The Limit: LAND and GOR or What to Value Absolutely

This week I led an AP Calculus Summer Institute in Virginia. An instructive differential equation problem came up that took us back to Algebra 1 and the definition of absolute value.

You often see in Algebra 1 the solution of the equation |x | = 3 given in one step as

Now, this is correct, but solving inequalities this way presents major problems as soon as you get to things like

If you go with the plus and minus now you have to deal with

So students are told things like for Less then use and or (LAND)

and for Greater then use or or GOR

The problem is there is no attempt to understand the concept. The are three situations, =, <, >, and three different ways to approach them which need to be memorized.

The right way, which works for any, every, and all absolute value problems is to use the definition of absolute value:

If an expression is positive or zero, then its absolute value is the same expression;
if the expression is negative, the absolute value is its opposite.

Then every absolute value is broken into two cases:

Students should learn the definition in words and how to apply it. (And they should say "its opposite" rather than "its negative.") Then every situation is handled the same way: replace the absolute value with two expressions, or if you know which case the expression falls into, the nuse that part of the definition.

Now for the differential equation. The problem was to solve

with the initial condition that y(2)=0.

After separating the variables and antidifferentiating we arrive at

The question was whether to evaluate the constant immediately or to first continue on to

Substituting at this point gives

the correct solution.

While substituting the initial condition into

Apparently gives

This differs from the earlier attempt by the "+" sign. There is something wrong and there is no obvious way of realizing it.

This mistake is this: Near the initial condition point (2,0),