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Solving inequalities/ tgx<=-3

tgx<=-3 inequation

A inequation with variable

The solution

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tan(x) <= -3
tan⁡(x)≤−3\tan{\left(x \right)} \leq -3 
tan(x) <= -3
Detail solution
Given the inequality:
tan⁡(x)≤−3\tan{\left(x \right)} \leq -3
To solve this inequality, we must first solve the corresponding equation:
tan⁡(x)=−3\tan{\left(x \right)} = -3
Solve:
Given the equation
tan⁡(x)=−3\tan{\left(x \right)} = -3
- this is the simplest trigonometric equation
This equation is transformed to
x=πn+atan⁡(−3)x = \pi n + \operatorname{atan}{\left(-3 \right)}
Or
x=πn−atan⁡(3)x = \pi n - \operatorname{atan}{\left(3 \right)}
, where n - is a integer
x1=πn−atan⁡(3)x_{1} = \pi n - \operatorname{atan}{\left(3 \right)}
x1=πn−atan⁡(3)x_{1} = \pi n - \operatorname{atan}{\left(3 \right)}
This roots
x1=πn−atan⁡(3)x_{1} = \pi n - \operatorname{atan}{\left(3 \right)}
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}
=
(πn−atan⁡(3))+−110\left(\pi n - \operatorname{atan}{\left(3 \right)}\right) + - \frac{1}{10}
=
πn−atan⁡(3)−110\pi n - \operatorname{atan}{\left(3 \right)} - \frac{1}{10}
substitute to the expression
tan⁡(x)≤−3\tan{\left(x \right)} \leq -3
tan⁡(πn−atan⁡(3)−110)≤−3\tan{\left(\pi n - \operatorname{atan}{\left(3 \right)} - \frac{1}{10} \right)} \leq -3
-tan(1/10 - pi*n + atan(3)) <= -3

the solution of our inequality is:
x≤πn−atan⁡(3)x \leq \pi n - \operatorname{atan}{\left(3 \right)}
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Solving inequality on a graph
0-80-60-40-2020406080-50005000
Rapid solution [src] 
   /                   pi    \
And|x <= pi - atan(3), -- < x|
   \                   2     /
x≤π−atan⁡(3)∧π2<xx \leq \pi - \operatorname{atan}{\left(3 \right)} \wedge \frac{\pi}{2} < x 
(pi/2 < x)∧(x <= pi - atan(3))
Rapid solution 2 [src] 
 pi               
(--, pi - atan(3)]
 2
x in (π2,π−atan⁡(3)]x\ in\ \left(\frac{\pi}{2}, \pi - \operatorname{atan}{\left(3 \right)}\right] 
x in Interval.Lopen(pi/2, pi - atan(3))
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