log(x)*(3*x-1)>1 inequation

Given the inequality:
$$\left(3 x - 1\right) \log{\left(x \right)} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(3 x - 1\right) \log{\left(x \right)} = 1$$
Solve:
$$x_{1} = 1.37649594984106$$
$$x_{1} = 1.37649594984106$$
This roots
$$x_{1} = 1.37649594984106$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1.37649594984106$$
=
$$1.27649594984106$$
substitute to the expression
$$\left(3 x - 1\right) \log{\left(x \right)} > 1$$
$$\left(-1 + 1.27649594984106 \cdot 3\right) \log{\left(1.27649594984106 \right)} > 1$$

0.690731135522226 > 1

Then
$$x < 1.37649594984106$$
no execute
the solution of our inequality is:
$$x > 1.37649594984106$$

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