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Source: — Data Sufficiency |
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by gmatmachoman » Fri Dec 17, 2010 3:57 am
st1 :
X= - |X|

This means that X will ALWAYS be a negative number irrespective the sign.

ex :
case 1 : x= 3
|X| = |3| = 3
Since X= - |X| ------> X = - |3| = -3

case 2

X = -3
|X| = | -3| = 3
Since X= - |X| ------> X = - |-3| = -3

But the prompt asks for a "specific value" ....so st 1 in insufficient

St 2 is sufficient

Pick B
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by goyalsau » Fri Dec 17, 2010 4:37 am
gmatmachoman wrote:st1 :
X= - |X|

This means that X will ALWAYS be a negative number irrespective the sign.

ex :
case 1 : x= 3
|X| = |3| = 3
Since X= - |X| ------> X = - |3| = -3
I agree with you answer, and i have some where in the forum that when x = - | x |

X is always negative,

but in your example you are considering a +ve value of x

Don't you think your both statements contradict.
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by Rahul@gurome » Fri Dec 17, 2010 8:07 am
x = -|x|
x = -|x| implies |x| = -x, which is nothing but the definition of absolute value of x whenever x < 0. Thus this statement tells us x is negative.
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by fskilnik@GMATH » Fri Dec 17, 2010 8:12 am
goyalsau wrote:This means that X will ALWAYS be a negative number irrespective the sign.
Hi goyalsau!

Watch out, |x| = -x does NOT mean x is negative... |x| = -x means that x is NULL OR NEGATIVE.

goyalsau wrote: Don't you think your both statements contradict.
I´ve already explained lengthly in another BTG post some WRONG understanding on this matter (made by Experts, too). To put it shortly: the fact that one of the statements ALONE is sufficient to answer the question asked (in this case, sttm 2) makes the Data Sufficiency problem to be FINISHED BEFORE the need to use both statements together. In THIS scenario, statements (1) and (2) does NOT need to be "coherent", "harmonious", whatever.

(It´s not a matter of opinion here. It´s a matter of mathematical logic.)

Best Regards,
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by fskilnik@GMATH » Fri Dec 17, 2010 8:15 am
Rahul@gurome wrote:|x| = -x, which is nothing but the definition of absolute value of x whenever x < 0. Thus this statement tells us x is negative.
The usual definition of |x| really is:

|x| = x if x is positive or null
|x| = -x if x is negative

BUT this is only because it would be less "elegant" to define as:

|x| = x if x is positive
|x| = -x if x is negative or null

OR EVEN

|x| = x if x is positive or null
|x| = -x if x is negative or null

In other words, 0 is the only real number that coincides with its opposite (0 = -0), therefore we have to take special care when dealing with 0 when modulus is concerned.

Regards,
Fabio.

P.S.: it´s not a matter of opinion, here. If you try x=0 in the statement |x| = -x you will see that it makes the statement TRUE. In other words, x=0 satisfies statement (1). PERIOD.
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by pradeepkaushal9518 » Fri Dec 17, 2010 8:22 am
/x/ is always positive hence -/x/ will be always negative

am i right experts?
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by fskilnik@GMATH » Fri Dec 17, 2010 8:36 am
pradeepkaushal9518 wrote:/x/ is always positive hence -/x/ will be always negative

am i right experts?
NO!

|x| is always NULL OR POSITIVE therefore -|x| is always NULL or NEGATIVE!

Important:

01. |x| will be zero if and only if x is zero.
02. -|x| will be zero if and only if x is zero.

Got it? :)
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by pradeepkaushal9518 » Fri Dec 17, 2010 8:55 am
thats correct thanks sir
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by fskilnik@GMATH » Fri Dec 17, 2010 9:06 am
pradeepkaushal9518 wrote:thats correct thanks sir
My pleasure, pradeepkaushal9518.

Let me ask you a question just for both of us be sure we understood each other, may I?

The question is: If in a certain question the examiner asks whether sqrt(-x*|x|) < 9 (where sqrt = square root), please tell me:

01. What is implicitly given in relation to x? (I mean: is x certainly positive? Certainly negative? etc)

02. What are the values of x that satisfies (the implicit condition on item 01. above AND) the inequality given?

(In other words, if you would "simplifly the question asked", you would get this: "Is A < x <= B ?" and I want to know the values of A and B...)

I am waiting your answers, ok?!

(Don´t be stressed. They are HARD, but if you get them right OR if you understand my after-your-response explanation, THEN you will have seen a "bigger picture" and I will consider "the case closed"...ok?)

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by fskilnik@GMATH » Fri Dec 17, 2010 10:18 am
Well, I don´t want to make you (pradeepkaushal9518) unconfortable, nor anyone, so let me show you (all) what I have in mind!

Question 01. From the fact that the square root is a radical with an even index (2), that means the radicand must be positive or null, therefore we have to assume that -x*|x| >=0 and that means x is NULL or NEGATIVE.

Conclusion: the answer for question 01 is "x is implicitly assumed to be negative or zero."

More detailed explanation is below:

-------------------------------------------------------------
Please note that

> if x is null, -x*|x| = 0 and this is ok to be "under the square root" (that is, to be the radicand of a radical with even index) ;

> if x is negative, -x is positive and |x| is also positive, therefore -x*|x| is positive and, again, this is ok to be "under the square root".

> If x is positive, then |x| is positive BUT -x is negative, therefore -x*|x| is negative and this is a not-allowed scenario.
-------------------------------------------------------------


Question 02: assuming (as you should) that x is negative or null (as explained in question 01´s answer, above), we know that |x| = -x, therefore:

sqrt(-x*|x|) = sqrt (-x*(-x)) = sqrt (x^2) = |x| = -x, therefore -x < 9 implies x > -9 !!!

Conclusion: we know (from question 01) that x < = 0 and (from question 01 and 02) that x > -9, therefore we have:

-9 < x <= 0, and hence I can conclude the values of A and B I´ve asked for!!


Please feel free (all of you) to ask me if something is not 100% understood, ok?

Regards,
Fabio.
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by goyalsau » Fri Dec 17, 2010 7:45 pm
Thanks Fabio,

Great Question and Even a much better explanation,

x = - | x | { When x is negative and Zero }

I must say Zero is creating a lot of problems............
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