Bunuel wrote:
If \(y=|x+5|-|x-5|\), then \(y\) can take how many integer values?
A. 5
B. 10
C. 11
D. 20
E. 21
Kudos for a correct solution.
There are a lot of good mathematical solutions here, but I'll be honest - even as somebody with a 790 GMAT, that's absolutely not what I'd do. I'm not confident in my ability to consistently draw correct graphs for complex absolute value functions like this one. Especially not within 2 minutes.
I would (and I did) actually just start plugging in numbers to see what happens. The biggest answer choice is 21, so there can't possibly be more than 21 values to count. That really isn't that many, if I work quickly. The numbers in the problem aren't tough either (just adding and subtracting 5). I might be able to find some patterns along the way that would speed it up as well. Plus,
if I come up with an easy strategy for a problem that I know will eventually work, that means I can safely spend a little extra time on the problem. I know that I'll figure it out, so I might spend as much as 3 minutes on this one if I really had to.
Okay, let's try it:
x = 0: y = 5 - 5 = 0
x = 1: y = 6 - 4 = 2
x = 2: y = 7 - 3 = 4
Interesting - I notice that there's a pattern there. To save time, I'm going to guess that the values keep increasing until I hit x = 5, because |x + 5| will keep getting bigger, and |x - 5| will keep getting smaller. I'm going to assume that x = 3, x = 4, and x = 5 give me the values 6, 8, and 10.
If I go bigger than x = 5, what happens? How about x = 6?
|6 + 5| - |6 - 5| = 10 - interesting - that's a value I already got.
|7 + 5| - |7 - 5| = 10 - same value again. Boring. I'm going to stop testing bigger numbers.
Right now, I've got on my paper: 0, 2, 4, 6, 8, 10.
Now let's test negative numbers, since we're dealing with absolute values.
x = -1: |-1 + 5| - |-1 - 5| = 4 - 6 = -2
x = -2: |-2 + 5| - |-2 - 5| = 3 - 7 = -4
Okay, I'm going to guess that it's symmetrical, and that I can get the values -2, -4, -6, -8, and -10. If I had extra time, I could check that.
I'm almost done, but let me think just a little bit more. Is there any way I could possibly get a different value? Maybe I should try some fractions, like x = 1/2... just to see what happens.
x = 1/2: |1/2 + 5| - |1/2 - 5| = 5.5 - 4.5 = 1
Interesting! That gave me a value I didn't have before. How about x = 1/3?
x = 1/3: |1/3 + 5| - |1/3 - 5| = 5.333 - 4.6666 = not an integer
I'm thinking right now that I have to work with halves, since when I subtract, it has to come out to an integer. So x = 1/2 gives me a value of 1. x = 3/2 gives a value of 3, x = 5/2 gives a value of 5... I can probably also get -1, -3, -5, ... just by dealing with negative numbers instead.
Finally, how big can these numbers get? What about x = 9/2?
x = 9/2: |9/2 + 5| - |9/2 - 5| = 9.5 - 0.5 = 9
x = 11/2: |11/2 + 5| - |11/2 - 5| = 10.5 - 0.5 = 10
x = 13/2: |13/2 + 5| - |13/2 - 5| = 11.5-1.5 = 10
Okay, it gets 'stuck' at 10 again.
Looks like my values are:
0, 2, 4, 6, 8, 10
-2, -4, -6, -8, -10
1, 3, 5, 7, 9
-1, -3, -5, -7, -9
For a total of
21 values.
I did a lot of writing and a lot of arithmetic, but I didn't take that much time on this one. That's because I went with a plan I felt pretty confident about right from the beginning. I took the 'easy way out' - that's what you want to do on the GMAT!
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