What is the value of y? (1) 3|x^2 - 4| = y - 2 (2) |3 - y| = 11

What is the value of y?

(1) 3|x2 – 4| = y – 2

(2) |3 – y| = 11


My question is, that in the Manhattan Gmat guide, they always ask you to find solutions of the absolute value equation, and then PLUG both values in to the original equation. In case if one of those values dont match up, then that solution can be eliminated. If in either of the equations , we would find a value which could not work, then could we safely count that as sufficiency for that statement, or would we have to still assume that both are possible so not sufficient?

Thanks

As for your question: in DS questions, when we are asked to determine value of an unknown, statement is sufficient if it gives single numerical value of this unknown. So if you have two solutions for a variable out of which one is not valid for some reason then you are left with only one solution and thus the statement is sufficient (of course if you are asked to find the value of this variable).

C for sure is the answer ,
Bunuel your explanations are simply awsome.
Thanks

Answer C, Bunuel's explaination has definitely helped me better understand absolute value questions.

Sorry I can't to figure out why y >= 2........

1) 3x^2 - 4 = y-2 and -3x^2 + 4 = y-2 and then ??'

thanks

Again solving intuitively, and reaching the answer faster.

Picking the statement 2 first as it is the simpler of the two.
(2) |3 – y| = 11

we get two values of Y; one +ve(14) and one -ve(-8). Not sufficient.

(1) 3|x^2 – 4| = y – 2
or, y= 3|x^2-4|+2 (RHS expression can never be -ve under any circumstance, since it involves a mod expression and '+2').
Therefore the value of Y is always +ve. But the statement in itself is not sufficient since we don't know the value of x.

Hence combining the two statements we can select the +ve value of Y from statement 2.

Answer is C.

Thanks.

thanks Bunuel... it was really helpful in understanding the concept of absolute values even more clerared now.. thank you

Hi,

Can I solve statement 1 like this:

3|x^2-4|=y-2

Now since this is an absolute value

I would 1st solve for x

x^2-4=0
x2=4
and x=+/-2
now if I substituent the value of x in the above expression
If x= +2
3|x^2-4|=y-2
3|(2)^2-4|=y-2
3|0|=y-2
therefore y=2

now if x=-2
3|x^2-4|=y-2
3|(-2)^2-4|=y-2
3|0|=y-2
and therefore y=2

In both the cases I will get the same value for Y.

Can someone please explain what is wrong with this approach.

We don't know whether x^2-4=0, thus all your further steps are based on that false assumption. If we knew that x^2-4=0, then x^2-4=0=y-2 --> y-2=0 --> y=2.

Also, you can notice that your approach is not correct from the fact that on the GMAT, two data sufficiency statements always provide TRUE information and these statements never contradict each other. From (2) we have that y is -8 or 14, and if from (1) you get that y is 2 it would mean that the statements clearly contradict.

Does this make sense?

No. When a DS question asks about the value of some variable, then the statement(s) is sufficient ONLY if you can get the single numerical value of this variable. So, the second statement is NOT sufficient because we can two possible values of y, not one.

Hope it's clear.

I'm still pretty confused after reading through all the posts in this thread. Bunuel EducationAisle VeritasKarishma Could you please help me understand this? Thanks!

What's wrong with this approach below? Is there any other way if we can't infer that (y-2) >= 0 (i.e. y >= 2) from stmt 1?

Stmt 1:

3 |x^2 -4| = y - 2
y = 3 |x^2 -4| + 2

Case 1:

x^2 - 4 >= 0 => x^2 >= 2 => x <= -2 or x >= 2
y = 3 (x^2 - 4) + 2 = 3x^2 -10

Case 2:

x^2 -4 < 0 => x^2 < 4 => -2 < x < 2
y = 3 (-x^2 + 4) + 2 = -3x^2 + 14

Value of y dependent on value of x => Not sufficient

Stmt 2:

|3 - y| = 11
|y - 3| = 11

Case 1:

y - 3 >= 0 => y >= 3
y - 3 = 11 => y = 14

Case 2:

y - 3 < 0 => y < 3
-y + 3 = 11 => y = -8

2 values of y => Not sufficient

Stmt 1 + Stmt 2:

Y can only be -8 or 14.

If x = 0, (i.e. value in case 2 from stmt 1):
y = -3x^2 + 14 = 14

We could still find some value of x from the 2 equations (though not sure which one we'll use) in stmt 1 that could give y = -8 since x doesn't have to be an integer. Shouldn't the answer be E? Let's say that I couldn't infer y >= 2 from stmt 1 and I want to follow my approach mentioned above. Can't seem to figure out what's wrong in that approach?

This is the whole point - there is no value of x for which y can be -8.

