If $x$ and $y$ are irrational but $x + y$ is rational then $x - y$ is irrational

Question

Prove that if $x$ and $y$ are irrational but $x + y$ is rational then $x - y$ is irrational.

I can understand how it works in my head, I don't know how to prove it though.

Answer 1

Suppose $x$ and $y$ are irrational while $x+y$ is rational. Assume on the contrary $x-y$ was rational. Then $x = \frac{(x+y)+(x-y)}{2}$ is rational. Contradiction.

Answer 2

Hint:

$y = \frac{(x+y) - (x-y)}{2}$

Answer 3

Note that $$x-y = (x+y)-2y$$ which is a difference between a rational and an irrational number.

Answer 4

The set of rational numbers is closed under addition and subtraction, multiplication and division. This means $$ ∀a,b ∈ \Bbb Q, a + b = c ⇒ c∈\Bbb Q $$ Let z=x+y be rational, then z-2*y is rational if and only if 2*y is rational. If and only if y is irrational, then y+y = 2*y is also irrational. Combining the two, it follows that (x+y)-(y+y) is irrational if (x+y) is rational but y is irrational.