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# If x/|x|<x which of the following must be true about x?

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Senior Manager
Joined: 28 Dec 2010
Posts: 302
Location: India
Re: If x/|x|<x which of the following must be true about x? [#permalink]

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27 May 2012, 05:05
Bunuel, the question doesn't mention anywhere that x is an integer. so why have we not considered the values of 0<x<1 which also doesnt satisfy the equation? If it has been considered then the solution should be A and not B. x>1 will always hold true aka must be true! while x>-1 is sometimes true.. aka can be true.
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Posts: 44566
Re: If x/|x|<x which of the following must be true about x? [#permalink]

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27 May 2012, 05:16
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Expert's post
vibhav wrote:
Bunuel, the question doesn't mention anywhere that x is an integer. so why have we not considered the values of 0<x<1 which also doesnt satisfy the equation? If it has been considered then the solution should be A and not B. x>1 will always hold true aka must be true! while x>-1 is sometimes true.. aka can be true.

I think you don't understand the question.

Given: $$-1<x<0$$ and $$x>1$$. Question: which of the following must be true?

A. $$x>1$$. This opinion is not always true since $$x$$ can be $$-\frac{1}{2}$$ which is not more than 1.
B. $$x>-1$$. This option is always true since any $$x$$ from $$-1<x<0$$ and $$x>1$$ is more than -1.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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30 May 2012, 05:23
Hi Shouvik,

But I am not dividing .. I first consider that x<0 then cross multiply and flip the sign. I am still not clear with where did I go wrong.
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Joined: 08 Apr 2012
Posts: 125
Re: If x/|x|<x which of the following must be true about x? [#permalink]

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30 May 2012, 05:36
1
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pavanpuneet wrote:
Hi Shouvik,

But I am not dividing .. I first consider that x<0 then cross multiply and flip the sign. I am still not clear with where did I go wrong.

Ok,

Let me explain step by step:

Note that we have considered x<0. So anything we divide or multiply by a negative number x will change the signs of an inequality.

1. Since x<0, x/|x| < x => x/(-x) < x
2. Cross-multiplying both sides by (-x). Now since x<0, (-x)>0. So if we cross multiply it doesn't change the sign.
x < -x^2
3. Now, we divide both sides by x. Since x<0, this changes the sign of the inequality.
1 > -x
4. Simplifying further,
-1<x

This is exactly what bunuel had got.

Regards,

Shouvik.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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12 Jun 2012, 04:31
1
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Expert's post
There was a lot of confusion between options (A) and (B). Therefore, I would like to explain why option (B) is correct using diagrams.

Forget this question for a minute. Say instead you have this question:

Question 1: x > 2 and x < 7. What integral values can x take?
I guess most of you will come up with 3, 4, 5, 6. That’s correct. I can represent this on the number line.
Attachment:

Ques3.jpg [ 4.45 KiB | Viewed 8653 times ]

You see that the overlapping area includes 3, 4, 5 and 6.

Now consider this:

Question 2: x > 2 or x > 5. What integral values can x take?
Let’s draw that number line again.
Attachment:

Ques4.jpg [ 4.24 KiB | Viewed 8649 times ]

So is the solution again the overlapping numbers i.e. all integers greater than 5? No. This question is different. x is greater than 2 OR greater than 5. This means that if x satisfies at least one of these conditions, it is included in your answer. Think of sets. AND means it should be in both the sets (i.e. overlapping). OR means it should be in at least one of the sets. Hence, which values can x take? All integral values starting from 3 onwards i.e. 3, 4, 5, 6, 7, 8, 9 …

Now go back to this question. The solution is a one liner.

