Well, perhaps I muddied the water for some of you. Sorry about that. In terms of skills, when you solve |u|<a where u is any expression and a is another expression [I thought it was for constants, but it turns out to always work], you can solve by finding the intersection (and) of the solutions to u<a and to -u<a, which we write u<a and -u<a. When you solve |u|>a, you find the union (or) of the solutions to u>a and to -u>a, which we write u>a or -u>a.
My explanation in the supplemental handout was to motivate why this works. After all, the course is not just about skills, but it is also about justification. The absolute value is a piecewise-defined function. That is, there are different rules depending on the value of the expression being worked with. The skill-based method that works for absolute value does not work for other piecewise-defined functions. But thinking about each of the "pieces" separately and joining them properly will always work.
Since the handout was a first edition, I'm curious where you found the biggest issues.