log(3)*(2*x+1)<2 inequation

Given the inequality:
$$\left(2 x + 1\right) \log{\left(3 \right)} < 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(2 x + 1\right) \log{\left(3 \right)} = 2$$
Solve:
Given the equation:

log(3)*(2*x+1) = 2

Expand expressions:

2*x*log(3) + log(3) = 2

Reducing, you get:

-2 + 2*x*log(3) + log(3) = 0

Expand brackets in the left part

-2 + 2*x*log3 + log3 = 0

Move free summands (without x)
from left part to right part, we given:
$$2 x \log{\left(3 \right)} + \log{\left(3 \right)} = 2$$
Divide both parts of the equation by (2*x*log(3) + log(3))/x

x = 2 / ((2*x*log(3) + log(3))/x)

We get the answer: x = -1/2 + 1/log(3)
$$x_{1} = - \frac{1}{2} + \frac{1}{\log{\left(3 \right)}}$$
$$x_{1} = - \frac{1}{2} + \frac{1}{\log{\left(3 \right)}}$$
This roots
$$x_{1} = - \frac{1}{2} + \frac{1}{\log{\left(3 \right)}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(- \frac{1}{2} + \frac{1}{\log{\left(3 \right)}}\right)$$
=
$$- \frac{3}{5} + \frac{1}{\log{\left(3 \right)}}$$
substitute to the expression
$$\left(2 x + 1\right) \log{\left(3 \right)} < 2$$
$$\left(2 \left(- \frac{3}{5} + \frac{1}{\log{\left(3 \right)}}\right) + 1\right) \log{\left(3 \right)} < 2$$

/  1     2   \           
|- - + ------|*log(3) < 2
\  5   log(3)/           

the solution of our inequality is:
$$x < - \frac{1}{2} + \frac{1}{\log{\left(3 \right)}}$$

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