tanx/(cosx+√(3)sinx)≥0 inequation

Given the inequality:
$$\frac{\tan{\left(x \right)}}{\sqrt{3} \sin{\left(x \right)} + \cos{\left(x \right)}} \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\tan{\left(x \right)}}{\sqrt{3} \sin{\left(x \right)} + \cos{\left(x \right)}} = 0$$
Solve:
Given the equation
$$\frac{\tan{\left(x \right)}}{\sqrt{3} \sin{\left(x \right)} + \cos{\left(x \right)}} = 0$$
transform
$$\frac{\tan{\left(x \right)}}{2 \sin{\left(x + \frac{\pi}{6} \right)}} = 0$$
$$\frac{\tan{\left(x \right)}}{\sqrt{3} \sin{\left(x \right)} + \cos{\left(x \right)}} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation:
$$\frac{\tan{\left(x \right)}}{\sqrt{3} w + \cos{\left(x \right)}} = 0$$
Multiply the equation sides by the denominator w*sqrt(3) + cos(x)
we get:
$$\tan{\left(x \right)} = 0$$
Expand brackets in the left part

tanx = 0

This equation has no roots
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 0$$
$$x_{1} = 0$$
This roots
$$x_{1} = 0$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10}$$
=
$$- \frac{1}{10}$$
substitute to the expression
$$\frac{\tan{\left(x \right)}}{\sqrt{3} \sin{\left(x \right)} + \cos{\left(x \right)}} \geq 0$$
$$\frac{\tan{\left(- \frac{1}{10} \right)}}{\sqrt{3} \sin{\left(- \frac{1}{10} \right)} + \cos{\left(- \frac{1}{10} \right)}} \geq 0$$

         -tan(1/10)               
-----------------------------     
    ___                       >= 0
- \/ 3 *sin(1/10) + cos(1/10)     
     

but

         -tan(1/10)              
-----------------------------    
    ___                       < 0
- \/ 3 *sin(1/10) + cos(1/10)    
    

Then
$$x \leq 0$$
no execute
the solution of our inequality is:
$$x \geq 0$$

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