logx(4x-3)>=2 inequation

Given the inequality:
$$\left(4 x - 3\right) \log{\left(x \right)} \geq 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(4 x - 3\right) \log{\left(x \right)} = 2$$
Solve:
$$x_{1} = 1.69623401783029$$
$$x_{1} = 1.69623401783029$$
This roots
$$x_{1} = 1.69623401783029$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1.69623401783029$$
=
$$1.59623401783029$$
substitute to the expression
$$\left(4 x - 3\right) \log{\left(x \right)} \geq 2$$
$$\left(\left(-1\right) 3 + 4 \cdot 1.59623401783029\right) \log{\left(1.59623401783029 \right)} \geq 2$$

1.58295559157076 >= 2

but

1.58295559157076 < 2

Then
$$x \leq 1.69623401783029$$
no execute
the solution of our inequality is:
$$x \geq 1.69623401783029$$

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