This seems like a stupid question . I just don’t understand why the algebra textbooks I see don’t really address this with students. I boy that I am tutoring brought it up and I was slightly embarrassed because I couldn’t give a quick and logical explanation for why he got it wrong.

Assume $a<b$. If $x<a$ or $x>b$, if we combine them we can write this as




Are these “wrong”? And if so, why?

  • $\begingroup$ I didn’t think so. I argued that he was correct and he should challenge her “-2”’s that she gave him. I also showed him why he was correct by rewriting as two separate inequalities. It was upsetting though that the teacher didn’t know he was right. $\endgroup$ Oct 30 '19 at 22:28
  • $\begingroup$ So i see conflicting solutions here with both answers. Why don't the textbooks ever address this? Shouldn't there be somewhere in the mathematical literature an indepth coverage of this? $\endgroup$ Oct 31 '19 at 3:52
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    $\begingroup$ I don't know what textbooks you've been using, but this seems to be a pretty standard algebra/precalculus topic covered in many textbooks. $\endgroup$ Oct 31 '19 at 20:37
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    $\begingroup$ I must say that this issue -- students not reading chained $a * b * c$ for relations $*$ as meaning $a * b$ and $b * c$ (and the transitivity inferences we can make) -- is a common frustration I deal with every day, even in a sophomore-level Discrete Math course for math & computing majors. I wish this were addressed/tested at some point in the curriculum, but I can't find time for it in my own courses. $\endgroup$ Nov 1 '19 at 3:46

The convention is that "$b<x<a$" means "$b$ is less than $x$ and $x$ is less than $a$." What you suggest is only wrong in that it goes against the shorthand that everyone (?) has already agreed on.

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    $\begingroup$ I would surmise the convention is because the number like is traditionally grows to the right. But it can grow to the left as well if you specify the arrowhead on the left instead of on the right side of the line. $\endgroup$
    – Rusty Core
    Oct 30 '19 at 22:02
  • $\begingroup$ I sort of see that argument, because the number line is naturally ordered left to right. It also seems that the fact that writing it as two separate inequalities gives the geometric impression that the graph of the solution set has two branches. But this kid’s teacher marked him wrong and I want him to dispute it because what he wrote is technically mathematically accurate. $\endgroup$ Oct 30 '19 at 22:19
  • $\begingroup$ The common core uses 'or'. corestandards.org/Math/Content/6/EE/B/8 However, I don't see anything wrong with the shorthand approach. a < x < b. I would put the smallest number on the left though to avoid confusion. $\endgroup$
    – matthew
    Oct 31 '19 at 16:55
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    $\begingroup$ @Eleven "I want him to dispute it because what he wrote is technically mathematically accurate." If you think so, you need to relook at your own mathematical education again. It is mathematically inaccurate in all senses of the phrase. $\endgroup$
    – YiFan
    Nov 7 '19 at 10:09

As a different view, I would say that this is "wrong" in the sense that we usually expect transitivity with (many of) our relations. E.g. if I write $a=b=c$ then usually we would say $a=c$ as well.

There are certainly counterexamples for more general relations, such as if $aRb$ means $a$ and $b$ share a hobby, so that $aRbRc$ wouldn't necessarily mean $a$ and $c$ share a hobby (maybe $b$ just has lots of hobbies).

But we don't use notation that has an unambiguous interpretation as an ordering for that. And $<$ definitely has that interpretation, so unless you want $$x<a<b<x<a<b<x\cdots$$ then the notation you suggest is too open to misinterpretation.

Interesting side note: this ambiguity is actually relied on in notation for outcomes of voting methods in the literature. In that case, $A\succ B\succ C\succ A$ has meaning when $A\succ B$ means $A$ defeats $B$ in a head-to-head vote - because we don't always get transitive election results. So sometimes our notation has to be careful depending upon who is reading it.


Although, because there is no $x$ that satisfies both inequalities simultaneously, it would not be unreasonable to give the expression $1 < x < 0$ the interpretation that $x$ is not between $1$ and $0$, something that could equally well be written as $x < 0$ or $x > 1$, this is definitely not standard practice and is surely more confusing for students than simply writing $x < 0$ or $x > 1$. Standard practice is to interpret the expression $1 < x < 0$ as an absurdity, because standard practice is to interpret $p < q < r$ as implying $p < r$, which is false if $p = 1$ and $r = 0$. Departing from standard practice causes problems. A student who learns nonstandard notations will have a hard time reading textbooks, learning from other teachers, and communicating about mathematics.

There are many possible notations. It is bad practice as a student and as a teacher to invent new notations when there are well-established standard notations. Sometimes it is necessary to improve on existing notation, but this needs real justification and should be done rarely. This is a pragmatic matter, because experience shows that (unnecessary) proliferation of notation generates far more confusion than it remedies.


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