3x-7x+4/6<_3 inequation

Given the inequality:
$$\left(- 7 x + 3 x\right) + \frac{2}{3} \leq 3$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 7 x + 3 x\right) + \frac{2}{3} = 3$$
Solve:
Given the linear equation:

3*x-7*x+4/6 = 3

Looking for similar summands in the left part:

2/3 - 4*x = 3

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

x = 7/3 / (-4)

$$x_{1} = - \frac{7}{12}$$
$$x_{1} = - \frac{7}{12}$$
This roots
$$x_{1} = - \frac{7}{12}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{7}{12} + - \frac{1}{10}$$
=
$$- \frac{41}{60}$$
substitute to the expression
$$\left(- 7 x + 3 x\right) + \frac{2}{3} \leq 3$$
$$\frac{2}{3} + \left(\frac{\left(-41\right) 3}{60} - \frac{\left(-41\right) 7}{60}\right) \leq 3$$

17/5 <= 3

but

17/5 >= 3

Then
$$x \leq - \frac{7}{12}$$
no execute
the solution of our inequality is:
$$x \geq - \frac{7}{12}$$

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