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Solving inequalities/ cot(x+(pi/3))<-1

cot(x+(pi/3))<-1 inequation

A inequation with variable

The solution

You have entered [src] 
   /    pi\     
cot|x + --| < -1
   \    3 /
cot⁡(x+π3)<−1\cot{\left(x + \frac{\pi}{3} \right)} < -1 
cot(x + pi/3) < -1
Detail solution
Given the inequality:
cot⁡(x+π3)<−1\cot{\left(x + \frac{\pi}{3} \right)} < -1
To solve this inequality, we must first solve the corresponding equation:
cot⁡(x+π3)=−1\cot{\left(x + \frac{\pi}{3} \right)} = -1
Solve:
x1=−7π12x_{1} = - \frac{7 \pi}{12}
x1=−7π12x_{1} = - \frac{7 \pi}{12}
This roots
x1=−7π12x_{1} = - \frac{7 \pi}{12}
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0<x1x_{0} < x_{1}
For example, let's take the point
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
−7π12−110- \frac{7 \pi}{12} - \frac{1}{10}
=
−7π12−110- \frac{7 \pi}{12} - \frac{1}{10}
substitute to the expression
cot⁡(x+π3)<−1\cot{\left(x + \frac{\pi}{3} \right)} < -1
cot⁡((−7π12−110)+π3)<−1\cot{\left(\left(- \frac{7 \pi}{12} - \frac{1}{10}\right) + \frac{\pi}{3} \right)} < -1
    /1    pi\     
-cot|-- + --| < -1
    \10   4 /

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

Then
x<−7π12x < - \frac{7 \pi}{12}
no execute
the solution of our inequality is:
x>−7π12x > - \frac{7 \pi}{12}
         _____  
        /
-------ο-------
       x1
Rapid solution 2 [src] 
      /  ___     ___\       
      |\/ 2  + \/ 6 |  2*pi 
(-atan|-------------|, ----]
      |  ___     ___|   3   
      \\/ 2  - \/ 6 /
x in (−atan⁡(2+6−6+2),2π3]x\ in\ \left(- \operatorname{atan}{\left(\frac{\sqrt{2} + \sqrt{6}}{- \sqrt{6} + \sqrt{2}} \right)}, \frac{2 \pi}{3}\right] 
x in Interval.Lopen(-atan((sqrt(2) + sqrt(6))/(-sqrt(6) + sqrt(2))), 2*pi/3)
Rapid solution [src] 
   /               /  ___     ___\    \
   |     2*pi      |\/ 2  + \/ 6 |    |
And|x <= ----, atan|-------------| < x|
   |      3        |  ___     ___|    |
   \               \\/ 6  - \/ 2 /    /
x≤2π3∧atan⁡(2+6−2+6)<xx \leq \frac{2 \pi}{3} \wedge \operatorname{atan}{\left(\frac{\sqrt{2} + \sqrt{6}}{- \sqrt{2} + \sqrt{6}} \right)} < x 
(x <= 2*pi/3)∧(atan((sqrt(2) + sqrt(6))/(sqrt(6) - sqrt(2))) < x)
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