Question 8
Let $x \gt 1$ a non integer number. Prove that
$$
\left( \frac{x+\{x\}}{[x]} - \frac{[x]}{x+\{x\}} \right) + \left( \frac{x+[x]}{\{x\}} - \frac{\{x\}}{x+[x]} \right) \gt \frac{9}{2},
$$
where $[x]$ and $\{x\}$ represents the integer and the fractional part of $x$.
$$
\left( \frac{x+\{x\}}{[x]} - \frac{[x]}{x+\{x\}} \right) + \left( \frac{x+[x]}{\{x\}} - \frac{\{x\}}{x+[x]} \right) \gt \frac{9}{2},
$$
where $[x]$ and $\{x\}$ represents the integer and the fractional part of $x$.