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If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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If \(x\neq{0}\) and \(\frac{x}{|x|}<x\), which of the following must be true? (A) \(x>1\) (B) \(x>-1\) (C) \(|x|<1\) (D) \(|x|>1\) (E) \(-1<x<0\) m09 q22Explanations required for this one.
Not convinced at all with the OA.
My range is -1<x<0 and x>1.
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Originally posted by Marcab on 04 Dec 2012, 04:38.
Last edited by Bunuel on 23 Dec 2017, 01:16, edited 4 times in total.
Renamed the topic and edited the question.
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If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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If \(x\neq{0}\) and \(\frac{x}{|x|}<x\), which of the following must be true?(A) \(x>1\) (B) \(x>-1\) (C) \(|x|<1\) (D) \(|x|>1\) (E) \(-1<x<0\) Notice that we are asked to find which of the options MUST be true, not COULD be true. Let's see what ranges does \(\frac{x}{|x|}< x\) give for \(x\). Two cases: If \(x<0\) then \(|x|=-x\), hence in this case we would have: \(\frac{x}{-x}<x\) --> \(-1<x\). But remember that we consider the range \(x<0\), so \(-1<x<0\); If \(x>0\) then \(|x|=x\), hence in this case we would have: \(\frac{x}{x}<x\) --> \(1<x\). So, \(\frac{x}{|x|}< x\) means that \(-1<x<0\) or \(x>1\).Only option which is ALWAYS true is B. ANY \(x\) from the range \(-1<x<0\) or \(x>1\) will definitely be more the \(-1\). Answer: B. As for other options: A. \(x>1\). Not necessarily true since \(x\) could be -0.5; C. \(|x|<1\) --> \(-1<x<1\). Not necessarily true since \(x\) could be 2; D. \(|x|>1\) --> \(x<-1\) or \(x>1\). Not necessarily true since \(x\) could be -0.5; E. \(-1<x<0\). Not necessarily true since \(x\) could be 2. P.S. Please read carefully and follow: http://gmatclub.com/forum/rules-for-pos ... 33935.html Please pay attention to the rules #3 and 6. Thank you.
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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04 Dec 2012, 05:14
If you choose 0.9, then B fails. I just reviewed m09 q22 on the forum. You have changed the answer choices in that thread but not yet on GC CAT.
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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04 Dec 2012, 05:18
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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That's what I am saying. Since x cannot take this value then how can B be answer. How can x>-1 when 0<x<1 is not accepted?
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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Marcab wrote: That's what I am saying. Since x cannot take this value then how can B be answer.
How can x>-1 when 0<x<1 is not accepted? Consider following: If \(x=5\), then which of the following must be true about \(x\):A. x=3 B. x^2=10 C. x<4 D. |x|=1 E. x>-10 Answer is E (x>-10), because as x=5 then it's more than -10. Or: If \(-1<x<10\), then which of the following must be true about \(x\):A. x=3 B. x^2=10 C. x<4 D. |x|=1 E. x<120 Again answer is E, because ANY \(x\) from \(-1<x<10\) will be less than 120 so it's always true about the number from this range to say that it's less than 120. The same with original question: If \(-1<x<0\) or \(x>1\), then which of the following must be true about \(x\):A. \(x>1\) B. \(x>-1\) C. \(|x|<1\) D. \(|x|>1\) E. \(-1<x<0\) As \(-1<x<0\) or \(x>1\) then ANY \(x\) from these ranges would satisfy \(x>-1\). So B is always true. \(x\) could be for example -1/2, -3/4, or 10 but no matter what \(x\) actually is, it's IN ANY CASE more than -1. So we can say about \(x\) that it's more than -1. Hope it's clear.
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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04 Dec 2012, 05:57
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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\(\frac{x}{|x|}<x\) Test the values for x. (1) x = 0 ==> No! Since it is given that x is not 0 (2) x = -1 ==> -1 < -1 ==> No! (3) x = 1 ==> 1 < 1 ==> No! (4) x = -2 ==> -1 < -2 ==> No! (5) x = \(\frac{-1}{4}\) ==> -1 < \(\frac{-1}{4}\) ===> Yes! (5) x = 2 ==> 1 < 2 ==> Yes! Our range: -1 < x < 0 or x > 1 What must always be true in that range? x > -1 always Answer: B
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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08 Dec 2012, 19:05
Marcab wrote: Hey thanks.
Its crystal clear now. Yes - same here. The word in that explanation that helped me the most is "satify". I think the difficulty of this question is good. The learning moment is also exactley what I needed. The language is what confused me on the first attempt. I think it would be understood by more people if the question had the english rephrased to: "... which of the following statements can be satisfied by all possible values of x". Having said that, I learnt a lot about absolute values on the number plane trying to get my head around this explanation, so maybe it's helping us learn in the best way possible 
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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16 Dec 2012, 11:23
Appreciate if someone could point out where I am going wrong here. x / |x| < x Since x is non zero, dividing by x on both sides 1 / |x| < 1 Taking reciprocal, |x| > 1 Then I just jumped into Choice D. Didn't even look at the others. Await your valued views.
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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Since you don't know about the sign of X, i.e. whether its positive or negative, you cannot multiply or divide.
