(1/2)*x-3*x+4>=0 inequation

Given the inequality:
$$\left(- 3 x + \frac{x}{2}\right) + 4 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 3 x + \frac{x}{2}\right) + 4 = 0$$
Solve:
Given the linear equation:

(1/2)*x-3*x+4 = 0

Expand brackets in the left part

1/2x-3*x+4 = 0

Looking for similar summands in the left part:

4 - 5*x/2 = 0

Move free summands (without x)
from left part to right part, we given:
$$- \frac{5 x}{2} = -4$$
Divide both parts of the equation by -5/2

x = -4 / (-5/2)

$$x_{1} = \frac{8}{5}$$
$$x_{1} = \frac{8}{5}$$
This roots
$$x_{1} = \frac{8}{5}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{8}{5}$$
=
$$\frac{3}{2}$$
substitute to the expression
$$\left(- 3 x + \frac{x}{2}\right) + 4 \geq 0$$
$$\left(- \frac{3 \cdot 3}{2} + \frac{3}{2 \cdot 2}\right) + 4 \geq 0$$

1/4 >= 0

the solution of our inequality is:
$$x \leq \frac{8}{5}$$

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