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Simple Algebra Question

  1. Sep 1, 2007 #1

    I am working through the superb book "50 challenging problems in probability." I have a slight problem with some of the algebra used in solving one of the problems. The algebra to solve is shown below:


    Since, for b > 0:
    \frac{r}{r+b} > \frac{r-1}{r+b-1}

    \left(\frac{r}{r+b}\right)^2 > \frac{1}{2} > \left(\frac{r-1}{r+b-1}\right)^2

    [tex]\frac{r}{r+b} > \frac{1}{\sqrt{2}} > \frac{r-1}{r+b-1}[/tex]

    So (this is the part I have difficulty with):

    [tex]r > \frac{b}{\sqrt{2}-1} = (\sqrt{2}+1)b[/tex]

    I cant see how the two sides of the = sign can be transformed into each other, or how they are equivalent. Can anyone help with this?

  2. jcsd
  3. Sep 1, 2007 #2
    Divide both sides with [tex]\sqrt{2}+1[/tex] work the product in the denominator to yield one, and you'll see.
  4. Sep 1, 2007 #3


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    Science Advisor

    I would have multiplied both sides by [itex]\sqrt{2}-1[/itex]!:rolleyes:
  5. Sep 1, 2007 #4

    matt grime

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    Homework Helper

    But surely better than to verify, is to understand how to transform things like this in the future. It is rationalizing the denominator.
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