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tg(x)≥-√3/3 inequation

A inequation with variable

The solution

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             ___ 
          -\/ 3  
tan(x) >= -------
             3   
$$\tan{\left(x \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
tan(x) >= (-sqrt(3))/3
Detail solution
Given the inequality:
$$\tan{\left(x \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
Solve:
Given the equation
$$\tan{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(- \frac{\sqrt{3}}{3} \right)}$$
Or
$$x = \pi n - \frac{\pi}{6}$$
, where n - is a integer
$$x_{1} = \pi n - \frac{\pi}{6}$$
$$x_{1} = \pi n - \frac{\pi}{6}$$
This roots
$$x_{1} = \pi n - \frac{\pi}{6}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n - \frac{\pi}{6}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{\pi}{6} - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(x \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
$$\tan{\left(\pi n - \frac{\pi}{6} - \frac{1}{10} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
                           ___ 
    /1    pi       \    -\/ 3  
-tan|-- + -- - pi*n| >= -------
    \10   6        /       3   
                        

but
                          ___ 
    /1    pi       \   -\/ 3  
-tan|-- + -- - pi*n| < -------
    \10   6        /      3   
                       

Then
$$x \leq \pi n - \frac{\pi}{6}$$
no execute
the solution of our inequality is:
$$x \geq \pi n - \frac{\pi}{6}$$
         _____  
        /
-------•-------
       x1
Solving inequality on a graph
Rapid solution [src]
  /   /            pi\     /5*pi              \\
Or|And|0 <= x, x < --|, And|---- <= x, x <= pi||
  \   \            2 /     \ 6                //
$$\left(0 \leq x \wedge x < \frac{\pi}{2}\right) \vee \left(\frac{5 \pi}{6} \leq x \wedge x \leq \pi\right)$$
((0 <= x)∧(x < pi/2))∨((x <= pi)∧(5*pi/6 <= x))
Rapid solution 2 [src]
    pi     5*pi     
[0, --) U [----, pi]
    2       6       
$$x\ in\ \left[0, \frac{\pi}{2}\right) \cup \left[\frac{5 \pi}{6}, \pi\right]$$
x in Union(Interval.Ropen(0, pi/2), Interval(5*pi/6, pi))