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If x/|x|, which of the following must be true for all

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If x/|x|, which of the following must be true for all  [#permalink]

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New post Updated on: 09 Jul 2013, 08:56
1
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A
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D
E

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If \(\frac{x}{|x|} \lt x\), which of the following must be true about \(x\)? (\(x \ne 0\))

A. \(x\gt 2\)
B. \(x \in (-1,0) \cup (1,\infty)\)
C. \(|x| \lt 1\)
D. \(|x| = 1\)
E. \(|x|^2 \gt 1\)

M24

Originally posted by praveenvino on 15 Jan 2011, 11:44.
Last edited by Bunuel on 09 Jul 2013, 08:56, edited 1 time in total.
Renamed the topic and edited the question.
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Re: range of root - GMAT Club test - M24  [#permalink]

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New post 15 Jan 2011, 13:47
2
9
praveenvino wrote:
X/|X| < X . Which of the following must be true for all ?

a. X > 1
b. X is an element in (-1,0) U (1,inf)
c. |X| < 1
d. |X| = 1
e. |X|^2 > 1

Can some one explain how X can be zero for the above condition?


x is in the denominator so it can not equal to zero as division be zero is undefined.

Correct form of this question is below (m09 q22, discussed here: m09-q22-69937.html):

If \(\frac{x}{|x|} \lt x\), which of the following must be true about \(x\)? (\(x \ne 0\))
A. \(x\gt 2\)
B. \(x \in (-1,0) \cup (1,\infty)\)
C. \(|x| \lt 1\)
D. \(|x| = 1\)
E. \(|x|^2 \gt 1\)

\(\frac{x}{|x|}< x\)
Two cases:
A. \(x<0\) --> \(\frac{x}{-x}<x\) --> \(-1<x\). But as we consider the range \(x<0\) then \(-1<x<0\)

B. \(x>0\) --> \(\frac{x}{x}<x\) --> \(1<x\).

So the given inequality holds true in two ranges \(-1<x<0\) and \(x>1\).

Answer: B.

For more check: math-absolute-value-modulus-86462.html

Hope it helps.
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Re: range of root - GMAT Club test - M24  [#permalink]

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New post 15 Jan 2011, 14:03
Thanks Bunuel. X not equals zero condition was actually missing in the question in m24. Thanks for your help.
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Re: If x/|x|, which of the following must be true for all  [#permalink]

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New post 11 Dec 2014, 23:00
A must be true too.
If x>1 satisfy x/|x|<x
then x>2 will do too.
can anyone explain choice A? thanks!
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Re: If x/|x|, which of the following must be true for all  [#permalink]

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New post 12 Dec 2014, 05:51
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Re: If x/|x|, which of the following must be true for all  [#permalink]

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New post 22 Apr 2015, 18:22
Brunel, Can you please explain why option E is not feasible?
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Re: If x/|x|, which of the following must be true for all  [#permalink]

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New post 22 Apr 2015, 19:19
1
praveenvino wrote:
If \(\frac{x}{|x|} \lt x\), which of the following must be true about \(x\)? (\(x \ne 0\))

A. \(x\gt 2\)
B. \(x \in (-1,0) \cup (1,\infty)\)
C. \(|x| \lt 1\)
D. \(|x| = 1\)
E. \(|x|^2 \gt 1\)

M24



x < x*|x|
x-x*|x|< 0
roots of this equation are : -1,0,1
rest is explained in the attached image ...
Answer B.
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If x/|x|, which of the following must be true for all  [#permalink]

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New post 23 Apr 2015, 00:44
1
AverageGuy123 wrote:
Brunel, Can you please explain why option E is not feasible?


Dear AverageuGuy123

As Bunuel explained above,

Either -1 < x < 0
Or x > 1

Now, |x| as you know, represents the magnitude of x. Option E says that |x|^2 must be greater than 1.

Let's first consider the case when -1 < x < 0

A possible value of x in this case is -0.5
So, what is the value of |x|^2? It is equal to 0.25

Is it greater than 1? NO

Let's now consider the case when x > 1

A possible value of x in this case is 2.
So, what is the value of |x|^2? It's 4.

Is it greater than 1? YES

So, as we see, that |x|^2 CAN BE greater than 1. But can we say that |x|^2 MUST BE greater than 1? NO, because |x|^2 is not greater than 1 for all possible values of x.

So, the key takeaway from this discussion is that:

we need to be careful whether the question is asking about MUST BE TRUE statements or about CAN BE TRUE statements.

Hope this helped! :)

- Japinder
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Re: If x/|x|, which of the following must be true for all  [#permalink]

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New post 23 Apr 2015, 03:36
AverageGuy123 wrote:
Brunel, Can you please explain why option E is not feasible?


You can plug in numbers to eliminate options.

"which of the following must be true about x" means that every acceptable value of x must lie in the range given in the correct option. The acceptable values of x are the values for which x/|x| < x.

A. x>2
Must x be greater than 2?

This should make you check for 2.
2/|2| < 2
1 < 2 (True)
So 2 is an acceptable value of x. But 2 is not greater than 2.
So this option is not correct. This also makes you eliminate options (C) and (D).

E. |x|^2>1
Must x be greater than 1 or less than -1?

Check for 1/2
(1/2)/|1/2| < 1/2
1 < 1/2 (False)

Check for -1/2
(-1/2)/|-1/2| < -1/2
-1 < -1/2 (True)

So x = -1/2 is an acceptable value but it does not lie in this range. Hence option (E) is also incorrect.

Answer must be (B)
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Re: If x/|x|, which of the following must be true for all  [#permalink]

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New post 24 Apr 2015, 10:21
Hi All,

This question can be dealt with in a variety of ways. It's actually really susceptible to TESTing VALUES, which we can use to determine possibilities and eliminate answers.

We're told that X/|X| < X. The question asks what must be TRUE about X.

While this inequality looks complicated, you can quickly prove some things about X....

IF....
X = 1
1/|1| is NOT < 1
So X CANNOT be 1
Eliminate D.

IF.....
X = 2
2/|2| IS < 2
So X CAN be 2
Eliminate A and C.

IF....
X = -2
-2/|-2| is NOT < -2
So X CANNOT be -2
Eliminate E.

There's only one answer left....

Final Answer:

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Re: If x/|x|, which of the following must be true for all  [#permalink]

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New post 26 Jan 2018, 23:48
Given: x/|x| < x
since |x| >= 0 always
multiply both LHS and RHS by |x|
x < x|x|
=> x - x|x| < 0
=> x(1 - |x|) < 0
if x > 0, then 1 - |x| < 0 to hold the above inequality => |x| > 1 => x(since x is positive in this case) > 1
if x < 0, then 1 - |x| > 0 => |x| < 1 => -1 < x < 0 (to hold the above inequality)

Option B captures the above range perfectly
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Re: If x/|x|, which of the following must be true for all &nbs [#permalink] 26 Jan 2018, 23:48
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