ctg(x)<-sqrt(3) inequation

Given the inequality:
$$\cot{\left(x \right)} < - \sqrt{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(x \right)} = - \sqrt{3}$$
Solve:
Given the equation
$$\cot{\left(x \right)} = - \sqrt{3}$$
transform
$$\cot{\left(x \right)} - 1 + \sqrt{3} = 0$$
$$\cot{\left(x \right)} - 1 + \sqrt{3} = 0$$
Do replacement
$$w = \cot{\left(x \right)}$$
Expand brackets in the left part

-1 + w + sqrt3 = 0

Move free summands (without w)
from left part to right part, we given:
$$w + \sqrt{3} = 1$$
Divide both parts of the equation by (w + sqrt(3))/w

w = 1 / ((w + sqrt(3))/w)

We get the answer: w = 1 - sqrt(3)
do backward replacement
$$\cot{\left(x \right)} = w$$
substitute w:
$$x_{1} = - \frac{\pi}{6}$$
$$x_{1} = - \frac{\pi}{6}$$
This roots
$$x_{1} = - \frac{\pi}{6}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{\pi}{6} - \frac{1}{10}$$
=
$$- \frac{\pi}{6} - \frac{1}{10}$$
substitute to the expression
$$\cot{\left(x \right)} < - \sqrt{3}$$
$$\cot{\left(- \frac{\pi}{6} - \frac{1}{10} \right)} < - \sqrt{3}$$

    /1    pi\      ___
-cot|-- + --| < -\/ 3 
    \10   6 /   

but

    /1    pi\      ___
-cot|-- + --| > -\/ 3 
    \10   6 /   

Then
$$x < - \frac{\pi}{6}$$
no execute
the solution of our inequality is:
$$x > - \frac{\pi}{6}$$

         _____  
        /
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       x_1