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Inequalities and bounds

  1. Feb 5, 2012 #1
    1. The problem statement, all variables and given/known data
    I just want to show that given x<0, [tex]\frac{x-1}{x-2} <1[/tex].


    3. The attempt at a solution

    I don't know why I am having trouble with this! I feel like this is so easy!

    So if x<0, then we know [tex]x-1<-1, x-2<-2 [/tex]. So
    [tex]\frac{-1}{2}<\frac{1}{x-2}[/tex] and [tex]\frac{x-1}{x-2}<\frac{-1}{x-2}[/tex].

    I can't seem to get a good upper bound on [tex]\frac{1}{x-2}[/tex] that makes the entire thing less than one. Am I doing something illegal? Because now it looks like I should want to get[tex]\frac{1}{x-2} <-1[/tex] to make it all less than one, but clearly that is not true.
     
  2. jcsd
  3. Feb 5, 2012 #2

    SammyS

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    [itex]\displaystyle \frac{x-1}{x-2}=\frac{x-2+1}{x-2}=1+\frac{1}{x-2}[/itex]

    Can you show that 1/(x-2) < 0 ?
     
  4. Feb 5, 2012 #3
    Yes I can. Thank you so much! Was I doing anything illegal or just picking bad bounds? I can't seem to find a mistake in what I was doing before.
     
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