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.
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.
Edit Didn't read properly
$\displaystyle \frac{3+x}{3-x}=\frac{p}{q} \rightarrow q(3+x)=p(3-x) \rightarrow 3q+3x=3p-px \rightarrow (3+p)x=3(p-q)$ so $\displaystyle x=\frac{3(p-q)}{3+p}$
Which conclusion we can draw from here?
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 $\displaystyle (3-x)$
$\displaystyle q(3+x)=p(3-x)$