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
$$b<x<a$$
Or
$$a>x>b$$
Are these “wrong”? And if so, why?
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.
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.
