log(1/2)(x+1)<0 inequation

Given the inequality:
$$\left(x + 1\right) \log{\left(\frac{1}{2} \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x + 1\right) \log{\left(\frac{1}{2} \right)} = 0$$
Solve:
Given the equation:

log(1/2)*(x+1) = 0

Expand expressions:

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

Reducing, you get:

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

Expand brackets in the left part

-log2 - x*log2 = 0

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

x = 0 / ((-log(2) - x*log(2))/x)

We get the answer: x = -1
$$x_{1} = -1$$
$$x_{1} = -1$$
This roots
$$x_{1} = -1$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-1 + - \frac{1}{10}$$
=
$$- \frac{11}{10}$$
substitute to the expression
$$\left(x + 1\right) \log{\left(\frac{1}{2} \right)} < 0$$
$$\left(- \frac{11}{10} + 1\right) \log{\left(\frac{1}{2} \right)} < 0$$

log(2)    
------ < 0
  10      

but

log(2)    
------ > 0
  10      

Then
$$x < -1$$
no execute
the solution of our inequality is:
$$x > -1$$

         _____  
        /
-------ο-------
       x1