log(3)(x+2)< inequation

Given the inequality:
$$\left(x + 2\right) \log{\left(3 \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x + 2\right) \log{\left(3 \right)} = 0$$
Solve:
Given the equation:

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

Expand expressions:

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

Reducing, you get:

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

Expand brackets in the left part

2*log3 + x*log3 = 0

Divide both parts of the equation by (2*log(3) + x*log(3))/x

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

We get the answer: x = -2
$$x_{1} = -2$$
$$x_{1} = -2$$
This roots
$$x_{1} = -2$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-2 + - \frac{1}{10}$$
=
$$- \frac{21}{10}$$
substitute to the expression
$$\left(x + 2\right) \log{\left(3 \right)} < 0$$
$$\left(- \frac{21}{10} + 2\right) \log{\left(3 \right)} < 0$$

-log(3)     
-------- < 0
   10       

the solution of our inequality is:
$$x < -2$$

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