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Senior Manager  Joined: 06 Apr 2008
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If x/|x|<x which of the following must be true about x?  [#permalink]

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If $$\frac{x}{|x|}<x$$ which of the following must be true about $$x$$?

(A) $$x>1$$

(B) $$x>-1$$

(C) $$|x|<1$$

(D) $$|x|=1$$

(E) $$|x|^2>1$$
Math Expert V
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nmohindru wrote:
If $$\frac{x}{|x|}<x$$ which of the following must be true about $$x$$?

(A) $$x>1$$

(B) $$x>-1$$

(C) $$|x|<1$$

(D) $$|x|=1$$

(E) $$|x|^2>1$$

This question was well explained by Durgesh and Ian Stewart, but since there are still some doubts, I'll try to add my 2 cents.

First of all let's solve this inequality step by step and see what is the solution for it, or in other words let's see in which ranges this inequality holds true.

Two cases for $$\frac{x}{|x|}<x$$:

A. $$x<0$$ --> $$|x|=-x$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$ --> $$-1<x<0$$;

B. $$x>0$$ --> $$|x|=x$$ --> $$\frac{x}{x}<x$$ --> $$1<x$$.

So given inequality holds true in the ranges: $$-1<x<0$$ and $$x>1$$. Which means that $$x$$ can take values only from these ranges.

------{-1}xxxx{0}----{1}xxxxxx

Now, we are asked which of the following must be true about $$x$$. Option A can not be ALWAYS true because $$x$$ can be from the range $$-1<x<0$$, eg $$-\frac{1}{2}$$ and $$x=-\frac{1}{2}<1$$.

Only option which is ALWAYS true is B. ANY $$x$$ from the ranges $$-1<x<0$$ and $$x>1$$ will definitely be more the $$-1$$, all "red", possible x-es are to the right of -1, which means that all possible x-es are more than -1.

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VP  Joined: 07 Nov 2007
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For x < 0,

x / |x| < x
=> x / -x < x
=> x > -1
x < 0 & x > -1
-1<X<0

For x > 0,
x / |x| < x
=> x / x < x
=> x > 1
x>0&x > 1
----> x>1

Question says which one must be true..

Option A ) correct.. X>1 (X>1 implies that when X>0 )
this is always true

bet on A.
##### General Discussion
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Joined: 30 Apr 2008
Posts: 1554
Location: Oklahoma City
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To condense some of what the other thread states:

If you use -1/2, it works, so at first you might think the answer is clearly b), but "must be true about X" also means "must ALWAYS be true about X". If you use +1/2 (which is > -1 as in answer B) it doesn't work.

1/2 over 1/2 = 1 < x which is 1/2 so you get 1 < 1/2 and that's not true.

So answer a MUST ALWAYS be true because only positive X over abs(x) = 1 and X > 1, it will always be greater than 1 (which is x/|x| when x is positive).

Does this condense the other thread?

nmohindru wrote:
If X/|X| < X Which of the following must be true about X ?

a) X>1

b) X>-1

c) |X| < 1

d) |X| = 1

e) |X|^2 > 1
Senior Manager  Joined: 27 May 2008
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i think OA is correct....

the question is If x / |x| < x. so we have to consider only those values of x for which this inequalty is true and what are those values
1. when x is between -1 and 0
2. when x is more than 1

lets call these conditions our universe.

Now the question is for all values of x (in our universe) which of the following is true
option B, x > -1, has both conditions 1 and 2

now you may say what about x = 1/2 .... that wasnt even part of our universe... so even if x = 1/2 is satisfying option B and not the question stem, we dont have to worry... becuase we are not supposed to take it as an example ...

for all values of x in our universe, option B is ALWAYS true...
Option A is not always true...
Intern  Joined: 21 Jul 2008
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I believe (E) should be the answer.

we know that |x| = -x for x < 0
= x for x > 0

For x < 0,

x / |x| < x
=> x / -x < x
=> x > -1

For x > 0,

x / |x| < x
=> x / x < x
=> x > 1

Therefore |x| > 1 or |x|^2 > 1
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Joined: 30 Apr 2008
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Because the question states what MUST be true, I think the answer has to be A. We can't select only our "universe" of numbers that satisfy the selection. If the answer, like in B, says X > -1, then every single number greater than -1 must satisfy the inequality and that is not the case.

