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# M09-19

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Math Expert
Joined: 02 Sep 2009
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16 Sep 2014, 00:40
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25% (medium)

Question Stats:

70% (01:23) correct 30% (01:16) wrong based on 125 sessions

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Is $$|x - 5| \gt 4$$?

(1) $$x^2-4 \lt 0$$

(2) $$x^2-1 \lt 0$$

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Joined: 02 Sep 2009
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16 Sep 2014, 00:40
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Official Solution:

Is $$|x - 5| \gt 4$$?

Is $$|x-5| \gt 4$$? Opening the absolute value we get the question: is $$x \lt 1$$ or $$x \gt 9$$?

(1) $$x^2-4 \lt 0$$ and thus $$x^2 \lt 4$$ therefore $$|x| \lt 2$$ and we can deduce that $$-2 < x < 2$$, so we can have an YES answer as well as a NO answer (consider $$x=0$$ and $$x=1.5$$). Not sufficient.

(2) $$x^2-1 \lt 0$$ thus $$x^2 \lt 1$$ therefore $$|x| \lt 1$$ and finally, $$-1 \lt x \lt 1$$. Sufficient.

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12 Oct 2014, 11:36
Bunuel wrote:
Official Solution:

Is $$|x-5| \gt 4$$? Opening the absolute value we get the question: is $$x \lt 1$$ or $$x \gt 9$$?

(1) $$x^2-4 \gt 0$$ and thus $$x^2 \gt 4$$ therefore $$|x| \gt 2$$ and we can deduct that $$x \lt -2$$ or $$x \gt 2$$, so we can have an YES answer as well as a NO answer (consider $$x=-5$$ and $$x=5$$). Not sufficient.

(2) $$x^2-1 \lt 0$$ thus $$x^2 \lt 1$$ therefore $$|x| \lt 1$$ and finally, $$-1 \lt x \lt 1$$. Sufficient.

Hi Bunuel,
I have a doubt in inequality with absolute value questions where we are asked to find if the given statement satisfies the two required inequalities for instance in the above question:
we are required to find if $$x<1 or x >9$$, so as per the Second statement we get $$-1<x<1$$ so is this sufficient to answer the question?

Because $$x>9$$ or not is still left unattended.
SO what exactly is the approach in such questions?

Regards
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Joined: 02 Sep 2009
Posts: 47070

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12 Oct 2014, 11:48
earnit wrote:
Bunuel wrote:
Official Solution:

Is $$|x-5| \gt 4$$? Opening the absolute value we get the question: is $$x \lt 1$$ or $$x \gt 9$$?

(1) $$x^2-4 \gt 0$$ and thus $$x^2 \gt 4$$ therefore $$|x| \gt 2$$ and we can deduct that $$x \lt -2$$ or $$x \gt 2$$, so we can have an YES answer as well as a NO answer (consider $$x=-5$$ and $$x=5$$). Not sufficient.

(2) $$x^2-1 \lt 0$$ thus $$x^2 \lt 1$$ therefore $$|x| \lt 1$$ and finally, $$-1 \lt x \lt 1$$. Sufficient.

Hi Bunuel,
I have a doubt in inequality with absolute value questions where we are asked to find if the given statement satisfies the two required inequalities for instance in the above question:
we are required to find if $$x<1 or x >9$$, so as per the Second statement we get $$-1<x<1$$ so is this sufficient to answer the question?

Because $$x>9$$ or not is still left unattended.
SO what exactly is the approach in such questions?

Regards

The question asks whether x < 1 or x > 9 (those are two ranges which make |x - 5| > 4 to hold true).

(2) says that -1 < x < 1, so x IS less than 1, and therefore |x - 5| > 4 to holds true.
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12 Oct 2014, 11:54
Bunuel wrote:
earnit wrote:
Bunuel wrote:
Official Solution:

Is $$|x-5| \gt 4$$? Opening the absolute value we get the question: is $$x \lt 1$$ or $$x \gt 9$$?

(1) $$x^2-4 \gt 0$$ and thus $$x^2 \gt 4$$ therefore $$|x| \gt 2$$ and we can deduct that $$x \lt -2$$ or $$x \gt 2$$, so we can have an YES answer as well as a NO answer (consider $$x=-5$$ and $$x=5$$). Not sufficient.

(2) $$x^2-1 \lt 0$$ thus $$x^2 \lt 1$$ therefore $$|x| \lt 1$$ and finally, $$-1 \lt x \lt 1$$. Sufficient.

