2x/3-x+1/4<=--2 inequation

Given the inequality:
$$\left(- x + \frac{2 x}{3}\right) + \frac{1}{4} \leq 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- x + \frac{2 x}{3}\right) + \frac{1}{4} = 2$$
Solve:
Given the linear equation:

2*x/3-x+1/4 = --2

Looking for similar summands in the left part:

1/4 - x/3 = --2

Move free summands (without x)
from left part to right part, we given:
$$- \frac{x}{3} = \frac{7}{4}$$
Divide both parts of the equation by -1/3

x = 7/4 / (-1/3)

$$x_{1} = - \frac{21}{4}$$
$$x_{1} = - \frac{21}{4}$$
This roots
$$x_{1} = - \frac{21}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{21}{4} + - \frac{1}{10}$$
=
$$- \frac{107}{20}$$
substitute to the expression
$$\left(- x + \frac{2 x}{3}\right) + \frac{1}{4} \leq 2$$
$$\frac{1}{4} + \left(\frac{\left(- \frac{107}{20}\right) 2}{3} - - \frac{107}{20}\right) \leq 2$$

61     
-- <= 2
30     

but

61     
-- >= 2
30     

Then
$$x \leq - \frac{21}{4}$$
no execute
the solution of our inequality is:
$$x \geq - \frac{21}{4}$$

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