Question

Describe how you know if an inequality is an “and” or an “or” problem. (use -3 < x+5 < 8). Also, how do you know if an absolute value question is an “and” or an “or” problem? Lastly, will you always have two solutions to an absolute value problem?

Answer 1

first of all let's simplify the example
-3 < x+5 < 8
subtract 5 from each part

-8 < x < 3

In this notation, a < x < b, a is usually less than b
and x is any value "between" them.
As a matter of fact, I recall one text actually calling this the "between format", since we can read it as:
x lies between a and b
So the word "and" is always implied here
x > a AND x < b

if we were to say :
x > a OR x < b, then the entire number line would have to be included, which of course would not be true.

As to something like
|x| < 5
we have x < 5 ?? x > -5

Again, if we place OR between them, we get all the values of x
It must be AND to get the values of x between -5 and +5

In general, for
|stuff| < a, we have stuff < a AND stuff > -a, which is the same as:
-a < stuff < a

for
|stuff| > a , we have
stuff < -a OR stuff > a

If a > 0 , I cannot think of a case when there would not be two solutions.
If a < 0 , something like |stuff| < a would not make any sense, and there would be no solution.