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GCT Absolute Values - If y=|x+5|−|x−5|, then y can take

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by richachampion » Tue Nov 29, 2016 2:30 am
If y=|x+5|−|x−5|, then y can take how many integer values?

A. 5
B. 10
C. 11
D. 20
E. 21
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by richachampion » Tue Nov 29, 2016 2:30 am
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by GMATGuruNY » Tue Nov 29, 2016 5:44 am
I posted a solution here:
https://www.beatthegmat.com/tricky-modul ... 91962.html
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by [email protected] » Tue Nov 29, 2016 10:53 am
Hi richachampion,

GMAT questions are often built around a pattern of some kind (and sometimes more than one pattern). If you don't immediately see the pattern, then you might have to do a bit of work to figure out what the pattern is (the good news is that the work is rarely all that difficult, but you might have to do a lot of little calculations).

From the answer choices, we know that there are at least 5, but no more than 21, possible INTEGER values for Y. How hard would it be to find them all?

We can TEST VALUES here. Try TESTing X=0, X=1, X=2, etc. and note the results. What types of patterns are you noticing. What if you use negative values for X? What if you use non-integer values for X (such as 1/2, 3/2, etc.)?

On a certain level, the GMAT will reward you for being a 'worker', so don't be afraid of playing around with a prompt. You'd be amazed how quickly you can figure most of these questions out.

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by Matt@VeritasPrep » Thu Dec 08, 2016 9:02 pm
You could also think about this conceptually. |a - b| is the distance between a and b, and |a + b| is the distance between a and -b. So

|x + 5| = distance between x and -5

|x - 5| = distance between x and 5

We want the difference in those distances, or |x + 5| - |x - 5|. Since the difference between 5 and -5 is 10, the difference between |x + 5| and |x - 5| can be anything from -10 to 10, but nothing beyond that. (This isn't simply true for integers: we can get any real number from -10 to 10, but no real number outside of that range.)

To see why, imagine x at either end of the range. If x = -5, then we have 0 - 10, or -10. If x = 5, then we have 10 - 0, or 10. With 5 > x > -5, we can get any value between -10 and 10. If x > 5, then the difference simply becomes the difference between 5 and -5, or 10, and if x < -5, then the difference is simply -5 - 5, or -10, since the only relevant distance now is the distance from -5 to 5.

There are 21 integers in the set {-10, -9, ..., 0, ..., 9, 10}, so that's our pick.
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