Prove that if $x$ is irrational, then $\frac{x+1}{x-1}$ is irrational

Question

I have to prove that if $x$ is irrational, then $$\frac{x+1}{x-1}$$ is irrational too,
but I'm not sure where to start from.

Could someone give me a clue?

Answer 1

Let $$\frac{x+1}{x-1}=r\in\mathbb Q$$ Thus, $r\neq1$, $$x=\frac{r+1}{r-1},$$ which is a contradiction.

Answer 2

If $y$ is irrational then $y+q,q\cdot y,\frac{q}{y}$ are irrational, if $q\in\mathbb{Q}\setminus\{0\}.$

We have $$\frac{x+1}{x-1}=\frac{(x-1)+2}{x-1}=1+\frac{2}{x-1}.$$ From $x$ irrational follows $x-1$ irrational, therefore is $\frac{2}{x-1}$ irrational and so $\frac{2}{x-1}+1$ is irrational.