Solving inequalities (need some clarification)

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Homework Help Overview

The discussion revolves around solving the inequality x/(x-2) > 2, with a focus on rewriting it in the form P(x)/Q(x) > 0. Participants are exploring the implications of the inequality and the reasoning behind the cases presented.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the steps taken to manipulate the inequality and express confusion regarding the necessity of considering both positive and negative cases for the numerator and denominator. There is also a question about the meaning of having no solutions in certain cases.

Discussion Status

The conversation is active, with participants seeking clarification on the reasoning behind the OR cases in the inequality. Some have provided insights into the conditions under which the inequality holds, while others are questioning the implications of specific cases leading to no solutions.

Contextual Notes

Participants are navigating the constraints of the inequality and the implications of the conditions derived from it, particularly regarding the absence of solutions in certain scenarios.

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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 :)
 
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Nitrate said:

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.
 
Mark44 said:
Because numerator and denominator are either both positive OR both negative.

the rationale being that a (+/+) = + and (-/-) = +
right?
 
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
 
Because there aren't any numbers that satisfy x > 4 and x < 2.
 

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