There are two parts to this question.
1. I'm seeing the correct method to solve these types of inequalities as something to do with "transition points". I don't quite understand this method. How do we find the specific inequalities, and how do we work with the boundaries once we set them?
- I personally thought of a more systematic approach: Simplify one absolute value expression (convert to positive/negative cases), then, for each of those cases, split the second expression to get a total of 4 cases. Then, I string together the inequalities that I get after solving them all, via union or intersection as the inequality sign marks.
Edit: I've noticed the mistake in my process. In the Case 2 line, I forgot to switch the inequality sign. Still, I'm wondering why this isn't a viable strategy for solving these types of inequalities. Why is this method inferior to the method described in (1)?
|x-3|+|2x+5| > 6
|x-3| > 6-|2x+5|
Case 1: x-3 > 6-|2x+5|
|2x+5| > 9-x
Case 1a: 2x+5 > 9-x, 3x > 4, x > 4/3
Case 1b: 2x+5 < x-9, x < -14
Case 2: x-3 > |2x+5|-6
|2x+5| < x+3
Case 2a: 2x+5 < x+3, x < -2,
Case 2b: 2x+5 < -x-3, 3x < -8, x > -8/3
String together the 4 resulting inequalities:
((x < 4/3)U(x < -14))U((x < -2)U(x > -8/3))
So my answer would be x < -2 U x > -8/3
This is wrong. The answer would actually be x < -8/3 U x > -2.
What did I do wrong?