<s>[INST] 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. 
 [/INST] 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.
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