arctgx<=п/3 inequation

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

     /       ___\      
     |1    \/ 3 |    pi
-acot|-- - -----| <= --
     \10     3  /    3 
      

but

     /       ___\      
     |1    \/ 3 |    pi
-acot|-- - -----| >= --
     \10     3  /    3 
      

Then
$$x \leq \frac{\sqrt{3}}{3}$$
no execute
the solution of our inequality is:
$$x \geq \frac{\sqrt{3}}{3}$$

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