If $x-\left\lfloor x \right\rfloor +\frac { 1 }{ x } -\left\lfloor \frac { 1 }{ x } \right\rfloor =1$, then $x$ is irrational.
I am thinking of using the contrapositive: If $x$ is rational, then $x-\left\lfloor x \right\rfloor +\frac { 1 }{ x } -\left\lfloor \frac { 1 }{ x } \right\rfloor \neq 1$
However, I don't know how I would approach this after that point. My first guess it to create a rational number, $\frac { a }{ b } $ such that $a,b\in\mathbb{Z}$, but I don't know how to manipulate it from there to get to the right hand side. I guess the floor function is throwing me off.
I would appreciate a hint/nudge in the right direction so that I can arrive at the solution by myself.