Absolute Values in relation to TWR session

[ivanushk]

Hello Ron - I wanted to follow up on the Absolute Values TWR session held on February 18, 2010. The seminar was very helpful and to perfect my understanding of absolute value I just wanted to ask the following question:

How to best solve |QUANTITY|+|QUANTITY|=|QUANTITY| type question? There are quite a few questions of this type on the forums and I seem not to be able to do it quick enough or I simply get confused.

Example1 (Source Official Guide):

Is |3m - n| + |m - 2n| < |4m - 3n|?

(1) m > 0
(2) 2n < m

Example 2 (Source GMATPrep): If zy < xy < 0, is | x - z | + |x| = |z|?

(1) z < x
(2) y > 0

Example 3 (Source GMAT Club Tests) How many solutions will the equation |x+3| - |4-x| = |8+x| have?

A. 0
B. 1
C. 2
D. 3
E. 4

Please feel free to throw in a MGMAT example if you have one handy. I am not so much concerned about solving those particular problems, more than that, I want to be able to find a way to think about the question stems more clearly. What things should I keep in mind when I see this kind of structure? You have already mentioned that it is not useful to think about these questions in terms of theory or intervals, so I guess I am lost. And in this way it is a follow-up to the presentation you have made.

Would really appreciate your views on that Ron.

[RonPurewal]

If I said "don't think about intervals", I was almost certainly restricting that comment to equations with one absolute value on each side (i.e., |this| = |that|).
Those can be solved by just trying two cases (this = ±that), so there's little sense in expending the extra effort.

If you get something like #3 (which you won't on the official GMAT), then you're going to need to subdivide the number line into intervals.

[RonPurewal]

In any case, I'm going to frustrate you with a series of non-answers.
(:

"- #2 is, perhaps ironically, a faulty problem. (It's one of only 2 gmat prep problems I've seen that doesn't work correctly.)
The problem is that just the question prompt is already "sufficient". (If you know that zy < xy < 0 as stated, then the answer to the question is already narrowed to "yes".) You actually don't need either statement!
Overall I guess it's still technically fine (you'd pick D), but, yeah.

In any case, #2 is a bit weird, but it's a good place to test a couple of skills:
"- The inequality zy < xy < 0 breaks down into exactly two cases for where x, y, and z can lie on the number line. You should be able to spell out those cases.
"- If you understand that |this| is "how far away 'this' is from 0", and you also understand that |this - that| is "the distance between 'this' and 'that' ", then, once you've drawn those possibilities on the number line, you can bust through the rest of the problem in a hurry.
"- If you don't understand those things, then you can just test cases. You'll have plenty of time, since there are only 2 possible orderings of x/y/z.

Because you can very reasonably just test cases if you don't see the "distance" shortcut, there's nothing particularly obnoxious about this problem. You just have to be decisive"”"”if you don't IMMEDIATELY think of the shortcut, just start throwing numbers in there and see whether you can get a "yes" and a "no". (You can't get a "no".)

#3 is too technical"”"”and far too similar to the problems in an algebra textbook"”"”to show up on the GMAT. So, you can basically ignore it.

[RonPurewal]

"- And, is #1 really from the official guide? If so, what's the edition and problem number?
I can't find that problem in my OG's.

Although it's a bit more obnoxious than most, #1 is nothing you can't handle by simply testing cases.

If you are thinking "I need lots of obscure technical math knowledge to solve DS problems"... then it's almost certain that you aren't testing cases often enough.

[AbigailS228]

Just watched the TWR Absolute Values and I have a quick follow up question. I don't need clarification of the entire question I solved (got it from the 4 CAT), but the part I need help on looks like this...

"So you solve an equation and get:
x = -3y
If x and y are non-zero integers, we can deduce that they must have opposite signs: one positive, and the other negative. Therefore, this last equation could be rephrased as:
|x| = 3|y|"

Why did the negative go away for the 3? Is it because
|x| cannot = -|y|?

[RonPurewal]

Just think of |number| as the "size" or "magnitude" of the number. Basically, it represents how big the number is, irrespective of whether the number is positive or negative.

Those two equations are not equivalent; if you have |x| = 3|y|, then x = 3y and x = -3y are both possible. So, there must have been some further context, and/or some definite reason to turn something more specific into something less specific.

The point, though, is that x = -3y means that x is three times as "big" as y (and that the signs are opposite). By writing x = 3|y| we're paying attention only to the size/magnitude comparison, and not to the signs.