log(1/7)x<0 inequation

Given the inequality:
$$x \log{\left(\frac{1}{7} \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$x \log{\left(\frac{1}{7} \right)} = 0$$
Solve:
Given the linear equation:

log(1/7)*x = 0

Expand brackets in the left part

log1/7x = 0

Divide both parts of the equation by -log(7)

x = 0 / (-log(7))

$$x_{1} = 0$$
$$x_{1} = 0$$
This roots
$$x_{1} = 0$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10}$$
=
$$- \frac{1}{10}$$
substitute to the expression
$$x \log{\left(\frac{1}{7} \right)} < 0$$
$$\frac{\left(-1\right) \log{\left(\frac{1}{7} \right)}}{10} < 0$$

log(7)    
------ < 0
  10      

but

log(7)    
------ > 0
  10      

Then
$$x < 0$$
no execute
the solution of our inequality is:
$$x > 0$$

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