1. ## 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.

2. ## Re: Rational Problem

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

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?

3. ## 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.

4. ## 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.

Originally Posted by xmathlover
Prove that if (3+x)/(3-x) is rational, then x is rational.
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.

5. ## Re: Rational Problem

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

Which conclusion we can draw from here?

6. ## Re: Rational Problem

Originally Posted by Ruun

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

7. ## 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 $(3-x)$

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

8. ## Re: Rational Problem

Originally Posted by Ruun
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.