Re: How many values can the integer p=|x+3|-|x-3| assume?

I agree with Mike. 'Cannot be determined' is not a valid GMAT option. You probably wanted to say 'Infinite values'.

By the way, it's a very nice question. I think it has many subtle takeaways

p = |x+3| - |x-3|
First thing to realize here is that p needs to be an integer, not x.
Another thing, when you subtract two mods, the result takes the same value over a wide range.


____________________ -3 ___________x ___________3_________________________

<----------------------------

<-----------------------------------------------------------------


These are the two points -3 and 3 on the number line. We need to find
'the distance from -3' - 'the distance from 3' = p
i.e red line - green line.

Notice that the red line will cancel the part of green line to the left of -3 and hence red line - green line will always be -6 for all value to the left of -3.

Similarly, red line - green line will be 6 for all values to the right of 3.

The tricky values are the ones lying in between -3 and 3. When x = -3, we get p = -6. For some point between -3 and 3, we will get p = -5, -4, -3, -2.... 6. So there will be 13 values.
e.g. p = 5
If you move 0.5 to the right of -3, distance from -3 will be 0.5 and distance from 3 will be 5.5.
0.5 - 5.5 = -5

and so on...