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Math Help - Fraction proof

  1. #1
    Junior Member Freaky-Person's Avatar
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    Fraction proof

    I don't know, it has variables in it... Ugh finding a section is hard.

    Prove \frac{x+ab}{y+ab} \geq \frac{x}{y}

    because I moved countries a lot when I was little I think I missed out on most of the fraction axioms taught in elementary (took me to the end of high school to figure out cross multiplication >.>) so it might be that I'm missing some fundamental rule here...
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  2. #2
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    LEt x=1, y=2,a=1, b=-1

    So \frac{1-1}{2-1}\geq \frac{1}{2} is false,

    so what are the restrictions on x,y,a,b, are they all positive, are just a,b>0... we need more info otherwise I just gave you a counterexample meaning its false
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  3. #3
    Junior Member Freaky-Person's Avatar
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    Quote Originally Posted by artvandalay11 View Post
    LEt x=1, y=2,a=1, b=-1

    So \frac{1-1}{2-1}\geq \frac{1}{2} is false,

    so what are the restrictions on x,y,a,b, are they all positive, are just a,b>0... we need more info otherwise I just gave you a counterexample meaning its false
    oh, yeah, a,b > 0

    That was important, wasn't it >.>
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  4. #4
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    I'm thinking it's a proof by induction but I don't even know where to go with that >.> I mean, what am I adding 1 to? There are 4 variables here! And cases means too many cases...
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  5. #5
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    i dont think it's proof by induction, although we could just let c=ab for some c>0 but that is still 3 variables

    Let's go with this

    Consider y(x+ab)-x(y+ab)

    =yx+yab-xy-xab=yab-xab=ab(y-x)

    If ab(y-x)=0 then y=x, so \frac{x}{y}=1 and \frac{x+ab}{y+ab}=1 so the inequality holds


    If ab(y-x)>0, then y(x+ab)-x(y+ab)>0 so y(x+ab)>x(y+ab) and so (x+ab)>\frac{x(y+ab)}{y}

    And so \frac{x+ab}{y+ab}>\frac{x}{y} (note that everything we divided by was >0 so that argument really is valid)

    If ab(y-x)<0 then x>y since ab>0. So y(x+ab)-x(y+ab)<0 and just looking ahead this fails your inequality meaning we must place the restriction that x cannot be >y

    For example let x=4 a=1 b=1 and y=1

    Then \frac{4+1(1)}{1+1(1)}=\frac{5}{2}=2.5

    \frac{x}{y}=\frac{4}{1}=4 and 2.5 is not \geq 4
    Last edited by artvandalay11; September 27th 2009 at 05:24 PM.
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  6. #6
    Junior Member Freaky-Person's Avatar
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    That's supposed to be a greater than in the second part, yeah?
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  7. #7
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    yes sorry about that very important typo
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