I certainly agree that the "universe" must be -1 < X < 0 and 1 < x. Answer B defines one "universe" as all numbers that are greater than -1. Answer B includes all numbers between 0 and 1 which do not satisfy the inequality; therefore, B must not always be true. The answer should be A. I disagree with the OA.

nmohindru wrote:
If X/|X| < X Which of the following must be true about X ?

X>1

X>-1

|X| < 1

|X| = 1

|X|^2 > 1
Senior Manager  Joined: 27 May 2008
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guys i beg to differ.. i'll try to explain myself again...

our universe is defined in this line, if x/|x| < x this is given ... x only exits if it follows this condition ... if soemthing doesnt follow it then, its not x ... its y, may be z (doesnt matter) ...

if S is a set of possible values of x that will satisfy our condition x/|x| < x
S = [ -0.9, -0.8, -0.7, ....... -0.1, 1.1, 1.2, 1.3, 1.4 ....... upto infinite]

we have to find an answer that will always be true for all values of x,

lets see A, x > 1, this statement is not true for x = -0.9, -0.8 ..... -0.1

check B x > -1, this statement will "always" be true for all x ... clearly B is the answer...

i see your point that x = 0.5 will not satisfy the question condition, but thats the whole point 0.5 is not even x, its y
but if you select A, you are leaving some valid values of x.
Senior Manager  Joined: 27 May 2008
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"For which of the following conditions for x,
x/|x| < x is always true. "

now A will be the clear answer,
SVP  Joined: 29 Aug 2007
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nmohindru wrote:
If X/|X| < X Which of the following must be true about X ?

X>1

X>-1

|X| < 1

|X| = 1

|X|^2 > 1

For me it is disputable. I sometime go with B and sometime with A. B is broader than A but B again is not always a true case where as A is always true.

if we pay detail attention to MUST on "Which if the following must be true about X", it should be A because B is not a "must" true.
Senior Manager  Joined: 27 May 2008
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GMAT TIGER wrote:

x = 0, 1 , or any +ve fractions.

that is not even defined as x, our x is defined in the first line ....

i'll give an example,

x is a member of set S = [-4,-3,-2,-1, 4,5,6,7]
what is always true for all x
A. x >= 4
B. x >= -4

A or B... if you select A, then i cant explain it further..
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Joined: 24 Jun 2008
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9
I'm pretty sure Durgesh has already explained his answer convincingly, but just in case, I'll present a different problem:

If x is positive, what must be true?

I) x > 10
II) x > -10
III) x > 0

Of course, III) says exactly 'x is positive', so III) must be true. But if x is positive, x is obviously larger than -10. II) must also be true. I) doesn't need to be true; x could be 4, or 7, for example.

The same situation applies with the question that began this thread. We know that either -1 < x < 0, or x > 1. What must be true? Well, "x > 1" does not need to be true. We know that x might be -0.5, for example. On the other hand, "x > -1" absolutely must be true: if x satisfies the inequality x/|x| < x, then x is certainly larger than -1. If we had been asked whether "x > -1,000,000" must be true, the answer would also be yes. The question did not ask "What is the solution set of "x/|x| < x"; nor did it ask "For which values of x is x/|x| < x true?".
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Re: If x/|x|<x which of the following must be true about x?  [#permalink]

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Typical GMAT nonsense. A lot of people seem to talk themselves into a solution but in mathematics, there are no GMAT-truths. x>-1 must not be true, since it ignores the fact that 0 does not fulfill the requirement.
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Joined: 24 Jun 2008
Posts: 2015
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

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myfish wrote:
Typical GMAT nonsense. A lot of people seem to talk themselves into a solution but in mathematics, there are no GMAT-truths. x>-1 must not be true, since it ignores the fact that 0 does not fulfill the requirement.

No one is "talking themselves into a solution" here, and there's nothing wrong with the mathematics. I explained why earlier, but I can use a simpler example. If a question reads

If x = 5, what must be true?

I) x > 0

then clearly I) must be true; if x is 5, then x is certainly positive. It makes no difference that x cannot be equal to 12, or to 1000.

The same thing is happening in this question. We know that either -1 < x < 0, or that 1 < x. If x is in either of those ranges, then certainly x must be greater than -1. It makes no difference that x cannot be equal to 1/2, or to 0.

This is an important logical point on the GMAT (even though the question in the original post is not a real GMAT question), since it comes up all the time in Data Sufficiency. If a question asks

Is x > 0?

