Backwards Compound Inequalities? Some textbooks I've seen declare inequalities such as $-2>x>2$ to have no solution, or to be ill-defined, which I disagree with. I'm curious to know if anyone else thinks the same.
Inequalities can always be written two ways. For example, $x>2$ is the same as $2<x$. So far as I understand, the same applies to compound inequalities; for example, everyone would regard $-3<x<3$ to be well-defined, and it can be written "backwards" as $3>x>-3$.
When someone interprets $-3<x<3$, upon reflection, it is understood that there is an implicit intersection behind the scenes, as it can be read out-loud as "$-3<x$ and $x<3$." And when they interpret $3>x>-3$, it is the "backwards" version of $-3<x<3$. Both are two different, compact ways of expressing {$ x<3 $} $\cap$ {$ x>-3 $}.
So when I look at an inequality such as $-2>x>2$, I take it to mean there is an implicit union behind the scenes. In other words, $-2>x>2$ and $2<x<-2$ both refer to the same thing, namely {$ x<-2 $} $\cup$ {$ x>2 $}. Were I to read  $-2>x>2$ out-loud, I would read it as "$-2>x$ or $x>2$."
Am I crazy, or is there something wrong with this interpretation?
It seems to offer some advantages. For example, it makes the solution of certain absolute value inequalities very easy and natural.
 A: You can write inequalities any way you want if you think of the elements satisfying the inequality as members of a set combined with a truth table. There isn't any ambiguity in writing $-2>x>2$ or $-2>x<5$ as long as you read it left to right, or right to left in a PAIRWISE fashion: $-2>x$ and $x>2$ or $-2>x$ and $x<5$ respectively.. If you the write $\{x<-2\}\cap \{x>2\}$ you will realize that this intersection is the empty set, that there are no $x$ which satisfy the inequality. On the the other hand, if you write $-2>2$, this can be interpreted as a true or false statement, in this case being false. 
If you have something complicated like:
$-2>-3<5>2$, again it will be unambiguous if you read it left to right or right to left, in a pairwise fashion. In other words, $-2>-3$, $-3<5$, $5>2$. It WILL be ambigious otherwise, because for example, are you saying $-2>2$ as well as $-2>2$? 
A: I wouldn't say the system of inequalities $2<x<-2$ is "ill defined", but certainly it has no solutions: there is no number that is both bigger than $2$ and less than $-2$.
