There is a lot of truth in many of the answers here, but they are all addressing this from a pure math standpoint. The logic of absolute value isn't hard, though - even if there are many examples here of weak prior knowledge that can cause problems.
Absolute value is hard for students to learn because it is almost always taught in extremely abstract and boring ways that just repel the vast majority of people from math while powerfully discouraging sense-making.
Here is an example from the ironically named "Math is Fun - Absolute Value" website
How fun and interesting is this? If we were to poll 1000 students, would 5 of them enjoy making meaning of this? Would even one see absolute value as useful?
I doubt it. The normal reaction almost assuredly would be something like "This is just more proof that math is just a bunch of weird rules and steps to obey. More rote crap to memorize." With this as a foundation, there is almost no way to make sense of any of the other very clear and logical points made in this thread.
In order to get students interested in the meaning of absolute value they need context in which absolute value thinking is useful.
You are a purchasing agent at a pharmacy. You need to order a large
amount of a generic cancer drug and three different factories offer to
sell it to you. You ask for a sample of their pills to check their
quality. Each pill should have exactly 700mg of active ingredient, but
no factory is perfect.
Factories A and B are brand new and not ready to produce enough pills
for you. They send you pills just to get your feedback. Factories C
and D are ready to sell to you now. They send a much larger sample of
pills for you to analyze.
Your mission: (1) Recommend to your manager which factory to buy
from. (2) Justify your answer. (3) Develop a written process for evaluating
quality of all future pill samples.
Do great work because your patients' lives depends on it!
Pill Quality Case
Instructions are on the first sheet. Data is on the second sheet. Reflect explicitly to consolidate is on the third sheet.
Here's a little snippet of data.
First you can have them estimate the quality of each factory. B is clearly better than A.
Now, you can guide them through calculating, totaling, and averaging errors for each factory, in which case a pill being, say, 200mg over and another pill being 200mg under exactly cancels. And whaddaya know: Factory A is perfect and Factory B sucks! Math beats common sense! According to the total and mean error, we should go with Factory A. Maaaaaaybe grandma will scream in unnecessarily excruciating pain and the baby will go into overdose, but according to mean error calculations, that is OK. lolz
Then, the more outspoken ethical sticklers will say that they, in fact, don't want to torture grandma and kill a baby. (BUT, LIKE, MEAN ERROR CALCULATIONS YO, IT'S JUST MATH DUDE, GOTTA LISTEN TO DEM NUMBERS.)
Hmm. Maybe 200mg under and 200mg over should not cancel out when added up, because an overdosing baby doesn't really cancel out screaming grandma. So we could if we were very careful, always make sure to do:
- Error = 700mg - (measured active ingredient) when the mg of active ingredient is less than 700mg
- Error = (mg of active ingredient) - 700mg when the mg of active ingredient is greater than 700
And that way, we get no negative numbers. No more errors that offset each other.
We just have to be really careful on every single calculation. Every. Single. Pill.
That's pretty easy for Factories A and B.
Who wants to do all those calculations for the hundreds of pills from Factories C and D? Hands up for volunteers! Hands up please!
If only there were an easier way to get rid of those damn negative errors... Wouldn't that be awesome?
At this point, you can introduce absolute value (and clicking/dragging formulae in spreadsheets) and every student knows why it's necessary and what it means. You could go a little deeper here into this case, formally relate the two error formulae to absolute value, then gradually abstract away from this and towards all the other posts in this thread. It can now become obvious why $\lvert x-700 \rvert=400$ must have two solutions: An absolute error of 400mg can be caused by overage or underage.
This stuff can all make sense because absolute value has meaning in their minds.
Anyways, if you have any suggested updates for the case or other thoughts on how I teach absolute value, please reply. :-)