# Rational Problem

• Apr 3rd 2013, 06:50 AM
xmathlover
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.
• Apr 3rd 2013, 07:04 AM
emakarov
Re: Rational Problem
Quote:

Originally Posted by xmathlover
I know for x to be rational x = p/q

Every number can be represented as p / q. For example, $\displaystyle \pi=\pi / 1$. :)

Quote:

Originally Posted by xmathlover
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?
• Apr 3rd 2013, 07:07 AM
xmathlover
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.
• Apr 3rd 2013, 07:26 AM
emakarov
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
Prove that if (3+x)/(3-x) is rational, then x is rational.

Quote:

Originally Posted by xmathlover
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.
• Apr 3rd 2013, 07:26 AM
Ruun
Re: Rational Problem

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

Which conclusion we can draw from here?
• Apr 3rd 2013, 07:35 AM
xmathlover
Re: Rational Problem
Quote:

Originally Posted by Ruun

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

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)
• Apr 3rd 2013, 07:39 AM
Ruun
Re: Rational Problem
I assumed what we were given, that $\displaystyle \frac{3+x}{3-x}$ is rational. More carefully:

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

Multiply by $\displaystyle q$

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

Multiply by $\displaystyle (3-x)$

$\displaystyle q(3+x)=p(3-x)$
• Apr 3rd 2013, 07:41 AM
xmathlover
Re: Rational Problem
Quote:

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

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

Multiply by $\displaystyle q$

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

Multiply by $\displaystyle (3-x)$

$\displaystyle q(3+x)=p(3-x)$

ok, excellent. Makes sense.