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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16 Dec 2012, 21:04
But your dividing my the same number x on both sides, whether its positive or negative, implying both sides will simultaneously be negative together or positive together which doesnt change the sign. Right or not?
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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16 Dec 2012, 21:12
Okay, picked a few numbers and realized my mistake. Thanks Marcab. So, as to generalize, you can NEVER NEVER divide by any variable unless you know its greater than zero. Thanks
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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18 Dec 2012, 00:48
Ans: we take two cases as we see there is a modulus sign. The equation becomes x(1-|x|)<0 and then after solving for both cases we get x to be always greater than -1, so the answer is (B).
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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Quote: If \(x\neq{0}\) and \(\frac{x}{|x|}<x\), which of the following must be true?
(A) \(x>1\)
(B) \(x>-1\)
(C) \(|x|<1\)
(D) \(|x|>1\)
(E) \(-1<x<0\)
Hi Karishma
Can you pls help me with the answer to the above link.
I was able to solve the inequality
My answer after solving inequality is -1<x<0 or x>1
So how can be the answer not E
The point of elimination for option e in the official explanation is as given below:-How can x be 2 when the range is less than 0......
E. −1<x<0. Not necessarily true since x could be 2.
A 'must be true' question! They are absolutely straight forward if you get the fundamental but they can drive you crazy if you don't. "My answer after solving inequality is -1<x<0 or x>1" Perfect. That is the range of x for which the inequality works. So tell me, what values can x take? -1/2, -1/3, -2/3, 1.4, 2, 500, 123498 etc... Now the question is "which of the following must be true?" (A) \(x>1\) Are all these values greater than 1? No. (B) \(x>-1\) Are all these values greater than -1? Yes. The answer. Note that you dont have to establish that all value greater than -1 should work for the inequality. You only have to establish that all values which work for the inequality must satisfy this condition. (C) \(|x|<1\) Not true for all values of x. (D) \(|x|>1\) Not true for all values of x. (E) \(-1<x<0\) Not true for all values of x. x can take values 1.4, 2, 500 etc I wrote a post on this beautiful question sometime back: http://www.veritasprep.com/blog/2012/07 ... -and-sets/
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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rongali wrote: If x≠0 and x / |x| < x, which of the following must be true
A) x>1
B) x>−1
c) |x|<1
D) |x|>1
E) −1<x<0
why is B an answer, as the equation wont hold true for values between 0<x<1
Mark as a guessHide Answer \(\Rightarrow\) Given, x≠0 & \(\frac{x}{|x|}< x\) ................. So, Two cases will be formed here ........ i.e., When x < 0 & when x > 0. Now, First when x<0, in this case we have, \(\frac{x}{-x}< x\) Therefore, \(x> -1\) Now, when x > 0, in this case we have, \(\frac{x}{x}< x\). Therefore, \(x> 1\). Hence, from both the conditions above, we can say that x must be greater than -1 . i.e., \(x> -1\) Hence, B.
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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01 May 2013, 21:28
rongali wrote: If x≠0 and x / |x| < x, which of the following must be true
A) x>1
B) x>−1
c) |x|<1
D) |x|>1
E) −1<x<0
why is B an answer, as the equation wont hold true for values between 0<x<1
Mark as a guessHide Answer I think the answer is absolutely correct. Firstly the question asks for a "MUST be true" option. Both the ranges for x, as rightly calculated above are : x>1 OR -1<x<0. However, none of the options subscribe to MUST be true type.It is so because x could be 2 OR x could be -0.5 However, the option B, where x>-1 will always be true, irrespective of the two ranges. Any value lying in the range -1<x<0 IS always in tandem with x>-1 AND any value for x>1 WILL also have x>-1.
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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01 May 2013, 21:38
doe007 wrote: When x = 1/2, x is GT -1, but x / |x| is NOT < x
Thus we cannot say that x / |x| < x for all x > -1.
Option B is wrong. We are not saying that x / |x| < x for all x > -1. That defies the whole purpose of breaking down the given inequality into 2 ranges. All we are saying is that for the 2 given ranges, the value of x MUST BE x>-1. You are taking the value of x=0.5, which in the first place is invalid for the given problem.
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]
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01 May 2013, 21:52
doe007 wrote: vinaymimani wrote: doe007 wrote: When x = 1/2, x is GT -1, but x / |x| is NOT < x
Thus we cannot say that x / |x| < x for all x > -1.
Option B is wrong. We are not saying that x / |x| < x for all x > -1. That defies the whole purpose of breaking down the given inequality into 2 ranges. All we are saying is that for the 2 given ranges, the value of x MUST BE x>-1. You are taking the value of x=0.5, which in the first place is invalid for the given problem. The question is asking to find the rage in which all values of would satisfy the inequality x / |x| < x. As the example shown, definitely some values in the range x > -1 do not satisfy the inequality. Hence, we CANNOT say that x > -1 must be true to satisfy x / |x| < x. The question at NO point of time has asked "to find the rage in which all values of x would satisfy the inequality x / |x| < x", It just says which of the following MUST be TRUE. You don't assume that the given options encompass all the valid ranges. You find the valid ranges, then look for a common thread which binds them together and MUST BE TRUE, irrespective of the range(s).
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true?
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