-6*x-(3/2)>0 inequation

Given the inequality:
$$- 6 x - \frac{3}{2} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 6 x - \frac{3}{2} = 0$$
Solve:
Given the linear equation:

-6*x-(3/2) = 0

Expand brackets in the left part

-6*x-3/2 = 0

Move free summands (without x)
from left part to right part, we given:
$$- 6 x = \frac{3}{2}$$
Divide both parts of the equation by -6

x = 3/2 / (-6)

$$x_{1} = - \frac{1}{4}$$
$$x_{1} = - \frac{1}{4}$$
This roots
$$x_{1} = - \frac{1}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{4} + - \frac{1}{10}$$
=
$$- \frac{7}{20}$$
substitute to the expression
$$- 6 x - \frac{3}{2} > 0$$
$$- \frac{3}{2} - \frac{\left(-7\right) 6}{20} > 0$$

3/5 > 0

the solution of our inequality is:
$$x < - \frac{1}{4}$$

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