sin(x)+sqrt(3)*cos(x)>-1 inequation

Given the inequality:
$$\sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} > -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} = -1$$
Solve:
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{5 \pi}{6}$$
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{5 \pi}{6}$$
This roots
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{5 \pi}{6}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{\pi}{2} - \frac{1}{10}$$
=
$$- \frac{\pi}{2} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} > -1$$
$$\sin{\left(- \frac{\pi}{2} - \frac{1}{10} \right)} + \sqrt{3} \cos{\left(- \frac{\pi}{2} - \frac{1}{10} \right)} > -1$$

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-cos(1/10) - \/ 3 *sin(1/10) > -1
     

Then
$$x < - \frac{\pi}{2}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{\pi}{2} \wedge x < \frac{5 \pi}{6}$$

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