In Humphreys' book Introduction to Lie algebras and Representation Theory(3rd printing, p.62), the Diophantine inequality $$\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$$ appeared while classifying all Coxeter graphs of irreducible root systems. Under the assumption $p \geq q \geq r \geq 2 $, the possible triples turn out to be $(p,2,2), (3,3,2),(4,3,2),(5,3,2)$, which correspond to the Coxeter graph $\text{D}_{p+2}, \text{E}_6, \text{E}_7, \text{E}_8$, respectively.
It seems quite straightforward to solve the inequality in positive integers, but Humphreys said that
This inequality, by the way, has a long mathematical history.
without further comment. So I want to know if this Diophantine inequality has any important significance, or if it has appeared in other context (maybe relevant history) before.
I don't know what Humphreys exactly had in mind, but here is one piece of information which might be useful.
This inequality has some relationship with Diophantine equations of the form $Ax^p + By^q = Cz^r$, where $A, B, C, p, q, r$ are fixed integers and we look for integer solutions $(x, y, z)$.
See e.g. this answer, which gives details and references.