Solve for x under the assumption that x> 0

x-7/x < -6

Good evening:

Assuming (x - 7)/x < -6

We need x ≠ 0.

Adding 6 to both sides:



We put the expression in the same denominator:



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