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

$$\frac{x}{|x|}$$ is either 1 or -1.
So x > 1 or x > -1
So which values can x take? All values that are included in at least one of the sets. Therefore, x > -1.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Senior Manager Joined: 06 Aug 2011 Posts: 369 Re: If x/|x|<x which of the following must be true about x? [#permalink] ### Show Tags 13 Jun 2012, 13:04 VeritasPrepKarishma wrote: There was a lot of confusion between options (A) and (B). Therefore, I would like to explain why option (B) is correct using diagrams. Forget this question for a minute. Say instead you have this question: Question 1: x > 2 and x < 7. What integral values can x take? I guess most of you will come up with 3, 4, 5, 6. That’s correct. I can represent this on the number line. Attachment: Ques3.jpg You see that the overlapping area includes 3, 4, 5 and 6. Now consider this: Question 2: x > 2 or x > 5. What integral values can x take? Let’s draw that number line again. Attachment: Ques4.jpg So is the solution again the overlapping numbers i.e. all integers greater than 5? No. This question is different. x is greater than 2 OR greater than 5. This means that if x satisfies at least one of these conditions, it is included in your answer. Think of sets. AND means it should be in both the sets (i.e. overlapping). OR means it should be in at least one of the sets. Hence, which values can x take? All integral values starting from 3 onwards i.e. 3, 4, 5, 6, 7, 8, 9 … Now go back to this question. The solution is a one liner. 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) $$|x|^2>1$$ $$\frac{x}{|x|}$$ is either 1 or -1. So x > 1 or x > -1 So which values can x take? All values that are included in at least one of the sets. Therefore, x > -1. But wat if x will be zero?? that will be infinitive..!!.. I think answer a is correct.. m still confused . _________________ Bole So Nehal.. Sat Siri Akal.. Waheguru ji help me to get 700+ score ! Current Student Joined: 29 Mar 2012 Posts: 318 Location: India GMAT 1: 640 Q50 V26 GMAT 2: 660 Q50 V28 GMAT 3: 730 Q50 V38 Re: If x/|x|<x which of the following must be true about x? [#permalink] ### Show Tags 13 Jun 2012, 13:56 sanjoo wrote: But wat if x will be zero?? that will be infinitive..!!.. I think answer a is correct.. m still confused . Hi, x can't be zero, it will lead to 0/0 form, which is not defined. Refer this post: http://gmatclub.com/forum/topic-92348.html#p1089298 Regards, Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8024 Location: Pune, India Re: If x/|x|<x which of the following must be true about x? [#permalink] ### Show Tags 13 Jun 2012, 22:05 sanjoo wrote: But wat if x will be zero?? that will be infinitive..!!.. I think answer a is correct.. m still confused . The range x > -1 does not imply that every value greater than -1 will satisfy this inequality. It implies that every value that satisfies this inequality will be greater than -1. 0 does not satisfy this inequality because the LHS is not defined for x = 0 so it is immaterial. Answer is (B). _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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10 Jul 2012, 16:40
Bunuel wrote:

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

Hi Bunuel,

How did you derive $$-1<x<0$$ from $$-1<x$$ without further calculation (quoting from your solution quoted above)?

Thanks,
Diana
Math Expert
Joined: 02 Sep 2009
Posts: 44566

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11 Jul 2012, 01:44
dianamao wrote:
Bunuel wrote:

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

Hi Bunuel,

How did you derive $$-1<x<0$$ from $$-1<x$$ without further calculation (quoting from your solution quoted above)?

Thanks,
Diana

For case A we consider range when $$x<0$$ and get that $$-1<x$$, so when we combine we get $$-1<x<0$$.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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11 Jul 2012, 09:14
sanjoo wrote:
VeritasPrepKarishma wrote:
There was a lot of confusion between options (A) and (B). Therefore, I would like to explain why option (B) is correct using diagrams.

Forget this question for a minute. Say instead you have this question:

Question 1: x > 2 and x < 7. What integral values can x take?
I guess most of you will come up with 3, 4, 5, 6. That’s correct. I can represent this on the number line.
Attachment:
Ques3.jpg

You see that the overlapping area includes 3, 4, 5 and 6.

Now consider this:

Question 2: x > 2 or x > 5. What integral values can x take?
Let’s draw that number line again.
Attachment:
Ques4.jpg

So is the solution again the overlapping numbers i.e. all integers greater than 5? No. This question is different. x is greater than 2 OR greater than 5. This means that if x satisfies at least one of these conditions, it is included in your answer. Think of sets. AND means it should be in both the sets (i.e. overlapping). OR means it should be in at least one of the sets. Hence, which values can x take? All integral values starting from 3 onwards i.e. 3, 4, 5, 6, 7, 8, 9 …

Now go back to this question. The solution is a one liner.

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

$$\frac{x}{|x|}$$ is either 1 or -1.
So x > 1 or x > -1
So which values can x take? All values that are included in at least one of the sets. Therefore, x > -1.

But wat if x will be zero?? that will be infinitive..!!..
I think answer a is correct.. m still confused .

Stating that we must have $$x > -1$$, it doesn't mean that all the numbers with this property will satisfy the inequality given in the question.
Also, the question is not asking for the set of solutions of the given inequality (which means all the values for which the inequality holds).
It asks for a MUST or necessary condition. And we can easily see that another necessary condition, besides B, is $$x$$ being non-zero.
If we assume that $$x\leq -1$$, then we get
$$\frac{x}{|x|}=\frac{x}{-x}=-1<x$$
which contradicts $$x\leq -1$$.
So, necessarily $$x$$ must be greater than $$-1$$.