1) x = 5

that is exactly the same question as the one I asked above, but now it's phrased as a DS question. This question is really asking, when we use Statement 1, "If x = 5, must it be true that x > 0?" Clearly the answer is yes. If you misinterpret this question, and think it's asking "can x have any positive value at all", you would make a mistake on this question and on most GMAT DS algebra questions.
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Re: If x/|x|<x which of the following must be true about x?  [#permalink]

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IanStewart wrote:
myfish wrote:
Typical GMAT nonsense. A lot of people seem to talk themselves into a solution but in mathematics, there are no GMAT-truths. x>-1 must not be true, since it ignores the fact that 0 does not fulfill the requirement.

No one is "talking themselves into a solution" here, and there's nothing wrong with the mathematics. I explained why earlier, but I can use a simpler example. If a question reads

If x = 5, what must be true?

I) x > 0

then clearly I) must be true; if x is 5, then x is certainly positive. It makes no difference that x cannot be equal to 12, or to 1000.

The same thing is happening in this question. We know that either -1 < x < 0, or that 1 < x. If x is in either of those ranges, then certainly x must be greater than -1. It makes no difference that x cannot be equal to 1/2, or to 0.

This is an important logical point on the GMAT (even though the question in the original post is not a real GMAT question), since it comes up all the time in Data Sufficiency. If a question asks

Is x > 0?

1) x = 5

that is exactly the same question as the one I asked above, but now it's phrased as a DS question. This question is really asking, when we use Statement 1, "If x = 5, must it be true that x > 0?" Clearly the answer is yes. If you misinterpret this question, and think it's asking "can x have any positive value at all", you would make a mistake on this question and on most GMAT DS algebra questions.

Dear Ian,
I truly appreciate your efforts on here. 'Must be true' is a condition without exceptions. And when i plug in 0, the inequality is NOT true. That five apples are more than 0 apples is clear to me. However, the question asks for what 'Must be true'. Several ranges that make the inequality true makes this question inaccurate. Unless, if the GMAT translates 'Must be true' into 'may or may not be true' then explanation with the ranges make sense. For me, these type of questions make the GMAT into a lottery and I am not alone since many test takers have trouble with a logic that ignores exceptions. I have another example, fresh from Kaplan.

If it is true to -6<= n <= 10, which of the following must be true?

n<8
n=-6
n>-8
-10<n<7
none of the above

Same case. Official solution is n>-8, however n= -7 does not fulfill the first requirement, it therefore CAN BE TRUE - but not MUST BE TRUE

Again, I am no stranger to logic but, I am sure many will agree, these kind of questions are nonsense, especially when one considers the official (you and others) translation of the question into "Are 3 apples more than 2?" - what kind of a question is that?
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Re: If x/|x|<x which of the following must be true about x?  [#permalink]

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myfish wrote:

Dear Ian,
I truly appreciate your efforts on here. 'Must be true' is a condition without exceptions. And when i plug in 0, the inequality is NOT true. That five apples are more than 0 apples is clear to me. However, the question asks for what 'Must be true'. Several ranges that make the inequality true makes this question inaccurate. Unless, if the GMAT translates 'Must be true' into 'may or may not be true' then explanation with the ranges make sense. For me, these type of questions make the GMAT into a lottery and I am not alone since many test takers have trouble with a logic that ignores exceptions. I have another example, fresh from Kaplan.

If it is true to -6<= n <= 10, which of the following must be true?

n<8
n=-6
n>-8
-10<n<7
none of the above

Same case. Official solution is n>-8, however n= -7 does not fulfill the first requirement, it therefore CAN BE TRUE - but not MUST BE TRUE

Again, I am no stranger to logic but, I am sure many will agree, these kind of questions are nonsense, especially when one considers the official (you and others) translation of the question into "Are 3 apples more than 2?" - what kind of a question is that?

I've tried to explain the logic behind this question twice, so I won't try again, but I can assure you that every mathematician in the world would agree with the answer to this question - this has nothing to do with some kind of logic exclusive to the GMAT. The same is true of the Kaplan question you quote; if n is greater than -6, it is surely true that n is greater than -8. You seem to be looking at these problems backwards: you're assuming n > -8 is true, and asking if it needs to be true that -6 < n < 10. That's the opposite of what the question is asking you to do.
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Re: If x/|x|<x which of the following must be true about x?  [#permalink]

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I think answer should be A and not B because x cannot be equal to zero. If x equals to zero then the equation will lead to infinity.
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Re: If x/|x|<x which of the following must be true about x?  [#permalink]

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gautammalik wrote:
I think answer should be A and not B because x cannot be equal to zero. If x equals to zero then the equation will lead to infinity.