Hi Bunuel,
I have a doubt in inequality with absolute value questions where we are asked to find if the given statement satisfies the two required inequalities for instance in the above question:
we are required to find if $$x<1 or x >9$$, so as per the Second statement we get $$-1<x<1$$ so is this sufficient to answer the question?

Because $$x>9$$ or not is still left unattended.
SO what exactly is the approach in such questions?

Regards

The question asks whether x < 1 or x > 9 (those are two ranges which make |x - 5| > 4 to hold true).

(2) says that -1 < x < 1, so x IS less than 1, and therefore |x - 5| > 4 to holds true.

So as basically if either of the range is satisfied by the given statements conclusively, then we do not have to worry about the other range ie x>9 ?
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12 Oct 2014, 12:03
Yes, because x from either of the ranges gives an yes answer to the question.
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12 Oct 2014, 12:08
Got it. Thank you.

If possible, could we find similar questions with two ranges like above question and two statements that proves their sufficiency?
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12 Oct 2014, 12:09
earnit wrote:
Got it. Thank you.

If possible, could we find similar questions with two ranges like above question and two statements that proves their sufficiency?

Please go through inequalities questions in out questions bank: viewforumtags.php
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24 Mar 2016, 06:27
While the Algebra approach is perfectly fine, I was just wondering whether a Plug-In approach (picking up numbers & then matching with the inequalities) could be used in such cases or not. Is it going to be more confusing or more time confusing than solving the three given inequalities?
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05 Apr 2016, 06:25
Bunuel wrote:
Official Solution:

Is $$|x-5| \gt 4$$? Opening the absolute value we get the question: is $$x \lt 1$$ or $$x \gt 9$$?

(1) $$x^2-4 \gt 0$$ and thus $$x^2 \gt 4$$ therefore $$|x| \gt 2$$ and we can deduct that $$x \lt -2$$ or $$x \gt 2$$, so we can have an YES answer as well as a NO answer (consider $$x=-5$$ and $$x=5$$). Not sufficient.

(2) $$x^2-1 \lt 0$$ thus $$x^2 \lt 1$$ therefore $$|x| \lt 1$$ and finally, $$-1 \lt x \lt 1$$. Sufficient.

Dear Bunuel,
I solved this question using algebra and got the question right.
But there is a discrepancy in the result of x between your solution and mine.

X^2-4>0 can be written as (x-2)(x+2)>0
therefore x>2 or X>-2,
But if I write mod X>2
then X>2
or -X >2 ==> X<-2

How can the same equation have two different solutions ?
Thanks
Teja
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Joined: 02 Sep 2009
Posts: 47070

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05 Apr 2016, 06:40
evst wrote:
Bunuel wrote:
Official Solution:

Is $$|x-5| \gt 4$$? Opening the absolute value we get the question: is $$x \lt 1$$ or $$x \gt 9$$?

(1) $$x^2-4 \gt 0$$ and thus $$x^2 \gt 4$$ therefore $$|x| \gt 2$$ and we can deduct that $$x \lt -2$$ or $$x \gt 2$$, so we can have an YES answer as well as a NO answer (consider $$x=-5$$ and $$x=5$$). Not sufficient.

(2) $$x^2-1 \lt 0$$ thus $$x^2 \lt 1$$ therefore $$|x| \lt 1$$ and finally, $$-1 \lt x \lt 1$$. Sufficient.

Dear Bunuel,
I solved this question using algebra and got the question right.
But there is a discrepancy in the result of x between your solution and mine.

X^2-4>0 can be written as (x-2)(x+2)>0
therefore x>2 or X>-2,But if I write mod X>2
then X>2
or -X >2 ==> X<-2

How can the same equation have two different solutions ?
Thanks
Teja

How did you get x>2 or x>-2? It does not make sense.
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20 May 2016, 08:54
Bunuel wrote:
Official Solution:

Is $$|x-5| \gt 4$$? Opening the absolute value we get the question: is $$x \lt 1$$ or $$x \gt 9$$?

Can you explain "opening the absolute value"? I get what you mean in the end result as I can see if you add 5 to the other side you get 9 and if you subtract 5 you get -1 (how'd it become a positive 1?). But how did you know which one to flip the sign on?

I imagine this is much easier done than said.
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20 May 2016, 10:08
1
redfield wrote:
Bunuel wrote:
Official Solution:

Is $$|x-5| \gt 4$$? Opening the absolute value we get the question: is $$x \lt 1$$ or $$x \gt 9$$?

