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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. So far as I understand, the same applies to compound inequalities; for example, everyone would regard $-3 to be well-defined, and it can be written "backwards" as $3>x>-3$.

When someone interprets $-3, upon reflection, it is understood that there is an implicit intersection behind the scenes, as it can be read out-loud as "$-3 and $x<3$." And when they interpret $3>x>-3$, it is the "backwards" version of $-3. Both are two different, compact ways of expressing {$ 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 $2 both refer to the same thing, namely {$ 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.

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    You are right about the implicit "intersection," though I think of it as logical "and" (usual notation $\land$). But then for the inequalities running the other way, intead of saying intersection, which would be right, you switch to **union**, which is not right.2012-11-30
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    Help me to understand why it is not right to read an implicit union.2012-11-30
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    In the case $3\gt x \gt -3$ you recognized that we are dealing with an intersection. Now the expression $-2\gt x\gt 2$ has exactly the same **shape** as the previous one, the numbers are a bit different. So if the first is an intersection, so is the second. It so happens that the intersection of the two sets $\{x: -2\gt x\}$ and $\{x: x\gt 2\}$ is empty, but that's irrelevant to how one interprets the logic of the situation.2012-11-30
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    Thanks for the quick reply. I suppose that when I consider something of the form a>x>b, I am first noting whether a>b or b>a. In the former case, I read an implicit intersection, and when b>a, an implicit union. I should have made that explicit.2012-11-30
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    It is intersection independently of the relationship between $a$ and $b$. Of course, if $a\le b$ it is not very interesting, but it still is an intersection.2012-11-30
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    There are other instances where we work with inequalities on a conditional basis. For example, if a>b, ac>bc only when c>0. But I can't think of how to explain why a>x>b is an intersection, regardless. I would be so grateful if you would help me connect the dots, because to be direct, treating it as an implicit union when b>a seems to work so well.2012-12-01
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    It looks as if I will not succeed. You are comfortable with interpreting $1\lt x\lt 5$ as an intersection, (or conjunction). You want the *meaning* of a formula $F(x, a,b)$ that involves the parameters $x$, $a$, and $b$ to depend on the *values* of these parameters. That is not the way things are ordinarily done, and it is unlikely, at least in the short run, that you will alter standard mathematical practice.2012-12-01
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    I really appreciate your comments, many thanks. I will continue to think about this.2012-12-01
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    One more question, please: I feel compelled to ask, regarding your generalization: is not a piecewise function something which changes what it means depending on the values of the parameters?2012-12-01
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    The *meaning* of the expression $f(x)=x^2$ if $x\lt a$, and $f(x)=x^3$ if $x\gt a$ is independent of $a$. It means square $x$ if $x\t a$, $\dots$. Of course the function changes. Similarly, the meaning of $a \lt x\lt b$ does not change, but the actual set being described does.2012-12-01
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    There is a typo in your last comment. If you please, fix it, so I can see the complete comment. Thanks in advance...2012-12-01
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    "It means square $x$ if $x\lt a$, and $\dots$.'We need to distinguish between a computer program, which (usually) does not change, and what it does given certain inputs, which of course is input-dependent.2012-12-01

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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 $-2>2$?

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    Regarding the example in your last paragraph, I agree it makes perfect sense if read in a pairwise fashion, but I would be reluctant to leap to the conclusion that it also means -2>2, because the direction is not consistent throughout. What I mean is, a>b>c>d certainly means a>d, for example. But the chain is broken when one of the symbols is reversed, no?2012-11-30
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    The fact that you have to ask whether the chain is broken when one of the symbols is reversed implies there is ambiguity involved. I will say that if there is to be a universal standard, then it should be read pairwise to avoid any ambiguity.2012-11-30
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I wouldn't say the system of inequalities $2 is "ill defined", but certainly it has no solutions: there is no number that is both bigger than $2$ and less than $-2$.

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    I saw your comment, but see that Alex already stated an example of what I was trying to convey, and did so much better than I! So, my answer has been deleted, accordingly.2012-11-30