log(2)x>=1 inequation

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

log(2)*x = 1

Expand brackets in the left part

log2x = 1

Divide both parts of the equation by log(2)

x = 1 / (log(2))

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

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

but

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

Then
$$x \leq \frac{1}{\log{\left(2 \right)}}$$
no execute
the solution of our inequality is:
$$x \geq \frac{1}{\log{\left(2 \right)}}$$

         _____  
        /
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       x1