log½x≤−3. inequation

Given the inequality:
$$x \log{\left(\frac{1}{2} \right)} \leq -3$$
To solve this inequality, we must first solve the corresponding equation:
$$x \log{\left(\frac{1}{2} \right)} = -3$$
Solve:
Given the linear equation:

log(1/2)*x = -3

Expand brackets in the left part

log1/2x = -3

Divide both parts of the equation by -log(2)

x = -3 / (-log(2))

$$x_{1} = \frac{3}{\log{\left(2 \right)}}$$
$$x_{1} = \frac{3}{\log{\left(2 \right)}}$$
This roots
$$x_{1} = \frac{3}{\log{\left(2 \right)}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{3}{\log{\left(2 \right)}}$$
=
$$- \frac{1}{10} + \frac{3}{\log{\left(2 \right)}}$$
substitute to the expression
$$x \log{\left(\frac{1}{2} \right)} \leq -3$$
$$\left(- \frac{1}{10} + \frac{3}{\log{\left(2 \right)}}\right) \log{\left(\frac{1}{2} \right)} \leq -3$$

 /  1      3   \             
-|- -- + ------|*log(2) <= -3
 \  10   log(2)/             

but

 /  1      3   \             
-|- -- + ------|*log(2) >= -3
 \  10   log(2)/             

Then
$$x \leq \frac{3}{\log{\left(2 \right)}}$$
no execute
the solution of our inequality is:
$$x \geq \frac{3}{\log{\left(2 \right)}}$$

         _____  
        /
-------•-------
       x1