I have been presenting compound inequalities like
$3 < x < 7$
as being a shorter way of saying
$3 < x$ and $x < 7$.
From this point of view, though, I end up having to admit that it is okay to write
$7 > x < 4$
even though it "simplifies" down to just $x < 4$.
Is there any better way to formalize compound inequalities that would rule out ever writing "$7 > x < 4$," or should I embrace the fact that maybe this kind of weird inequality is a good exercise for students to see strange things and unpack the definitions?