(4*x+5)/(6-5*x)>0 inequation

Given the inequality:
$$\frac{4 x + 5}{6 - 5 x} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{4 x + 5}{6 - 5 x} = 0$$
Solve:
Given the equation:
$$\frac{4 x + 5}{6 - 5 x} = 0$$
Multiply the equation sides by the denominator 6 - 5*x
we get:
$$- \frac{\left(6 - 5 x\right) \left(4 x + 5\right)}{5 x - 6} = 0$$
Expand brackets in the left part

-5-4*x6+5*x-6+5*x = 0

Looking for similar summands in the left part:

-(5 + 4*x)*(6 - 5*x)/(-6 + 5*x) = 0

Move free summands (without x)
from left part to right part, we given:
$$- \frac{\left(6 - 5 x\right) \left(4 x + 5\right)}{5 x - 6} + 6 = 6$$
Divide both parts of the equation by (6 - (5 + 4*x)*(6 - 5*x)/(-6 + 5*x))/x

x = 6 / ((6 - (5 + 4*x)*(6 - 5*x)/(-6 + 5*x))/x)

$$x_{1} = - \frac{5}{4}$$
$$x_{1} = - \frac{5}{4}$$
This roots
$$x_{1} = - \frac{5}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{5}{4} + - \frac{1}{10}$$
=
$$- \frac{27}{20}$$
substitute to the expression
$$\frac{4 x + 5}{6 - 5 x} > 0$$
$$\frac{\frac{\left(-27\right) 4}{20} + 5}{6 - \frac{\left(-27\right) 5}{20}} > 0$$

-8/255 > 0

Then
$$x < - \frac{5}{4}$$
no execute
the solution of our inequality is:
$$x > - \frac{5}{4}$$

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