Case 1:

x^2 - 4 >= 0 => x^2 >= 2 => x <= -2 or x >= 2
y = 3 (x^2 - 4) + 2 = 3x^2 -10

This relation holds only when x is >= 2 or <= -2.
When x = 2, y = 3*2^2 - 10 = 2
When x = 3, y = 3*3^2 - 10 = 17
and similarly for -2, -3 etc...

Case 2:

x^2 -4 < 0 => x^2 < 4 => -2 < x < 2
y = 3 (-x^2 + 4) + 2 = -3x^2 + 14

This relation holds only when x is -2 < x < 2
When x = 2, y = -3*2^2 + 14 = 2
When x = 1, y = 11
When x = 0, y = 14
and similarly for -1 etc

y is always positive and its value is 2 or greater.

Or as Bunuel simply concludes in the solution: "we can conclude only that y≥2, as LHS is absolute value which is never negative, hence RHS also cannot be negative. "
So y - 2 ≥ 0
y ≥ 2

Target question: What is the value of y?

Statement 1: 3|x² – 4| = y – 2
Notice that there are INFINITELY MANY solutions to this equation. Just choose any value of x, and you will find a corresponding value of y.
Consider these two possible cases:
Case a: x = 1, which means y = 11 (once you plug x = 1 into the equation and then solve for y). In this case, the answer to the target question is y = 3
Case b: x = 2, which means y = 2. In this case, the answer to the target question is y = 2
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: |3 – y| = 11
Useful property: If |x| = k, then x = k or x = -k
So, EITHER 3 – y = 11 OR 3 – y = -11
Let's examine each case:
Case a: If 3 – y = 11, then y = -8. In this case, the answer to the target question is y = -8
Case b: If 3 – y = -11, then y = 14. In this case, the answer to the target question is y = 14
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 2 tells us that EITHER y = -8 OR y = 14
Great! There are only TWO possible values of y.
At this point, we need only determine whether each of these y-values will yield an actual solution for the statement 1 equation (3|x² – 4| = y – 2 )

Case a: y = -8, which means we get: 3|x² – 4| = (-8) – 2
Simplify: 3|x² – 4| = -10
So we get: |x² – 4| = -10/3
Since the absolute value of an expression is always greater than or equal to 0, we can conclude that this equation has ZERO solutions.

At this point, we can see that y = -8 is NOT a proper solution, which means it MUST be the case that y = 14, which means the COMBINED statements are sufficient.
However, let's see why by examining what happens when we test y = 14...

Case b: y = 14, which means we get: 3|x² – 4| = 14 – 2
Simplify: 3|x² – 4| = 12
Divide both sides by 3 to get: |x² – 4| = 4
From the above property, we can conclude that EITHER x² – 4 = 4 OR x² – 4 = -4
If x² – 4 = 4, then x² = 8, so x = √8 OR x = -√8
In other words, one possible solution is x = √8 and y = 12
Another possible solution is x = -√8 and y = 12

Likewise, if x² – 4 = -4, then x² = 0, which means x = 0
So another possible solution is x = 0 and y = 12

Notice that, for ALL THREE possible solutions, the answer to the target question is always the same: y = 12
So, it MUST be the case that y = 12
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

There isn't much given in the premise hence we can directly move to the statements.

Statement 1

(1) \(3|x^2 - 4| = y - 2\)

Clearly the value of y depends on the value of x, as x is not known this statement is insufficient.

However before we move away from the statement, we can conclude that y is a non negative number as the LHS will always be non negative once we move 2 from the RHS to the LHS

\(3|x^2 - 4| + 2= y \)

Thus y will be always greater than 0 (it will be greater than or equal to 2, but even if we are not able to figure that out, atleast we can conclude that y will be a non negative number)

As we have concluded, the statement is not sufficient. Hence eliminate A and D.

Statement 2

(2) |3 - y| = 11

y can be -8 or 14

Hence not sufficient to answer. Eliminate B

Combined

We know that y is non negative, hence only 14 holds true.

We have a unique value of y, Hence the answer is C.

KarishmaB

In S2, neither the positive case nor negative case we take for the absolute value satisfy their initial condition. Does this not mean that both are extraneous solutions? Why are we then taking either solutions into consideration when combining the statements?

Thanks in advance!

|3 - y| = 11

|y - 3| = 11
means distance of y from 3 is 11. This means that y is 14 or -8. We have 2 solutions.

If you use algebra instead,

When y >= 3, you get y - 3 = 11 so y = 14 (which is greater than 3)
and same for the second case. Both satisfy their original conditions.


If you got confused with |3 - y|, keep in mind that it is the same as |y - 3|. You can flip the sign inside the absolute value with 0 other changes.
Why? Because we know |x| = |-x|

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.