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# M09-22

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Math Expert
Joined: 02 Sep 2009
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12 Jul 2017, 21:32
sahil1425 wrote:
I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. what if x=0.5?

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15 Jul 2017, 08:47
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Bunuel wrote:
lshoshiashvili wrote:
Hello, thanks for interesting question.

how can |x| = -x when x<0? if x <0 then |-x|=x?

I think when x<0, -x/|x|<-x. isn't it?

Please explain the flaws in my logic.

thank u!

This is a property of an absolute value:

When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$.

You should brush-up fundamentals on modulus:

Hi Brunnel,

I sometimes get confused about checking which one and consider what as a constraint!.. What I was thinking is, since x>-1 can have 1 so X/|X| < X cannot be true. Well, in some case you can check the other way round. But Sometimes I get confused regarding this. Any idea whats the way around this?

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17 Aug 2017, 11:32
I agree with explanation. How come we aren't counting X>0.
I see that in the explanation part, but can it be elaborated a little more. What if x = 1/4.
In this case, 1 > 1/4.

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20 Aug 2017, 00:39
Bunuel wrote:
Official Solution:

If $$x \ne 0$$ and $$\frac{x}{|x|} \lt x$$, which of the following must be true?

A. $$x \gt 1$$
B. $$x \gt -1$$
C. $$|x| \lt 1$$
D. $$|x|&gt;1$$
E. $$-1 \lt x \lt 0$$

Notice that we are asked to find which of the options MUST be true, not COULD be true.

Let's see what ranges does $$\frac{x}{|x|} \lt x$$ give for $$x$$. Two cases:

If $$x \lt 0$$ then $$|x|=-x$$, hence in this case we would have: $$\frac{x}{-x} \lt x$$, which is the same as $$-1 \lt x$$. But remember that we consider the range $$x \lt 0$$, so $$-1 \lt x \lt 0$$;

If $$x \gt 0$$ then $$|x|=x$$, hence in this case we would have: $$\frac{x}{x} \lt x$$, which is the same as $$1 \lt x$$.

So, $$\frac{x}{|x|} \lt x$$ means that $$-1 \lt x \lt 0$$ or $$x \gt 1$$.

Only option which is ALWAYS true is B. ANY $$x$$ from the range $$-1 \lt x \lt 0$$ or $$x \gt 1$$ will definitely be more the $$-1$$.

As for other options:

A. $$x \gt 1$$. Not necessarily true since $$x$$ could be -0.5;

C. $$|x| \lt 1$$, so $$-1 \lt x \lt 1$$. Not necessarily true since $$x$$ could be 2;

D. $$|x| \gt 1$$, so $$x \lt -1$$ or $$x \gt 1$$. Not necessarily true since $$x$$ could be -0.5;

E. $$-1 \lt x \lt 0$$. Not necessarily true since $$x$$ could be 2.

Bunnel though i totally agree with your solution but in selection of option i do not agree. the question asks which of the following 'must be true"

if we are selecting B i.e., x>-1 then if x=1 the inequality does not hold true, in 0<x<1 also the inequality doesnt hold true.
But in E for every value of x in that interval the inequality holds true. i accept for x>1 also the inequality holds true but for every value in -1<x<0 the inequality holds true where as in B :x>-1 at some places the inequality is true and in some places the inequality does not hold true. B can be selected in case of "could be true" question.

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20 Aug 2017, 02:44
aamir89 wrote:
Bunuel wrote:
Official Solution:

If $$x \ne 0$$ and $$\frac{x}{|x|} \lt x$$, which of the following must be true?

A. $$x \gt 1$$
B. $$x \gt -1$$
C. $$|x| \lt 1$$
D. $$|x|&gt;1$$
E. $$-1 \lt x \lt 0$$

Notice that we are asked to find which of the options MUST be true, not COULD be true.

Let's see what ranges does $$\frac{x}{|x|} \lt x$$ give for $$x$$. Two cases:

If $$x \lt 0$$ then $$|x|=-x$$, hence in this case we would have: $$\frac{x}{-x} \lt x$$, which is the same as $$-1 \lt x$$. But remember that we consider the range $$x \lt 0$$, so $$-1 \lt x \lt 0$$;

If $$x \gt 0$$ then $$|x|=x$$, hence in this case we would have: $$\frac{x}{x} \lt x$$, which is the same as $$1 \lt x$$.

So, $$\frac{x}{|x|} \lt x$$ means that $$-1 \lt x \lt 0$$ or $$x \gt 1$$.

