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Manhattan GMAT 700+ Challenge Problem - Absolute

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by Kevin » Mon Jun 19, 2006 7:19 am
Here is another example of a 700+ Data Sufficiency problem. Try to solve it (I'll post the solution next Monday).

Absolutely Fabulous
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

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
Kevin Fitzgerald
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Source: — Data Sufficiency |
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by dkiran01 » Tue Jun 20, 2006 10:32 pm
is it D..not very sure.. :wink:
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by lan0583 » Wed Jun 21, 2006 4:49 am
the answer is E...

A is not sufficient because we need the value of K
B is not sufficient because we need the value of X

Combining A & B. Since the value of X is less than 0, X could be -1, -2, and so on..

Let's try -1 for x
y=lx-kl - lx+kl
y=l-1-3l - l-1+3l
y=l-4l - l2l
y= 2

Let's try -2 for x
y=lx-kl - lx+kl
y=l-2-3l - l-2+3l
y=l-5l - l1l
y= 4

We can see that the value of y varies. So the answer is E.
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by cjas » Wed Jun 21, 2006 7:39 am
E

y = mod(x-k) - mod(x+k)

3 cases

k>x
y = -2x
k<x
y = -2k
k=x
y = -2x


1) x<0
gives y>2*mod(x) and y < -2k

2) k = 3
y = -2k = -6
y= -2x (where -2x would be > -6)

in either case we do not know x
even upon considering 1)and 2) together, we do not know x

so E
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oops i didnt read it carefully

by aim-wsc » Thu Jun 22, 2006 12:09 am
quite tough
but answer is E
oops i didnt read it carefully earlier.:shock:
ok if its max it has to be B.
Last edited by aim-wsc on Sun Jun 25, 2006 1:38 pm, edited 2 times in total.
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i think its statement 2 only wat u say?

by dudeasc » Sat Jun 24, 2006 8:50 am
i think its statement 2 only wat u say?

wats the real answer?
hi
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by jbsears » Sun Jun 25, 2006 9:01 am
I'm getting B. The maximum seems to max out at 6. Let us know, quite curious. These types of q's are the most difficult for me.
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Answer

by Kevin » Tue Jun 27, 2006 7:16 am
Answer
(1) INSUFFICIENT: Statement (1) is insufficient because y is unbounded when both x and k can vary. Therefore y has no definite maximum.

To show that y is unbounded, let's calculate y for a special sequence of (x, k) pairs. The sequence starts at (-2, 1) and doubles both values to get the next (x, k) pair in the sequence.

y1 = | -2 – 1 | – | -2 + 1 | = 3 – 1 = 2
y2 = | -4 – 2 | – | -4 + 2 | = 6 – 2 = 4
y3 = | -8 – 4 | – | -8 + 4 | = –12 + 4 = 8
etc.

In this sequence y doubles each time so it has no definite maximum, so statement (1) is insufficient.

(2) SUFFICIENT: Statement (2) says that k = 3, so y = | x – 3 | – | x + 3 |. Therefore to maximize y we must maximize | x – 3 | while simultaneously trying to minimize | x + 3 |. This state holds for very large negative x. Let's try two different large negative values for x and see what happens:

If x = -100 then:
y = |-100 – 3| – |-100 + 3|
y = 103 – 97 = 6

If x = -101 then:
y = |-101 – 3| – |-101 + 3|
y = 104 – 98 = 6

We see that the two expressions increase at the same rate, so their difference remains the same. As x decreases from 0, y increases until it reaches 6 when x = –3. As x decreases further, y remains at 6 which is its maximum value.

The correct answer is B.
Kevin Fitzgerald
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by krisraam » Thu Jul 23, 2009 6:17 pm
On a number line the points will be like this

-------- -k --------0----------k----------

For y to have maximum value x needs to be either equal to or less than -k

y is max when x<=-k

y = 2k

We need k value to find the answer for y

B is sufficient.

Thanks
Raama
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by ketkoag » Fri Jul 24, 2009 3:16 am
nice one kevin!!
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