tgx<=-sqrt3 inequation

Given the inequality:
$$\tan{\left(x \right)} \leq - \sqrt{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(x \right)} = - \sqrt{3}$$
Solve:
Given the equation
$$\tan{\left(x \right)} = - \sqrt{3}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(- \sqrt{3} \right)}$$
Or
$$x = \pi n - \frac{\pi}{3}$$
, where n - is a integer
$$x_{1} = \pi n - \frac{\pi}{3}$$
$$x_{1} = \pi n - \frac{\pi}{3}$$
This roots
$$x_{1} = \pi n - \frac{\pi}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n - \frac{\pi}{3}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{\pi}{3} - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(x \right)} \leq - \sqrt{3}$$
$$\tan{\left(\pi n - \frac{\pi}{3} - \frac{1}{10} \right)} \leq - \sqrt{3}$$

    /1    pi       \       ___
-tan|-- + -- - pi*n| <= -\/ 3 
    \10   3        /    

the solution of our inequality is:
$$x \leq \pi n - \frac{\pi}{3}$$

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