5/x<-2 inequation

Given the inequality:
$$\frac{5}{x} < -2$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{5}{x} = -2$$
Solve:
Given the equation:
$$\frac{5}{x} = -2$$
Use proportions rule:
From a1/b1 = a2/b2 should a1*b2 = a2*b1,
In this case

a1 = 1

b1 = 1/2

a2 = 1

b2 = -x/5

so we get the equation
$$- \frac{x}{5} = \frac{1}{2}$$
$$- \frac{x}{5} = \frac{1}{2}$$
Divide both parts of the equation by -1/5

x = 1/2 / (-1/5)

We get the answer: x = -5/2
$$x_{1} = - \frac{5}{2}$$
$$x_{1} = - \frac{5}{2}$$
This roots
$$x_{1} = - \frac{5}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{5}{2} + - \frac{1}{10}$$
=
$$- \frac{13}{5}$$
substitute to the expression
$$\frac{5}{x} < -2$$
$$\frac{5}{- \frac{13}{5}} < -2$$

-25      
---- < -2
 13      

but

-25      
---- > -2
 13      

Then
$$x < - \frac{5}{2}$$
no execute
the solution of our inequality is:
$$x > - \frac{5}{2}$$

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