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Math Help - Question about inequations

  1. #1
    Junior Member
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    Question about inequations

    Hi !
    Can you help me how to solve inequations of this form:
    \frac{a}{b}<\frac{c}{d}<\frac{e}{f}
    a,b,c,d,e and f are polynomials
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  2. #2
    A Plied Mathematician
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    Sure. What you've got there are really two inequalities that are logically AND'ed together. That is you're solving the following simultaneous inequalities:

    \frac{a}{b}<\frac{c}{d} AND

    \frac{c}{d}<\frac{e}{f}.

    Can you write up a specific example to work on?
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  3. #3
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    Once you have the problem reduced to \frac{a}{b}< \frac{c}{d}, where a, b, c, and d are polynomials, you need to realize that "<" can change to ">" only where (1) they are "=" or (2) where at least one of the denominators is 0.

    So: first solve the equation \frac{a}{b}= \frac{c}{d} as well as b= 0 and d= 0. Those point separate "<" from ">". You can choose one point in each interval to see whether "<" or ">" is correct in that interval.

    Another way is to rewrite the problem as \frac{a}{b}- \frac{c}{d}= \frac{ac- bd}{bd}< 0, and then factor ac- bd as well as bd. For each interval between zeros of ac- bd, b, and d, you can count the number of negative factors to see if "> 0 " or "< 0" is correct.
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