Prove that if $(2+x)/y$ is rational, then $x$ is rational or $y$ is irrational.

Question

Full Question: Let $ x,y$ be real numbers with $y$ not equal to $0$. If $(2+x)/y$ is rational, then $x$ is rational or $y $ is irrational.

I'm taking an if $p$ then $(q \lor r)$ and making it a $(p\wedge\neg q)$ then $r$. I've decided to make my "$y$ is irrational" the $q $ so that I'm only dealing with rational numbers. So my proof now becomes...

Let $x,y$ be real numbers and $y$ does not equal $0$. Assume $(2+x)/y$ is rational and $y$ is rational. Prove that $x $ is rational.

So, $y=a/b$ for some $a,b $ integers and $ b$ does not equal $0$. Then $(2+x)/y = b(2+x)/a$

I get stuck here. I need to show that $ x=a_n$ integer over and integer.

I also tried $(2+x)/y = c/d$ for some integers $c, d$ and $ d$ does not equal $0$. But frankly I'm stuck now.

Ugh... will I ever get proofs???

Answer 1

Let $t = \frac{2+x}{y}$ and solve for $x$ in terms of $y$ and $t$. Since you are assuming $y$ and $t$ are rational, you will be able to conclude that $x$ is rational from the formula you get.