log(6)x>1 inequation

Given the inequality:
$$x \log{\left(6 \right)} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$x \log{\left(6 \right)} = 1$$
Solve:
Given the linear equation:

log(6)*x = 1

Expand brackets in the left part

log6x = 1

Divide both parts of the equation by log(6)

x = 1 / (log(6))

$$x_{1} = \frac{1}{\log{\left(6 \right)}}$$
$$x_{1} = \frac{1}{\log{\left(6 \right)}}$$
This roots
$$x_{1} = \frac{1}{\log{\left(6 \right)}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{1}{\log{\left(6 \right)}}$$
=
$$- \frac{1}{10} + \frac{1}{\log{\left(6 \right)}}$$
substitute to the expression
$$x \log{\left(6 \right)} > 1$$
$$\left(- \frac{1}{10} + \frac{1}{\log{\left(6 \right)}}\right) \log{\left(6 \right)} > 1$$

/  1      1   \           
|- -- + ------|*log(6) > 1
\  10   log(6)/           

Then
$$x < \frac{1}{\log{\left(6 \right)}}$$
no execute
the solution of our inequality is:
$$x > \frac{1}{\log{\left(6 \right)}}$$

         _____  
        /
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       x_1