Inequality

[amardeep.singh]

If x and y are non-zero integers and |x| + |y| = 32, what is xy?

(1) -4x - 12y = 0

(2) |x| - |y| = 16
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

For statement 1 : manhattan explanation is

1) SUFFICIENT: Statement (1) can be rephrased as follows:

-4x - 12y = 0
-4x = 12y
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|

We don’t know whether x or y is negative, but we do know that they have the opposite signs. Converting both variables to absolute value cancels the negative sign in the expression x = -3y.

We are left with two equations and two unknowns, where the unknowns are |x| and |y|:

|x| + |y| = 32
|x| - 3|y| = 0

Subtracting the second equation from the first yields

4|y| = 32
|y| = 8

Substituting 8 for |y| in the original equation, we can easily determine that |x| = 24. Because we know that one of either x or y is negative and the other positive, xy must be the negative product of |x| and |y|, or -8(24) = -192.

How ever i do not get this statement that

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|

I do understant that x and y have opposite signs. So keeping this fact if i put a mod value then one of the solution could be
x = 3y.
An explanation about this point would be really helpful.

[tim]

i assume this problem comes from our CAT exams; please be sure to put your questions in the appropriate forum in the future..

if x = -3y, they HAVE to be opposite signs. if they are both positive, then the left side of the equation is positive and the right side is negative. if they are both negative, the left side is negative and the right side is positive. so they MUST have opposite signs..

[kouranjelika]

Hi Tim,

But how do we jump from x=-3y to |x|=-3|y|?

Thanks!

[jnelson0612]

Hi Tim,

But how do we jump from x=-3y to |x|=-3|y|?

Thanks!

Have you tried plugging some values in to show yourself why that works? If not, definitely try some!

[kouranjelika]

Ok, sure I understand how we can prove it via number plugging but is this a property of any equation then?
Meaning can we re-phrase any equation to have the variables be in absolute value form and signs & anything else would remain outside?

[RonPurewal]

Ok, sure I understand how we can prove it via number plugging but is this a property of any equation then?
Meaning can we re-phrase any equation to have the variables be in absolute value form and signs & anything else would remain outside?

* No.

* In 99.99% of situations, there would be no reason to try this anyway. It's only done here because absolute values are an integral part of the question already.
In other words, you would never, ever do this sort of thing at random.

* More importantly, this isn't really a "re-phrasing", because |x| = 3|y| doesn't mean the same thing as x = -3y.
|x| = 3|y| means x = ±3y. I.e., it means that the magnitude of x is 3 times the magnitude of y, but we don't know the sign of either one.

[RonPurewal]

Also note that |x| = -3|y| is impossible unless both x and y are zero. (If you don't see why, then plug in a couple of nonzero numbers and watch what happens.)

[kouranjelika]

Yea this I'm very clear on. As you mentioned in one if your lectures, "we could mine this for some serious gold," if it had equations on each side with on variable on either side and we could just let both of those equal zero and find values for the variables (given its linear or a perfect square quadratic or something like that).

But I still don't really follow how this leap is made here. I mean in retrospect I plug in numbers into the absolute value equation and it works but I would never think to do this myself. In fact, I haven't looked at this since the weekend and before reading your response I just did the question again and it still didn't occur to me and I picked C. Can you explain how to arrive at this logic properly so I don't make such a mistake again or rather think of doing it this way in the future.

Thank you

[RonPurewal]

What, specifically, are you asking about?
Which steps, for which statement(s)?

[kouranjelika]

1) SUFFICIENT: Statement (1) can be rephrased as follows:

-4x - 12y = 0
-4x = 12y
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|

We don’t know whether x or y is negative, but we do know that they have the opposite signs. Converting both variables to absolute value cancels the negative sign in the expression x = -3y.

This part here. I mean this is quite a leap in my opinion, I get that it works, but how would you think of even doing that.

More on that, can stat (1) be proven sufficient without doing this operation?

[RonPurewal]

I actually wouldn’t make that leap, either. I’d just do the "brute force" / "grind" method.

Specifically, |x| is either x or -x. Likewise, |y| is either y or -y. So, the second equation must be equivalent to one of the following four:
x + y = 32
x - y = 32
-x + y = 32
-x - y = 32

That gives four systems to solve: each of these four, paired with the first equation. Annoying, sure, but not that time-consuming. (Most people badly overestimate the time they’ll need for a task like this, especially since the solution process is essentially the same for all four systems.)

Then, for each system, plug the solution back into the original (with the absolute values), and see whether it actually works. If it doesn’t, drop it; it’s a fake solution. If it does, keep it.

If you only get one "survivor", then, sufficient.

[kouranjelika]

Got it and found one (well actually two, but it's just the same values for x & y with reversed signs, which yield the same product in the end).

And Statement 2 is insufficient because we wouldn't know the specific signs of each number, could be the same or could be different, yes?

Thanks again! I am in fact going to skip my advanced hmw Section tomorrow to come to your study hall. Believe me, I am all over the recordings (actually listening to one right now as I write this), but the live experience is much better. I get crazy when I have a question that wasn't asked... :)

[RonPurewal]

And Statement 2 is insufficient because we wouldn't know the specific signs of each number, could be the same or could be different, yes?

Yes.

NB"”According to the timestamp, you wrote "I'm going to come to your study hall tomorrow" sometime on Sunday. That's four days before the study hall, not 1 day.

[kouranjelika]

Yea, I meant Thursday. I'm getting delirious from all this studying :)

See you TONIGHT!

[jnelson0612]

Yea, I meant Thursday. I'm getting delirious from all this studying :)

See you TONIGHT!

:-)