Absolute Value ..MGMAT CAT Question

[aps_asks]

Hi Instructors ,

I encountered the following Question in one of my MGMAT CAT .


What is the value of y?

(1) 3|x2 - 4| = y - 2

(2) |3 - y| = 11

MGMAT Explanation :

(1) INSUFFICIENT: Since this equation contains two variables, we cannot determine the value of y. We can, however, note that the absolute value expression |x2 - 4| must be greater than or equal to 0. Therefore, 3|x2 - 4| must be greater than or equal to 0, which in turn means that y - 2 must be greater than or equal to 0. If y - 2 > 0, then y > 2.

(2) INSUFFICIENT: To solve this equation for y, we must consider both the positive and negative values of the absolute value expression:

If 3 - y > 0, then 3 - y = 11
y = -8

If 3 - y < 0, then 3 - y = -11
y = 14

Since there are two possible values for y, this statement is insufficient.

(1) AND (2) SUFFICIENT: Statement (1) tells us that y is greater than or equal to 2, and statement (2) tells us that y = -8 or 14. Of the two possible values, only 14 is greater than or equal to 2. Therefore, the two statements together tell us that y must equal 14.

The correct answer is C.

My Question :

I have a doubt with regard to the Statement A )
When to consider the positive and negative cases of the absolute value and when to consider MGMAT Approach for statement 1 ) ?

I mean that if i consider the +v e and -ve cases of the Statement A ), then
1) 3( x2-4) = y-2
3x2 -12 = y-2
3x2 -10 = y

2) -3(x2-4) = y-2
-3x2 + 4 = y-2
y = 6 - 3x2

y is insufficient

Should i consider the above approach or the one as per the MGMAT explanation?

But if i consider the Positive , Negitive Cases approach...Then it is difficult to come to an answer

Please let me know your comments.

[jnelson0612]

Hi aps_asks,
That is a good question. For me, personally, I try to get to the right answer while doing as little work as possible, and taking as little time as possible. Since running those different positive/negative scenarios takes time and work, I would only do it if I have to. Here's how I would think through this problem:

Start with Statement 2. There is only one variable and it does make sense to consider the two scenarios. I find that y is either -8 or 14. Not sufficient.

Statement 1: this is scary! The variable inside the absolute value sign is squared, of all things. Ugh, I really don't want to have to look at all the scenarios for this. Fortunately, I have two variables, x and y. I can't possibly know what y is when I don't know what x is. Not sufficient, and I haven't done any work.

Together: y is either -8 or 14. Let's use logic and reason before we do any math. The left side HAS to be zero or positive. That means that y cannot be -8. y has to be 14. Sufficient!

Whenever you find that you start having to do a crazy amount of work, it's time to look for a shortcut. I hope that this helps! :-)

[yogesh.kc20]

I could guess the answer correctly by the approach mentioned above but I still don't understand why - "3|x2 - 4| must be greater than or equal to 0." (as it mentioned in the explanation)

Explanation says:
Since this equation contains two variables, we cannot determine the value of y. We can, however, note that the absolute value expression |x2 - 4| must be greater than or equal to 0. Therefore, 3|x2 - 4| must be greater than or equal to 0, which in turn means that y - 2 must be greater than or equal to 0. If y - 2 > 0, then y > 2.

[yogesh.kc20]

Sorry, my bad. It is talking about "absolute value expression |x2 - 4| must be greater than or equal to 0" and not the value of (x2-4)

[RonPurewal]

Sorry, my bad. It is talking about "absolute value expression |x2 - 4| must be greater than or equal to 0" and not the value of (x2-4)

Yeah.
|blah blah blah| -- where "blah blah blah" is anything in the entire mathematical world -- is always 0 or greater.