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Math Help - Rational Problem

  1. #1
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    Rational Problem

    Need some help, I know for x to be rational x = p/q and I'm pretty sure the statement holds true.

    Prove that if (3+x)/(3-x) is rational, then x is rational.
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  2. #2
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    Re: Rational Problem

    Quote Originally Posted by xmathlover View Post
    I know for x to be rational x = p/q
    Every number can be represented as p / q. For example, \pi=\pi / 1.

    Quote Originally Posted by xmathlover View Post
    Prove that if (3+x)/(3-x) is rational, then x is rational.
    Have you tried applying the definition of rationality and solving the resulting equation for x?
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    Re: Rational Problem

    Suppose x is rational. x= P/Q for some ints p and q where q does not equal zero. (3+P/Q)/(3-P/Q).

    This is where I get stuck.
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  4. #4
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    Re: Rational Problem

    Imagine that an implication "If A, then B", or A ⇒ B in symbolic form, works as a store that sells B's for the price of an A. If A ⇒ B has already been proven and can be used, it's like a store in your neighborhood. To use it, you need to have an A ready, and when you come to the store, you exchange it for a B. It's a different story if you would like to create your own business and open a store, i.e., if you would like to prove A ⇒ B. Then you need to have an infrastructure so that when a customer comes with an A, you can use it to manufacture, buy wholesale or otherwise obtain a B and give it to the customer.

    To summarize: If A ⇒ B is proven or assumed, you can give it an A and obtain a B. If you want to prove A ⇒ B, you need to be ready to accept an A and convert it into a B.

    Quote Originally Posted by xmathlover View Post
    Prove that if (3+x)/(3-x) is rational, then x is rational.
    Quote Originally Posted by xmathlover View Post
    Suppose x is rational. x= P/Q for some ints p and q where q does not equal zero. (3+P/Q)/(3-P/Q).

    This is where I get stuck.
    You need to prove that [(3+x)/(3-x) is rational] ⇒ [x is rational]. A customer comes to you with a proof of [(3+x)/(3-x) is rational]. Instead, you demand from him/her a proof that x= P / Q, i.e., what you are supposed to deliver! You and the customer are facing each other in bewilderment.
    Last edited by emakarov; April 3rd 2013 at 07:32 AM. Reason: Explained the notation A ⇒ B
    Thanks from Ruun
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    Re: Rational Problem

    Edit Didn't read properly

    \frac{3+x}{3-x}=\frac{p}{q} \rightarrow q(3+x)=p(3-x) \rightarrow 3q+3x=3p-px \rightarrow (3+p)x=3(p-q) so x=\frac{3(p-q)}{3+p}

    Which conclusion we can draw from here?
    Last edited by Ruun; April 3rd 2013 at 07:32 AM.
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    Re: Rational Problem

    Quote Originally Posted by Ruun View Post
    Edit Didn't read properly

    \frac{3+x}{3-x}=\frac{p}{q} \rightarrow q(3+x)=p(3-x) \rightarrow 3q+3x=3p-px \rightarrow (3+p)x=3(p-q) so x=\frac{3(p-q)}{3+p}

    Which conclusion we can draw from here?
    x is rational. So you assumed 3+x/3-x is rational? I don't follow how you went from 3+x/3-x to q(3+x) = p(3-x)
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  7. #7
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    Re: Rational Problem

    I assumed what we were given, that \frac{3+x}{3-x} is rational. More carefully:

    \frac{3+x}{3-x}=\frac{p}{q}

    Multiply by q

    q\left(\frac{3+x}{3-x}\right)=p

    Multiply by (3-x)

    q(3+x)=p(3-x)
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    Re: Rational Problem

    Quote Originally Posted by Ruun View Post
    I assumed what we were given, that \frac{3+x}{3-x} is rational. More carefully:

    \frac{3+x}{3-x}=\frac{p}{q}

    Multiply by q

    q\left(\frac{3+x}{3-x}\right)=p

    Multiply by (3-x)

    q(3+x)=p(3-x)
    ok, excellent. Makes sense.
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