DS Absolute value equation: What is the value of y?

[goenkavivek]

What is the value of y?

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

(2) |3 - y| = 11

May I please request for your help with this. Thanks a million!

This is what I tried to do:

(1) INSUF because has another variable X

(2) Within the absolute value bracket, the sign can be changed, thus the eqn becomes
|y - 3| = 11

Case 1: Y-3>0

Y-3=11
Y=14

Case 2:
Y-3<0
Y-3=-14
Y= -11

Insuff

So, I chose (E)

How wrong is wrong?

[RonPurewal]

first, you made a calculation mistake -- i've fixed it in blue:

What is the value of y?

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

(2) |3 - y| = 11

May I please request for your help with this. Thanks a million!

This is what I tried to do:

(1) INSUF because has another variable X

(2) Within the absolute value bracket, the sign can be changed, thus the eqn becomes
|y - 3| = 11

Case 1: Y-3>0

Y-3=11
Y=14

Case 2:
Y-3<0
Y-3=-11
Y= -8

Insuff

So, I chose (E)

How wrong is wrong?

you didn't think about combining the two statements -- apparently, you went straight to (e) from figuring out that the two individual statements weren't sufficient.
BUT
from statement (1), notice that y = 3(absolute value) + 2. this means that y is AT LEAST 0 + 2 = 2, because absolute values can't be negative.
in combination with statement (2), this rules out the case y = -8, and so you only have y = 14 left.

therefore (c)

[aps_asks]

Hi Ron ,

How to solve Statement 1) in an algebraic way and not consider
that |x2 - 4| is positive.

Please let me know your comments.

[RonPurewal]

Hi Ron ,

How to solve Statement 1) in an algebraic way and not consider
that |x2 - 4| is positive.

Please let me know your comments.

you can't!

basically, what you are asking me amounts to the following: "how can i use this absolute value, while ignoring the most salient property of absolute value?"
you have to consider the fact that absolute values are positive; that is the primary defining characteristic of absolute values, so you're not going to get around it with any neat little tricks.

it's possible to use algebra to split this equation up into two or more sub-equations, but to do so would be ultimately pointless -- it would be substantially more work, and you would still need to realize that the absolute-value expression is positive in order to do it.

[aps_asks]

Thanks Ron !

[RonPurewal]

Thanks Ron !

sure.

[gewoonkevinn]

I still do not quite understand this problem. The fact is, is that the term inside the absolute value brackets can either be positive or negative, although that is what I understand. And that is why Y in statement 2 has two answers. Can you please explain it to me in more detail concerning the fact that it is always positive!

[tim]

i don't quite understand your question. are you asking why the absolute value of something is always positive?

[abdul_tt]

What is the value of y?

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

(2) |3 - y| = 11

Starting with statement 2,

Case 1,
+(3-y)= 11
y= -8
Case 2,
-(3-y)=11
y=14

Let us put y=-8 in equation (1) and analyze what happens,

3|x^2 - 4| = -10

or |x^2 - 4| = -10/3

NOW NO MATTER WHAT VALUE OF 'X' ONE PUTS INSIDE THE ABSOLUTE VALUE PARENTHISIS , THE RESULT WILL NEVER BE EQUAL TO - 10/3

THIS IS WHAT THE HOLY CANON OF ABSOLUTE VALUE IS.


Therefore 'Y' has to be 14.
I believe this is a classic problem to understand 'absolute value' concept.

[RonPurewal]

abdul_tt,
yes, that works -- as long as you've first determined that statement (1) is insufficient.
(although you may have considered this first, you didn't actually mention it in your solution; you just used statement (2) and then went directly to combining the two statements.)

[matt.mat6]

Hi Ron, I have a doubt becouse the 1st equation is indeed two of them

1)3x^2-12=y-2, it follows that 3x^2-10=y
2)-3x^+12=y-2, it follows that -3x^2+14=y

We can not find y with these equations?

3x^2-10=-3x^2+14
x^2=4

So 3[4-4]=y-2
y=2

So A is my answer. What I did wrong?
Thanks

[tim]

The problem here is that 1 and 2 are completely different and incompatible cases that you have equated. It would be the equivalent of doing the following:

|x| = y

1) x = y
2) -x = y

x = -x
2x = 0
x = 0
y = x
y = 0

Do you see that this result totally does not follow from the statement |x| = y?

[matt.mat6]

Thanks Tim!!

The problem here is that 1 and 2 are completely different and incompatible cases that you have equated. It would be the equivalent of doing the following:

|x| = y

1) x = y
2) -x = y

x = -x
2x = 0
x = 0
y = x
y = 0

Do you see that this result totally does not follow from the statement |x| = y?

[jnelson0612]

Great!

[SaurabhG512]

I am confused to understand when do we really consider an absolute value to be "absolute values are always positive" and "absolute values have two versions - positive as we as negative"?

Can someone please explain in context to this question?