ctg(x)<=-1 inequation

Given the inequality:
$$\cot{\left(x \right)} \leq -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(x \right)} = -1$$
Solve:
Given the equation
$$\cot{\left(x \right)} = -1$$
transform
$$\cot{\left(x \right)} + 1 = 0$$
$$\cot{\left(x \right)} + 1 = 0$$
Do replacement
$$w = \cot{\left(x \right)}$$
Move free summands (without w)
from left part to right part, we given:
$$w = -1$$
We get the answer: w = -1
do backward replacement
$$\cot{\left(x \right)} = w$$
substitute w:
$$x_{1} = - \frac{\pi}{4}$$
$$x_{1} = - \frac{\pi}{4}$$
This roots
$$x_{1} = - \frac{\pi}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{\pi}{4} - \frac{1}{10}$$
=
$$- \frac{\pi}{4} - \frac{1}{10}$$
substitute to the expression
$$\cot{\left(x \right)} \leq -1$$
$$\cot{\left(- \frac{\pi}{4} - \frac{1}{10} \right)} \leq -1$$

    /1    pi\      
-cot|-- + --| <= -1
    \10   4 /      

but

    /1    pi\      
-cot|-- + --| >= -1
    \10   4 /      

Then
$$x \leq - \frac{\pi}{4}$$
no execute
the solution of our inequality is:
$$x \geq - \frac{\pi}{4}$$

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