Data Sufficiency

Vemuri wrote:If k is a positive constant and y = |x - k| - |x + k|, what is the maximum value of y?
(1) x < 0
(2) k = 3

OA is B

I think mals might have thought there was a + sign, rather than a - sign, in the expression in the question (there are a few questions in official materials which are similar to the above, except with a plus sign instead of a minus between the two absolute value terms).

I don't much care for the wording of the question; it doesn't really make logical sense as a DS question, even if I can tell what the intention is of the question designer. After all, what kind of information would be sufficient to find the 'maximum value of y' in general? If you saw a question that asked 'what is the maximum value of y?' and one Statement read "y > 0", would that be sufficient, or insufficient? That's unclear.

In any case, looking at the expression in the question:

y = |x - k| - |x + k|

y = |x - k| - |x - (-k)|

then y is just the difference between the distance from x to k, and the distance from x to -k. Drawing the number line:

---------(-k)-----------------k----------------

let's consider the three 'zones' in which x might lie (to the right of k, between k and -k, or less than -k).

If x is greater than k, we have:

---------(-k)-----------------k-----------x-----

So y will be negative if x is to the right of k, since then the distance from x to k is less than the distance to -k.

If x is between -k and k, we have:

---------(-k)---x-------------k----------------

Notice then that the distance from x to k is less than the distance from -k to k, so is less than 2k. So y must be less than 2k in this case.

On the other hand, if x is less than -k, we have

-x--------(-k)----------------k----------------

Then, if we subtract the distance from x to -k from the distance from x to k, we're left with the distance from -k to k, which will always be 2k no matter what the value of x. That's certain to be larger than the values of y we found in the other two cases above.

So we have the largest value of y when x < -k, and the largest value is 2k. So if k=3, y will never be larger than 6. So for that reason, I imagine the question designer intends the answer to be B, though as I said above, I don't think the format makes logical sense. I'm curious about the source.