Good evening:
Assuming (x - 7)/x < -6
We need x ≠ 0.
Adding 6 to both sides:
[tex]\mathsf{\dfrac{x-7}{x}+6\ \textless \ -6+6}\\ \\ \mathsf{\dfrac{x-7}{x}+6\ \textless \ 0}[/tex]
We put the expression in the same denominator:
[tex]\mathsf{\dfrac{x-7}{x}+\dfrac{6x}{x} \ \textless \ 0}\\ \\ \\ \mathsf{\dfrac{x + 6x-7}{x}\ \textless \ 0}\\ \\ \\ \mathsf{\dfrac{7x - 7}{x} \ \textless \ 0}\\ \\ \\ \mathsf{7\cdot\dfrac{x - 1}{x} \ \textless \ 0} \ \ \ \ \ \ \mathsf{(\div 7)}\\ \\ \\ \mathsf{\dfrac{x-1}{x} \ \textless \ 0}[/tex]
Now we need to study the signal of the numerator and denominator:
x - 1 = 0
x = 1 → When N = 0, x = 1.
x - 1 > 0 → x > 1
x - 1 < 0 → x < 1
x = 0 → D = 0 when x = 0
x > 0 → D > 0 when x > 0
x < 0 → D < 0 when x < 0
Now we build a table with the numerator and denominator signals:
N: - - - - - - - - - - - - - - -- - 1 + + + + + +
D: - - - - - - - - 0 + + + + + + + + + + + +
N/D + + + + + 0 - - - - - - - - 1 + + + + + +
Therefore, we have that (x - 1) / x < 0 when 0 < x < 1
[tex]\texttt{Solution: \ x}\in\mathbb{R}\mathtt{: x}\in\texttt{(0,1)}[/tex]
