} \right)} \geq - \frac{1}{\sqrt{3}}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(5 x - \frac{\pi}{3} \right)} = - \frac{1}{\sqrt{3}}$$
Solve:
$$x_{1} = \frac{\pi}{30}$$
$$x_{1} = \frac{\pi}{30}$$
This roots
$$x_{1} = \frac{\pi}{30}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\pi}{30}$$
=
$$- \frac{1}{10} + \frac{\pi}{30}$$
substitute to the expression
$$\tan{\left(5 x - \frac{\pi}{3} \right)} \geq - \frac{1}{\sqrt{3}}$$
$$\tan{\left(- \frac{\pi}{3} + 5 \left(- \frac{1}{10} + \frac{\pi}{30}\right) \right)} \geq - \frac{1}{\sqrt{3}}$$

                   ___ 
    /1   pi\    -\/ 3  
-tan|- + --| >= -------
    \2   6 /       3   
                

but

                  ___ 
    /1   pi\   -\/ 3  
-tan|- + --| < -------
    \2   6 /      3   
               

Then
$$x \leq \frac{\pi}{30}$$
no execute
the solution of our inequality is:
$$x \geq \frac{\pi}{30}$$

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        /
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       x1