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Solving inequalities/ ctg(x)<=-3/3

ctg(x)<=-3/3 inequation

A inequation with variable

The solution

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cot(x) <= -1
cot⁡(x)≤−1\cot{\left(x \right)} \leq -1 
cot(x) <= -1
Detail solution
Given the inequality:
cot⁡(x)≤−1\cot{\left(x \right)} \leq -1
To solve this inequality, we must first solve the corresponding equation:
cot⁡(x)=−1\cot{\left(x \right)} = -1
Solve:
Given the equation
cot⁡(x)=−1\cot{\left(x \right)} = -1
transform
cot⁡(x)+1=0\cot{\left(x \right)} + 1 = 0
cot⁡(x)+1=0\cot{\left(x \right)} + 1 = 0
Do replacement
w=cot⁡(x)w = \cot{\left(x \right)}
Move free summands (without w)
from left part to right part, we given:
w=−1w = -1
We get the answer: w = -1
do backward replacement
cot⁡(x)=w\cot{\left(x \right)} = w
substitute w:
x1=−π4x_{1} = - \frac{\pi}{4}
x1=−π4x_{1} = - \frac{\pi}{4}
This roots
x1=−π4x_{1} = - \frac{\pi}{4}
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0≤x1x_{0} \leq x_{1}
For example, let's take the point
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
−π4−110- \frac{\pi}{4} - \frac{1}{10}
=
−π4−110- \frac{\pi}{4} - \frac{1}{10}
substitute to the expression
cot⁡(x)≤−1\cot{\left(x \right)} \leq -1
cot⁡(−π4−110)≤−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≤−π4x \leq - \frac{\pi}{4}
no execute
the solution of our inequality is:
x≥−π4x \geq - \frac{\pi}{4}
         _____  
        /
-------•-------
       x1
Rapid solution [src] 
   /3*pi             \
And|---- <= x, x < pi|
   \ 4               /
3π4≤x∧x<π\frac{3 \pi}{4} \leq x \wedge x < \pi 
(x < pi)∧(3*pi/4 <= x)
Rapid solution 2 [src] 
 3*pi     
[----, pi)
  4
x in [3π4,π)x\ in\ \left[\frac{3 \pi}{4}, \pi\right) 
x in Interval.Ropen(3*pi/4, pi)
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