Is this enough? Are there other conditions? Do all numbers greater than $$-1$$ satisfy the given inequality in the question?
This is not what the question is about. But for sure, if $$x$$ is not greater than $$-1$$, than the inequality cannot hold.
Therefore, $$x$$ MUST be greater than $$-1$$.
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31 Aug 2012, 01:23
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.

Bunuel just a small query here

just to stick to a standard, its better not to divide by a variable as we don't know its sign, could be positive or negative

so following that I did this sum , however was having difficulty at the end combining both the x ranges, please can you help

$$\frac{x}{|x|} < x$$
$$x<x|x|$$
$$x-x|x|<0$$

$$x(1-|x|)<0$$
so

first case

when $$x>0$$ then $$1-|x|<0$$
so $$1-|x|<0 =|x|>1$$

$$|x| >1 = -1 > x > 1$$

so we have $$x>0$$and $$-1 > x > 1$$

now here is my concern how to combine these two to get the final range of $$x$$

I am having difficulty combining these two to get $$x>1$$ , is there any technique ?

second case ( although this was easier to combine )

$$x(1-|x|)<0$$

when $$x<0$$then $$1-|x|>0$$

so $$1-|x|>0 = 1 >|x|$$

$$|x|<1 = -1< x < 1$$

we have $$x<0$$ and $$-1< x < 1$$ now again for combining these two is there any standard way?

logically i can arrive at $$-1<x<0$$
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Math Expert
Joined: 02 Sep 2009
Posts: 44566

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31 Aug 2012, 01:40
stne 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.

Bunuel just a small query here

just to stick to a standard, its better not to divide by a variable as we don't know its sign, could be positive or negative

so following that I did this sum , however was having difficulty at the end combining both the x ranges, please can you help

$$\frac{x}{|x|} < x$$
$$x<x|x|$$
$$x-x|x|<0$$

$$x(1-|x|)<0$$
so

first case

when $$x>0$$ then $$1-|x|<0$$
so $$1-|x|<0 =|x|>1$$

$$|x| >1 = -1 > x > 1$$

so we have $$x>0$$and $$-1 > x > 1$$

now here is my concern how to combine these two to get the final range of $$x$$

I am having difficulty combining these two to get $$x>1$$ , is there any technique ?

second case ( although this was easier to combine )

$$x(1-|x|)<0$$

when $$x<0$$ then 1-|x|>0

so $$1-|x|>0 = 1 >|x|$$

$$|x|<1 = -1< x < 1$$

we have $$x<0$$ and $$-1< x < 1$$ now again for combining these two is there any standard way?

logically i can arrive at $$-1<x<0$$

When you consider x<0, then you automatically have that |x|=-x. Similarly, when you consider x>0, then you automatically have that |x|=x.

$$x(1-|x|)<0$$:

x<0 --> 1-(-x)>0 --> x>-1 --> -1<x<0.
x>0 --> 1-x<0 --> x>1.

So, the given inequality holds for -1<x<0. and x>1.

Hope it's clear.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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01 Sep 2012, 05:33
My learning:

Since this Question is asking "must be true about x" so the answer should cover the whole range of x, even if there is a gap (in this case it is from 0 to 1)

If the question would have asked "what all values of x satisfy this equation", then the correct answer would be "x > 1" (considering 0 < x < 1 is missing from choices)

Am I correct!?
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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02 Sep 2012, 22:02
dexerash wrote:
My learning:

Since this Question is asking "must be true about x" so the answer should cover the whole range of x, even if there is a gap (in this case it is from 0 to 1)

If the question would have asked "what all values of x satisfy this equation", then the correct answer would be "x > 1" (considering 0 < x < 1 is missing from choices)

Am I correct!?

Yes, you are correct that 'must be true about x' means the answer should cover the entire range of x. There can be some values in that range which x cannot take.

Had the question asked for the 'all values of x that satisfy this equation', the correct answer would be -1 < x< 0 or x > 1. x can take the values from -1 to 0 too so it should also appear in the range.
x>1 gives only the partial range of x.

