4/(x-6)<=1 inequation

Given the inequality:
$$\frac{4}{x - 6} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{4}{x - 6} = 1$$
Solve:
Given the equation:
$$\frac{4}{x - 6} = 1$$
Use proportions rule:
From a1/b1 = a2/b2 should a1*b2 = a2*b1,
In this case

a1 = 4

b1 = -6 + x

a2 = 1

b2 = 1

so we get the equation
$$4 = x - 6$$
$$4 = x - 6$$
Move free summands (without x)
from left part to right part, we given:
$$0 = x - 10$$
Move the summands with the unknown x
from the right part to the left part:
$$- x = -10$$
Divide both parts of the equation by -1

x = -10 / (-1)

We get the answer: x = 10
$$x_{1} = 10$$
$$x_{1} = 10$$
This roots
$$x_{1} = 10$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 10$$
=
$$\frac{99}{10}$$
substitute to the expression
$$\frac{4}{x - 6} \leq 1$$
$$\frac{4}{-6 + \frac{99}{10}} \leq 1$$

40     
-- <= 1
39     

but

40     
-- >= 1
39     

Then
$$x \leq 10$$
no execute
the solution of our inequality is:
$$x \geq 10$$

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