Looking at your final fraction, it is less than zero for positive divided by negative

or for negative divided by positive,

so you have a few cases to examine.

The numerator is negative for x < 2,

but the denominator is positive for x and x+1 both positive or both negative.

Similarly, the numerator is positive for x > 2

and the denominator is always positive in this case.

Hence we cannot have positive divided by negative for this fraction.