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.

Printable View

- Apr 3rd 2013, 07:50 AMxmathloverRational 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. - Apr 3rd 2013, 08:04 AMemakarovRe: Rational Problem
- Apr 3rd 2013, 08:07 AMxmathloverRe: 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. - Apr 3rd 2013, 08:26 AMemakarovRe: 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.

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. - Apr 3rd 2013, 08:26 AMRuunRe: Rational Problem
Edit Didn't read properly

so

Which conclusion we can draw from here? - Apr 3rd 2013, 08:35 AMxmathloverRe: Rational Problem
- Apr 3rd 2013, 08:39 AMRuunRe: Rational Problem
I assumed what we were given, that is rational. More carefully:

Multiply by

Multiply by

- Apr 3rd 2013, 08:41 AMxmathloverRe: Rational Problem