When must it be that |x+1| - |x-3| = -4?

Math works.

We've built it that way.

Having been doing math "professionally" for quite some time now (long enough that I'm not going to bother counting the years) I tend to forget just how well-built the mathematical machine really is.

This semester's already forcing me to reflect on the inner workings of some simple math concepts that I've sadly learned to take for granted.

Like absolute values: how marvelously they work! You really can depend on |a-b| = a-b to hold as long as a > b. Who'd'a thought? From the textbook I pulled the following absolute value problem with which the students are soon (today or Monday) to grapple: "find a formula for |x+1|-|x-3|, for any x."

Just now I was sitting here at my desk working through the problem so that I know we won't encounter any roadblocks during class...and I was marveling at how well one fares if one forgets everything else one knows about absolute values beyond the its basic defining formula. It's all just adding and subtracting, after all: if x < -1, you get |x+1| = |x-(-1)| = -1-x, and |x-3| = 3-x, so the difference you seek is -1-x-(3-x) = -4. It's marvelous!

This is what I'd hoped teaching this precalculus course would do for me: it's bringing me back into touch with the bits and bobbins at the base of mathematics that I've too long taken for granted. It's making me rethink why one does the things one does, and reflect on what one's really doing when one thinks one is doing something one's not really doing at all.

It's marvelous.