Can you explain "opening the absolute value"? I get what you mean in the end result as I can see if you add 5 to the other side you get 9 and if you subtract 5 you get -1 (how'd it become a positive 1?). But how did you know which one to flip the sign on?

I imagine this is much easier done than said.

hi,
there are two ways ..
1) open the MOD..
$$|x-5| \gt 4$$..
first time let x-5 be +ive ... so $$x-5>4.....................x>9...$$
second time take x-5 to be -ive... so $$-(x-5)>4...$$ Multiply both sides by '-', the > sign will change........$$.....x-5<4..... x<1.$$
2) critical method...
here it will amount to same as above
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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20 May 2016, 10:11
chetan2u wrote:
redfield wrote:
Bunuel wrote:
Official Solution:

Is $$|x-5| \gt 4$$? Opening the absolute value we get the question: is $$x \lt 1$$ or $$x \gt 9$$?

Can you explain "opening the absolute value"? I get what you mean in the end result as I can see if you add 5 to the other side you get 9 and if you subtract 5 you get -1 (how'd it become a positive 1?). But how did you know which one to flip the sign on?

I imagine this is much easier done than said.

hi,
there are two ways ..
1) open the MOD..
$$|x-5| \gt 4$$..
first time let x-5 be +ive ... so $$x-5>4.....................x>9...$$
second time take x-5 to be -ive... so $$-(x-5)>4...$$ Multiply both sides by '-', the > sign will change........$$.....x-5<4..... x<1.$$
2) critical method...
here it will amount to same as above

For more check the links below:

Theory on Absolute Values: math-absolute-value-modulus-86462.html
The E-GMAT Question Series on ABSOLUTE VALUE: the-e-gmat-question-series-on-absolute-value-198503.html
Properties of Absolute Values on the GMAT: properties-of-absolute-values-on-the-gmat-191317.html
Absolute Value: Tips and hints: absolute-value-tips-and-hints-175002.html

DS Absolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Absolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Absolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

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28 Dec 2016, 21:39
1
I was of the impression that in DS questions the two statements provided cannot contradict each other. Is that true? If yes, aren't the two statements contradicting each other in this question?
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30 Dec 2016, 02:02
CircleOfLife wrote:
I was of the impression that in DS questions the two statements provided cannot contradict each other. Is that true? If yes, aren't the two statements contradicting each other in this question?

Yes, you are absolutely correct. Edited the question. Thank you.
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31 May 2017, 02:09
Bunuel I'm not sure I understand - if for statement 2 I plug in x=0,
we have on one hand x-5=0-5=-5 > 4? No.
But |x-5| also means -(x-5), correct? Then if I plug in x=0 here, it becomes -(0-5)=5>4? Yes. We get 2 different answers with x=0 if we go with statement 2. Where am I going wrong? Thanks.
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31 May 2017, 02:14
1
bianalyst wrote:
Bunuel I'm not sure I understand - if for statement 2 I plug in x=0,
we have on one hand x-5=0-5=-5 > 4? No.
But |x-5| also means -(x-5), correct? Then if I plug in x=0 here, it becomes -(0-5)=5>4? Yes. We get 2 different answers with x=0 if we go with statement 2. Where am I going wrong? Thanks.

First of all an absolute value is ALWAYS non-negative, again it CANNOT be negative.

So, |x - 5|, no matter what x is will be positive or 0. It'll be 0 if x = 5 and positive for ANY other values of x.

If x = 0, the |x - 5| = |0 - 5| = |-5| = 5.

Hope it's clear.
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03 Jun 2017, 09:46
For problem-related to mod, keep in mind if |a|>|b| than a^2>b^2
X2-10x+9>0 which will be rewritten as (x-9)(x-1)>0
Solution set for this equation is (-∞,1) U (9,∞)

Statement 1 says (x-2)(x+2)<0 which means (-2<x<2), and within this solution set stem will be positive as well as negative, NS

Statement 2 says (x-1)(x+1)<0 which means (-1<x<1), and within this solution set stem only be positive only. Sufficient

Hence B is the answer. Hope I am right and it's clear.
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14 Nov 2017, 01:55
Hi Bunuel,

Many thanks for this great question. I got the solution, the way you showed it. Only today, while I was attempting another question from the GMAT Club tests, I stumbled upon the sign method for solving inequalities. I was just wondering if you or someone else could kindly explain how to solve THIS question using the sign method. Many thanks!
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M09-19   [#permalink] 14 Nov 2017, 01:55

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