Determining whether three values can be the lengths of a triangle or not 
Let $x,y,z$ be three distinct real positive numbers, Determine whether or not the three real numbers $$ \left| \frac{x}{y} - \frac{y}{x}\right| ,\left| \frac{y}{z} - \frac{z}{y}\right |, \left| \frac{z}{x} - \frac{x}{z}\right| $$ can be the lengths of the sides of a triangle.

Hello! I hope everybody is doing well. Can anybody help with the above problem?
I am not able to approach this problem.

WLOG $x > y > z$. Hence $\frac{x}{y} > 1, \frac{y}{z} > 1 $ and $\frac{x}{z} > 1$. Let $a=\frac{x}{y},b=\frac{y}{z},ab=\frac{x}{z}$ and hence we want to show whether $a - \frac{1}{a}$, $b - \frac{1}{b}$ and $ab - \frac{1}{ab}$ can form the sides of a triangle. If I could somehow know the smallest or the greatest side and could work on it using the Triangle Inequality we could be done.

Any help would be appreciated. Thanks. 

The hint.
Let $x>y>z.$
Thus, show that
$$\frac{x^2-z^2}{xz}>\frac{x^2-y^2}{xy}$$ and
$$\frac{x^2-z^2}{xz}>\frac{y^2-z^2}{yz},$$ which says that it's enough to check
$$\frac{x^2-y^2}{xy}+\frac{y^2-z^2}{yz}>\frac{x^2-z^2}{xz}$$ or
$$\sum_{cyc}\frac{x^2-y^2}{xy}>0$$ or
$$\sum_{cyc}(x^2z-x^2y)>0$$ or
$$(x-y)(y-z)(z-x)>0,$$ which is wrong.
