Whether to use AND or OR in describing an inequality
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.
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$\begingroup$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.
$\endgroup$ $\begingroup$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"
$\endgroup$ $\begingroup$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?
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