Good question - the tricky thing with absolute values is that each has two solutions, so to do them algebraically you have to consider both. As an example, for statement 1:
abs[z] > abs[y]
means either that a positive value of z is greater than a positive value of y or that a negative value of z is less than a negative value of y.
The pure algebra is tough given that there are so many possibilities, so that's why plugging in numbers can be helpful so that you can make it more tangible. And I know that another of your posts this week mentioned that you felt plugging in numbers was time-consuming, so that's probably something you'll want to work on. For data sufficiency, I personally prefer to use algebra whenever possible, but plug in numbers primarily when:
1) My goal is to "prove insufficiency" by finding numbers that give me two different answers to the main question.
2) I know that a question asks more about the types of numbers (conceptual) than for specific numbers ("calculational") - for example, if the question stem asks "is x < 0", I know that they're asking whether x is negative, and I may just plug in numbers to see if x has to be negative, or if it could be positive or 0.
3) The terms are nebulous (like this one...too many possibilities with absolute variables, inequalities, and multiple variables) and I want to make the problem more concrete.
Practice using numbers to your advantage and I think you'll find it a useful part of your strategic arsenal.
In this case, plugging in numbers to prove insufficiency, you'll find that, even given both statements, you could have:
x = -1, y = 0, z = -2
abs[-1 -(-2)] = abs [-1 - 0] (1 = 1) ---> NO
x=-1, y = 0, z = 2
abs [-1 -(2)] > abs [-1 - 0] (3 > 2) ----> YES
Therefore, the correct answer is E.