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

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

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

Expand brackets in the left part

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:
$$- \frac{x}{3} = \frac{9}{4}$$
Divide both parts of the equation by -1/3

x = 9/4 / (-1/3)

$$x_{1} = - \frac{27}{4}$$
$$x_{1} = - \frac{27}{4}$$
This roots
$$x_{1} = - \frac{27}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{27}{4} + - \frac{1}{10}$$
=
$$- \frac{137}{20}$$
substitute to the expression
$$\frac{2 x}{3} + \left(- x - \frac{1}{4}\right) \leq 2$$
$$\frac{\left(- \frac{137}{20}\right) 2}{3} + \left(- \frac{1}{4} - - \frac{137}{20}\right) \leq 2$$

61     
-- <= 2
30     

but

61     
-- >= 2
30     

Then
$$x \leq - \frac{27}{4}$$
no execute
the solution of our inequality is:
$$x \geq - \frac{27}{4}$$

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