12/(3x+7)>3 inequation

Given the inequality:
$$\frac{12}{3 x + 7} > 3$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{12}{3 x + 7} = 3$$
Solve:
Given the equation:
$$\frac{12}{3 x + 7} = 3$$
Use proportions rule:
From a1/b1 = a2/b2 should a1*b2 = a2*b1,
In this case

a1 = 12

b1 = 7 + 3*x

a2 = 1

b2 = 1/3

so we get the equation
$$\frac{12}{3} = 3 x + 7$$
$$4 = 3 x + 7$$
Move free summands (without x)
from left part to right part, we given:
$$0 = 3 x + 3$$
Move the summands with the unknown x
from the right part to the left part:
$$\left(-3\right) x = 3$$
Divide both parts of the equation by -3

x = 3 / (-3)

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
$$\frac{12}{3 x + 7} > 3$$
$$\frac{12}{\frac{\left(-11\right) 3}{10} + 7} > 3$$

120    
--- > 3
 37    

the solution of our inequality is:
$$x < -1$$

 _____          
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       x1