Check out this post for a detailed discussion on this question: http://www.veritasprep.com/blog/2012/07 ... -and-sets/
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Manager Joined: 14 Nov 2011 Posts: 135 Location: United States Concentration: General Management, Entrepreneurship GPA: 3.61 WE: Consulting (Manufacturing) Re: If x/|x|<x which of the following must be true about x? [#permalink] ### Show Tags 30 May 2013, 22:32 VeritasPrepKarishma wrote: There was a lot of confusion between options (A) and (B). Therefore, I would like to explain why option (B) is correct using diagrams. Forget this question for a minute. Say instead you have this question: Question 1: x > 2 and x < 7. What integral values can x take? I guess most of you will come up with 3, 4, 5, 6. That’s correct. I can represent this on the number line. Attachment: Ques3.jpg You see that the overlapping area includes 3, 4, 5 and 6. Now consider this: Question 2: x > 2 or x > 5. What integral values can x take? Let’s draw that number line again. Attachment: Ques4.jpg So is the solution again the overlapping numbers i.e. all integers greater than 5? No. This question is different. x is greater than 2 OR greater than 5. This means that if x satisfies at least one of these conditions, it is included in your answer. Think of sets. AND means it should be in both the sets (i.e. overlapping). OR means it should be in at least one of the sets. Hence, which values can x take? All integral values starting from 3 onwards i.e. 3, 4, 5, 6, 7, 8, 9 … Now go back to this question. The solution is a one liner. 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) $$|x|^2>1$$ $$\frac{x}{|x|}$$ is either 1 or -1. So x > 1 or x > -1 So which values can x take? All values that are included in at least one of the sets. Therefore, x > -1. Hi Karishma/Bunnel, Had the question been 'Which of the values of x satisfies this inequality?' then the correct answer would have been A Now the question is essentially, 'Which of the following ranges consists of all values of x that satisfy this inequality?' Please correct me if i am wrong. Math Expert Joined: 02 Sep 2009 Posts: 44566 Re: If x/|x|<x which of the following must be true about x? [#permalink] ### Show Tags 31 May 2013, 02:52 cumulonimbus wrote: VeritasPrepKarishma wrote: There was a lot of confusion between options (A) and (B). Therefore, I would like to explain why option (B) is correct using diagrams. Forget this question for a minute. Say instead you have this question: Question 1: x > 2 and x < 7. What integral values can x take? I guess most of you will come up with 3, 4, 5, 6. That’s correct. I can represent this on the number line. Attachment: Ques3.jpg You see that the overlapping area includes 3, 4, 5 and 6. Now consider this: Question 2: x > 2 or x > 5. What integral values can x take? Let’s draw that number line again. Attachment: Ques4.jpg So is the solution again the overlapping numbers i.e. all integers greater than 5? No. This question is different. x is greater than 2 OR greater than 5. This means that if x satisfies at least one of these conditions, it is included in your answer. Think of sets. AND means it should be in both the sets (i.e. overlapping). OR means it should be in at least one of the sets. Hence, which values can x take? All integral values starting from 3 onwards i.e. 3, 4, 5, 6, 7, 8, 9 … Now go back to this question. The solution is a one liner. 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) $$|x|^2>1$$ $$\frac{x}{|x|}$$ is either 1 or -1. So x > 1 or x > -1 So which values can x take? All values that are included in at least one of the sets. Therefore, x > -1. Hi Karishma/Bunnel, Had the question been 'Which of the values of x satisfies this inequality?' then the correct answer would have been A Now the question is essentially, 'Which of the following ranges consists of all values of x that satisfy this inequality?' Please correct me if i am wrong. Yes, that's correct. _________________ Intern Joined: 21 May 2012 Posts: 1 Re: If x/|x|<x which of the following must be true about x? [#permalink] ### Show Tags 16 Feb 2014, 13:14 x cannot be zero and x cannot take values between 0<x<1 eg. lets try 1/2 in the equation 1/2/|1/2|<1/2 1<1/2 not possible so answer is x>1 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8024 Location: Pune, India Re: If x/|x|<x which of the following must be true about x? [#permalink] ### Show Tags 16 Feb 2014, 20:28 khannitw5 wrote: x cannot be zero and x cannot take values between 0<x<1 eg. lets try 1/2 in the equation 1/2/|1/2|<1/2 1<1/2 not possible so answer is x>1 That's not correct. Check the explanations given above or this post: http://www.veritasprep.com/blog/2012/07 ... -and-sets/ _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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07 Jul 2014, 01:32
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.

Bunnel, if I were to multiply the original stem with |x| (since |x| is always positive) it would result in x*(|x|-1) > 0.
This would mean x > 0 and |x| > 1
|x| > 1 would lead to x < -1 and x > 1 . This is completely different from the answer you've reached. I see that your method is accurate and the answer justified, but can you please correct my method here.
Re: If x/|x|<x which of the following must be true about x?   [#permalink] 07 Jul 2014, 01:32

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