# 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.

For example, here's the treatment in Sullivan, College Algebra, Sec. 1.5. Note that each permitted combination is explicitly described, and mixed-direction symbols are prohibited in the last line:

• +1 Exactly what I was thinking and was going to say when I read the question, but you beat me to it! (Actually, probably better than what I was going to say, now that I've read your answer again.) – Dave L Renfro Jan 23 '18 at 14:22
• +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
• @NickC: My stab at it would be (in line with the OP's example) it always reduces to some shorter, clearer, single relational expression. In the same sense that we have standardized (short) ways of writing fractions, polynomials, complex numbers, etc. – Daniel R. Collins Jan 23 '18 at 23:03

A statement such as $$a>b could be useful if $$a$$ and $$c$$ are not constants. I would interpret it as $$a>b\ \mathrm{and}\ b, i.e. $$b<\min(a, c)$$. In general, I'd interpret a statement of the form $$a★b‡c$$ as $$a★b\ \mathrm{and}\ b‡c$$. I'm pretty sure this is also how Python does it, for example.

"By default", all chained inequalities can be considered as illegal, because from an computer scientist's point of view and assuming that $$<$$ is left-associative:

$$a simplifies to $$\text{\{True or False\}} < c$$ which does not make sense ("type error").

To get around this, one defines the allowed combinations and their "expanded forms" as shown in Daniel's quote.