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My understanding is that the following:

$5 < x < 10$ is read as "x is greater than 5 AND less than 10,"

whereas the solution to $| x + 2 | > 4$,

which is $x > 2, x < -6$, should be read as "x is greater than 2 OR less than -6"

Am I using this correctly? I'm wondering because while teaching some kids, I realized that Khan Academy doesn't seem to be using this correctly and I was doubting whether I was using them correctly.

Thanks in advance for the help.

amWhy
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J.Ko
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    I occasionally run into this problem. I assume (may be generously) that my students know what they are doing. The reason is that the interpretation depends on how you read it! A) The inequality $|x+2|>4$ is true, if $x>2$ AND it is true, if $x<-6$. (But it is false for all other $x$.) B) The inequality $|x+2|>4$ is equivalent to [($x>2$) OR ($x<-6$)], are both correct IMHO. When compressing the sentence to a line on an answer sheet which parts of the sentence will you leave out? – Jyrki Lahtonen Sep 13 '15 at 06:05
  • The notation $5<x<10$ is, by definition, short hand for $5<x\land x<10$, (see this). Finding the set of numbers that satisfy $|x+2|>4$ might be described in a multitude of ways, including disjunctions, but this is something that has nothing to do with inequalities alone. – Git Gud Sep 13 '15 at 06:05
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    The same with equations. The solutions to $x^2+x-2=0$ are $x=1$ AND $x=-2$, but as a single variable predicate $x^2+x-2=0$ is equivalent to ($x=-2$) OR ($x=1$). In the first way of writing AND is not really a logical conjunction. It is just used as when you are listing the alternatives. – Jyrki Lahtonen Sep 13 '15 at 06:11

3 Answers3

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Your first example is definitely correct. The chained inequality $5 < x < 10$ formally has the same meaning as 5 < x & x < 10, that is, "5 is less than x and x is less than 10", equivalent to "x is greater than 5 and less than 10", or in interval notation $(5, 10)$.

The second example is ambiguous, in that the piece of writing "x > 2, x < -6", is really malformed (undefined) mathematical writing; specifically, using the comma there isn't syntactically meaningful. That said, your English statement is correct that the solution to $|x + 2| > 4$ is certainly "x is greater than 2 or less than -6", which could be properly written as as "x > 2 or x < -6", or via the union of intervals $(-\infty, -6) \cup (2, \infty)$.

In short: An "or" statement really needs a proper symbol written out for it, unlike a chained inequality which is defined to imply the "and" connector.

(Edit) One of Khan's weaknesses is that he churns out so many videos he tends to be sloppy about details like this, and the video format isn't susceptible to editing afterward to fix errors like written text is.

Daniel R. Collins
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In the first case, there are two inequalities and the value of x satisfying both the inequalities is the intersection of the sets satisfying each inequality i.e.

$$x<5 \implies x\in (-\infty,5)$$ $$x>10 \implies x\in (10,\infty)$$ $$5<x<10 \implies x\in(-\infty,5)\cap(5,\infty)$$ That's why we use "AND".

In the second case, we use union i.e.

$$x+2>4 \implies x\in(2,\infty)$$ $$x+2<-4 \implies x\in(-\infty,-6)$$ $$|x+2|<4 \implies x\in (2,\infty) \cup (-\infty,-6)$$

That's why we use "OR"

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The first case is definitely "and" -- both must be true. The second case is "or" -- one of the conditions needs to hold.

What is the example from KA?

P. Michaels
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