Only option which is ALWAYS true is B. ANY $$x$$ from the range $$-1 \lt x \lt 0$$ or $$x \gt 1$$ will definitely be more the $$-1$$.

As for other options:

A. $$x \gt 1$$. Not necessarily true since $$x$$ could be -0.5;

C. $$|x| \lt 1$$, so $$-1 \lt x \lt 1$$. Not necessarily true since $$x$$ could be 2;

D. $$|x| \gt 1$$, so $$x \lt -1$$ or $$x \gt 1$$. Not necessarily true since $$x$$ could be -0.5;

E. $$-1 \lt x \lt 0$$. Not necessarily true since $$x$$ could be 2.

Bunnel though i totally agree with your solution but in selection of option i do not agree. the question asks which of the following 'must be true"

if we are selecting B i.e., x>-1 then if x=1 the inequality does not hold true, in 0<x<1 also the inequality doesnt hold true.
But in E for every value of x in that interval the inequality holds true. i accept for x>1 also the inequality holds true but for every value in -1<x<0 the inequality holds true where as in B :x>-1 at some places the inequality is true and in some places the inequality does not hold true. B can be selected in case of "could be true" question.

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21 Aug 2017, 08:03
aamir89 wrote:
Bunuel wrote:
Official Solution:

If $$x \ne 0$$ and $$\frac{x}{|x|} \lt x$$, which of the following must be true?

A. $$x \gt 1$$
B. $$x \gt -1$$
C. $$|x| \lt 1$$
D. $$|x|&gt;1$$
E. $$-1 \lt x \lt 0$$

Bunnel though i totally agree with your solution but in selection of option i do not agree. the question asks which of the following 'must be true"

if we are selecting B i.e., x>-1 then if x=1 the inequality does not hold true, in 0<x<1 also the inequality doesnt hold true.
But in E for every value of x in that interval the inequality holds true. i accept for x>1 also the inequality holds true but for every value in -1<x<0 the inequality holds true where as in B >-1 at some places the inequality is true and in some places the inequality does not hold true. B can be selected in case of "could be true" question.

Hi,

This question created a lot of confusion for me in the past until an instructor started with the basic. What does the question mean?

This part 'If $$x \ne 0$$ and $$\frac{x}{|x|} \lt x$$' does NOT ask us to solve the question and find range of x but rather this part tell us a Truth or Fact, the other part asks us to state 'What Facts do we know about the solution of the inequality?

So you, me and other had successfully obtained the range. The question becomes if solution of inequality x/|x|<x is −1<x<0 or x>1, what MUST BE TRUE about EVERY X?

I quoted what he said below:
The OA does NOT imply that EVERY value greater than -1 is a valid solution for x/|x|<x.
It implies the reverse:
That every valid solution for x/|x|<x is greater than -1.

P.S.: This question or much similar to it appeared in GMATPrep or question pack 1 .

I hope it helps

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29 Aug 2017, 23:19
I think this is a poor-quality question and I don't agree with the explanation. Hi. Could you pls point me on mistake in my thoughts.
We have two options for X here:
x>1 or 0>x>-1
According to correct answer choice x>-1, x can be 1/2, for example. Let's check it for this solution 1<1/2 - seems not correct.

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29 Aug 2017, 23:25
StaicyT wrote:
I think this is a poor-quality question and I don't agree with the explanation. Hi. Could you pls point me on mistake in my thoughts.
We have two options for X here:
x>1 or 0>x>-1
According to correct answer choice x>-1, x can be 1/2, for example. Let's check it for this solution 1<1/2 - seems not correct.

This is explained several times on this thread: we are given that -1 < x < 0 or x > 1. Whatever the actual value of x could be (again it could be only -1 < x < 0 or x > 1) it would be correct to say that x is greater than -1.
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31 Aug 2017, 13:10
Bunuel wrote:
When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;
[/textarea]

Hi Bunuel,

I believe |x|=-x if x<0 and not if x≤0.

Please correct me if I am wrong.

Regards
Srinath

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31 Aug 2017, 21:26
krsrinath wrote:
Bunuel wrote:
When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;
[/textarea]

Hi Bunuel,

I believe |x|=-x if x<0 and not if x≤0.

Please correct me if I am wrong.

Regards
Srinath

|0| = -0 = 0. So, what I've written is correct.
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Re: M09-22   [#permalink] 31 Aug 2017, 21:26

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# M09-22

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