(x-4):(x-7)<=0 inequation

Given the inequality:
$$\frac{x - 4}{x - 7} \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{x - 4}{x - 7} = 0$$
Solve:
Given the equation:
$$\frac{x - 4}{x - 7} = 0$$
Multiply the equation sides by the denominator -7 + x
we get:
$$x - 4 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$x = 4$$
$$x_{1} = 4$$
$$x_{1} = 4$$
This roots
$$x_{1} = 4$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 4$$
=
$$\frac{39}{10}$$
substitute to the expression
$$\frac{x - 4}{x - 7} \leq 0$$
$$\frac{-4 + \frac{39}{10}}{-7 + \frac{39}{10}} \leq 0$$

1/31 <= 0

but

1/31 >= 0

Then
$$x \leq 4$$
no execute
the solution of our inequality is:
$$x \geq 4$$

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