ln(4x-3)>=ln9 inequation

Given the inequality:
$$\log{\left(4 x - 3 \right)} \geq \log{\left(9 \right)}$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(4 x - 3 \right)} = \log{\left(9 \right)}$$
Solve:
Given the equation
$$\log{\left(4 x - 3 \right)} = \log{\left(9 \right)}$$
$$\log{\left(4 x - 3 \right)} = \log{\left(9 \right)}$$
This equation is of the form:

log(v)=p

By definition log

v=e^p

then
$$4 x - 3 = e^{\frac{\log{\left(9 \right)}}{1}}$$
simplify
$$4 x - 3 = 9$$
$$4 x = 12$$
$$x = 3$$
$$x_{1} = 3$$
$$x_{1} = 3$$
This roots
$$x_{1} = 3$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 3$$
=
$$\frac{29}{10}$$
substitute to the expression
$$\log{\left(4 x - 3 \right)} \geq \log{\left(9 \right)}$$
$$\log{\left(-3 + \frac{4 \cdot 29}{10} \right)} \geq \log{\left(9 \right)}$$

log(43/5) >= log(9)

but

log(43/5) < log(9)

Then
$$x \leq 3$$
no execute
the solution of our inequality is:
$$x \geq 3$$

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