tg(5x-(п)/(3))>=-(sqrt(3))/(3) inequation

Given the inequality:
$$\tan{\left(5 x - \frac{\pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(5 x - \frac{\pi}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
Solve:
Given the equation
$$\tan{\left(5 x - \frac{\pi}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
transform
$$\frac{2 \cdot \left(3 \tan{\left(5 x \right)} - \sqrt{3}\right)}{3 \left(\sqrt{3} \tan{\left(5 x \right)} + 1\right)} = 0$$
$$\frac{\sqrt{3}}{3} + \frac{\tan{\left(5 x \right)} - \sqrt{3}}{\sqrt{3} \tan{\left(5 x \right)} + 1} = 0$$
Do replacement
$$w = \tan{\left(5 x \right)}$$
Given the equation:
$$\frac{w - \sqrt{3}}{\sqrt{3} w + 1} + \frac{\sqrt{3}}{3} = 0$$
Multiply the equation sides by the denominator 1 + w*sqrt(3)
we get:
$$2 w - \frac{2 \sqrt{3}}{3} = 0$$
Expand brackets in the left part

2*w - 2*sqrt3/3 = 0

Divide both parts of the equation by (2*w - 2*sqrt(3)/3)/w

w = 0 / ((2*w - 2*sqrt(3)/3)/w)

We get the answer: w = sqrt(3)/3
do backward replacement
$$\tan{\left(5 x \right)} = w$$
Given the equation
$$\tan{\left(5 x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$5 x = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$5 x = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
Divide both parts of the equation by
$$5$$
substitute w:
$$x_{1} = \frac{\pi n}{5} + \frac{\operatorname{atan}{\left(w_{1} \right)}}{5}$$
$$x_{1} = \frac{\pi n}{5} + \frac{\operatorname{atan}{\left(\frac{\sqrt{3}}{3} \right)}}{5}$$
$$x_{1} = \frac{\pi n}{5} + \frac{\pi}{30}$$
$$x_{1} = \frac{\pi}{30}$$
$$x_{1} = \frac{\pi}{30}$$
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
$$\tan{\left(5 x - \frac{\pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
$$\tan{\left(- \frac{\pi}{3} + 5 \left(- \frac{1}{10} + \frac{\pi}{30}\right) \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$

                   ___ 
    /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:
$$x \geq \frac{\pi}{30}$$

         _____  
        /
-------•-------
       x_1