Think again.
Every value greater than -1 need not satisfy the inequality but every value satisfying the inequality must be greater than -1.

x/|x| can take only 2 values: 1 or -1
If x is positive, x/|x| = 1
If x is negative, x/|x| = -1
x cannot be 0.

Now let's look at the question.
x > x/|x| holds for x > 1 (x is positive) or -1 < x < 0 (x is negative)

x can take many values e.g. -1/3, -4/5, 2, 5, 10 etc

Which of the following holds for every value that x can take?
(A) X > 1
(B) X > -1

I hope that you agree that X > 1 doesn't hold for every possible value of X whereas X > -1 holds for every possible value of X.

Mind you, every value greater than -1 need not be a possible value of x.
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Bunuel wrote:
nmohindru wrote:
If $$\frac{x}{|x|}<x$$ which of the following must be true about $$x$$?

(A) $$x>1$$

(B) $$x>-1$$

(C) $$|x|<1$$

(D) $$|x|=1$$

(E) $$|x|^2>1$$

This question was well explained by Durgesh and Ian Stewart, but since there are still some doubts, I'll try to add my 2 cents.

First of all let's solve this inequality step by step and see what is the solution for it, or in other words let's see in which ranges this inequality holds true.

Two cases for $$\frac{x}{|x|}<x$$:

A. $$x<0$$ --> $$|x|=-x$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$ --> $$-1<x<0$$;

B. $$x>0$$ --> $$|x|=x$$ --> $$\frac{x}{x}<x$$ --> $$1<x$$.

So given inequality holds true in the ranges: $$-1<x<0$$ and $$x>1$$. Which means that $$x$$ can take values only from these ranges.

------{-1}xxxx{0}----{1}xxxxxx

Now, we are asked which of the following must be true about $$x$$. Option A can not be ALWAYS true because $$x$$ can be from the range $$-1<x<0$$, eg $$-\frac{1}{2}$$ and $$x=-\frac{1}{2}<1$$.

Only option which is ALWAYS true is B. ANY $$x$$ from the ranges $$-1<x<0$$ and $$x>1$$ will definitely be more the $$-1$$, all "red", possible x-es are to the right of -1, which means that all possible x-es are more than -1.

isnt E true ? from the above 1<mod (x) which implies 1< mod(x) squared
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koro12 wrote:
Bunuel wrote:
nmohindru wrote:
If $$\frac{x}{|x|}<x$$ which of the following must be true about $$x$$?

(A) $$x>1$$

(B) $$x>-1$$

(C) $$|x|<1$$

(D) $$|x|=1$$

(E) $$|x|^2>1$$

This question was well explained by Durgesh and Ian Stewart, but since there are still some doubts, I'll try to add my 2 cents.

First of all let's solve this inequality step by step and see what is the solution for it, or in other words let's see in which ranges this inequality holds true.

Two cases for $$\frac{x}{|x|}<x$$:

A. $$x<0$$ --> $$|x|=-x$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$ --> $$-1<x<0$$;

B. $$x>0$$ --> $$|x|=x$$ --> $$\frac{x}{x}<x$$ --> $$1<x$$.

So given inequality holds true in the ranges: $$-1<x<0$$ and $$x>1$$. Which means that $$x$$ can take values only from these ranges.

------{-1}xxxx{0}----{1}xxxxxx

Now, we are asked which of the following must be true about $$x$$. Option A can not be ALWAYS true because $$x$$ can be from the range $$-1<x<0$$, eg $$-\frac{1}{2}$$ and $$x=-\frac{1}{2}<1$$.

Only option which is ALWAYS true is B. ANY $$x$$ from the ranges $$-1<x<0$$ and $$x>1$$ will definitely be more the $$-1$$, all "red", possible x-es are to the right of -1, which means that all possible x-es are more than -1.

isnt E true ? from the above 1<mod (x) which implies 1< mod(x) squared

We are told that $$x$$ can take values only from these ranges.

------{-1}xxxx{0}----{1}xxxxxx

Now, $$|x|^2>1$$ means that $$x^2>1$$ --> $$x<-1$$ or $$x>1$$. Since $$x<-1$$ is not true about $$x$$ (we know that $$-1<x<0$$ and $$x>1$$), then this option is not ALWAYS true.

Hope it's clear.
_________________ Re: Inequality   [#permalink] 27 May 2012, 01:37

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