log(1/2)x+1>=-1 inequation

Given the inequality:
$$x \log{\left(\frac{1}{2} \right)} + 1 \geq -1$$
To solve this inequality, we must first solve the corresponding equation:
$$x \log{\left(\frac{1}{2} \right)} + 1 = -1$$
Solve:
Given the linear equation:

log(1/2)*x+1 = -1

Expand brackets in the left part

log1/2x+1 = -1

Move free summands (without x)
from left part to right part, we given:
$$- x \log{\left(2 \right)} = -2$$
Divide both parts of the equation by -log(2)

x = -2 / (-log(2))

$$x_{1} = \frac{2}{\log{\left(2 \right)}}$$
$$x_{1} = \frac{2}{\log{\left(2 \right)}}$$
This roots
$$x_{1} = \frac{2}{\log{\left(2 \right)}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{2}{\log{\left(2 \right)}}$$
=
$$- \frac{1}{10} + \frac{2}{\log{\left(2 \right)}}$$
substitute to the expression
$$x \log{\left(\frac{1}{2} \right)} + 1 \geq -1$$
$$\left(- \frac{1}{10} + \frac{2}{\log{\left(2 \right)}}\right) \log{\left(\frac{1}{2} \right)} + 1 \geq -1$$

    /  1      2   \             
1 - |- -- + ------|*log(2) >= -1
    \  10   log(2)/             

the solution of our inequality is:
$$x \leq \frac{2}{\log{\left(2 \right)}}$$

 _____          
      \    
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       x1