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Solving inequalities (need some clarification)

  1. May 2, 2012 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations



    3. 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 :)
     
  2. jcsd
  3. May 2, 2012 #2

    Mark44

    Staff: Mentor

    Because numerator and denominator are either both positive OR both negative.
     
  4. May 2, 2012 #3
    the rationale being that a (+/+) = + and (-/-) = +
    right?
     
  5. May 2, 2012 #4
    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
     
  6. May 2, 2012 #5

    Mark44

    Staff: Mentor

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