Solving inequalities (need some clarification)

[Nitrate]

Homework Statement

Solve x/(x-2) > 2 by first rewriting it in the form P(x)/Q(x)>0

Homework Equations

The Attempt at a Solution

well... i got up to
x/(x - 2) > 2
x/(x - 2) - 2 > 0
x/(x - 2) - 2(x - 2)/(x - 2) > 0
x/(x - 2) - (2x - 4)/(x - 2) > 0
(x - 2x + 4)/(x - 2) > 0
(-x + 4)/(x - 2) > 0
-x + 4 > 0 and x - 2 > 0 OR -x + 4 < 0 and x - 2 < 0

However, i do not understand why there has to be an OR case if the inequality is only >

please help :)

 

[Mark44]

Homework Statement

Solve x/(x-2) > 2 by first rewriting it in the form P(x)/Q(x)>0

Homework Equations

The Attempt at a Solution

well... i got up to
x/(x - 2) > 2
x/(x - 2) - 2 > 0
x/(x - 2) - 2(x - 2)/(x - 2) > 0
x/(x - 2) - (2x - 4)/(x - 2) > 0
(x - 2x + 4)/(x - 2) > 0
(-x + 4)/(x - 2) > 0
-x + 4 > 0 and x - 2 > 0 OR -x + 4 < 0 and x - 2 < 0

However, i do not understand why there has to be an OR case if the inequality is only >

please help :)

Because numerator and denominator are either both positive OR both negative.

 

[Nitrate]

Because numerator and denominator are either both positive OR both negative.

the rationale being that a (+/+) = + and (-/-) = +
right?

 

[Nitrate]

okay another question
why would we do the following (the part with ***), what does it mean by no solutions: x/(x - 2) > 2
x/(x - 2) - 2 > 0
x/(x - 2) - 2(x - 2)/(x - 2) > 0
x/(x - 2) - (2x - 4)/(x - 2) > 0
(x - 2x + 4)/(x - 2) > 0
(-x + 4)/(x - 2) > 0
-x + 4 > 0 and x - 2 > 0 OR -x + 4 < 0 and x - 2 < 0
x < 4 and x > 2 OR x > 4 and x < 2
*****x > 4 and x < 2 has no solutions so discard it, leaving just

 

[Mark44]

Because there aren't any numbers that satisfy x > 4 and x < 2.