# Framework for Compound Inequalities

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?

• DRC's accepted answer already gets to the heart of this, but I view compound inequalities as and statements. This is true of other mathematical writing; e.g., if I see $\mathbb{R} \ni a \neq 0$, then I read it as: "The real numbers contain a number $a$ that is nonzero," i.e., $a$ is both a real number and nonzero. For $3<x<7$, the corresponding and statement holds, whereas such is not the case with $7>x<4$ (in which the intent is to express an or statement). – Benjamin Dickman Jan 27 '18 at 17:56

Just because you've defined a meaning for $a < b < c$ does not mean that any mishmash of other relational operators becomes equally well-defined as notation. Stick with what you've defined for a chained equality and don't permit arbitrary, nonstandard, off-track jaunts.
• +1: And I think the key is that $a<x<b$ is defined to mean "$x$ is between $a$ and $b$". It is then pointed out that this happens to be logically equivalent to saying "$a<x$ and $x<b$", but it is not explicitly defined as such. – Brendan W. Sullivan Jan 23 '18 at 18:55
• I agree with Sullivan's last piece of advice, but I figure the inquisitive student would ask why we should never mix inequality symbols. Other than wanting to avoid nonstandard formats, is there something objectionable about a thing like $7>x<4$? – Nick C Jan 23 '18 at 19:30
• @There is nothing fundamentally objectionable about it. When you define the composition of two relations you sometimes use notation like $a \, R \, b S \, c$ where $R$ and $S$ are arbitrary relations. – John Coleman Jan 23 '18 at 19:34