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# If |x|>3, which of the following must be true?

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If |x|>3, which of the following must be true? [#permalink]

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10 Sep 2012, 02:56
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If |x| > 3, which of the following must be true?

I. x > 3

II. x^2 > 9

III. |x - 1| > 2

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

[Reveal] Spoiler:
I can't understand how the official answer can be right. For me is B. Please respond and I'll provide official explanation! Thanks
[Reveal] Spoiler: OA

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Re: If |x|>3, which of the following must be true? [#permalink]

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03 Feb 2013, 21:36
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corvinis wrote:
If |x| > 3, which of the following must be true?

I. x > 3

II. X^2 > 9

III. |x-1|>2

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

I can't understand how the official answer can be right. For me is B. Please respond and I'll provide official explanation! Thanks

Responding to a pm:

|x| > 3 implies that x is a point whose distance from 0 is more than 3. So x could be greater than 3 or less than -3. Before you move further, think about the values x can take: 3.00001, 3.5, 4.2, 5.7, 67, 1000, -3.45, -4, -8, -100 etc. The only values it cannot take are -3 <= x <= 3

Which of the following must be true?

I. x > 3

For every value that x can take, must x be greater than 3? No. e.g. if x takes -3.45, -4 etc, it will not be greater than 3 so this is not true.

II. X^2 > 9
This is the same as |x| > 3 so it must be true

III. |x-1|>2
This implies that the distance of x from 1 must be greater than 2. So x is either greater than 3 or less than -1. Now, recall all the values that x can take. For each value, can be say that x is either greater than 3 or less than -1? Yes.
3.00001 - x is greater than 3
3.5 : x is greater than 3
4.2 : x is greater than 3
5.7 : x is greater than 3
67 : x is greater than 3
1000 : x is greater than 3
-3.45 : x is less than -1
-4 : x is less than -1
-8 : x is less than -1
-100 : x is less than -1

For every value that x can take, x will be either greater than 3 or less than -1. Note that we are not saying that every value less than -1 must be valid for x. We are saying that every value that is valid for x (found by using |x| > 3) will be either greater than 3 or less than -1. Hence |x-1|>2 must be true for every value that x can take.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17823 [15], given: 235 Senior Manager Joined: 24 Aug 2009 Posts: 497 Kudos [?]: 867 [6], given: 276 Schools: Harvard, Columbia, Stern, Booth, LSB, Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 10 Sep 2012, 03:17 6 This post received KUDOS 5 This post was BOOKMARKED corvinis wrote: If |x| > 3, which of the following must be true? I. x > 3 II. X^2 > 9 III. |x-1|>2 A. I only B. II only C. I and II only D. II and III only E. I, II, and III I can't understand how the official answer can be right. For me is B. Please respond and I'll provide official explanation! Thanks Original statement |x| > 3, which means either x>3 or x<-3 Now Check the options Option1- x > 3 - not always true as x can be smaller than -3 Thus option A,C & E is ruled out. Only B & D are left Option2- X^2 > 9 - Always true for x>3 or x<-3 To check - if x = 4,5,6,7.... or -4,-5,-6,-7 x^2>9 Option3-|x-1|>2, which means (x-1)>2 ---> x>3 (if x-1>0) - True it also means (x-1)<-2---->x<-1 (if x-1<0) X<-1 satisfies x<-3. Thus true Both 2 & 3 is true Thus Answer D Hope it helps _________________ If you like my Question/Explanation or the contribution, Kindly appreciate by pressing KUDOS. Kudos always maximizes GMATCLUB worth -Game Theory If you have any question regarding my post, kindly pm me or else I won't be able to reply Kudos [?]: 867 [6], given: 276 Manager Joined: 24 Jul 2011 Posts: 75 Kudos [?]: 155 [4], given: 5 Location: India Concentration: Strategy, General Management GMAT 1: 670 Q49 V33 WE: Asset Management (Manufacturing) Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 10 Sep 2012, 08:10 4 This post received KUDOS 2 This post was BOOKMARKED corvinis wrote: If |x| > 3, which of the following must be true? I. x > 3 II. X^2 > 9 III. |x-1|>2 A. I only B. II only C. I and II only D. II and III only E. I, II, and III I can't understand how the official answer can be right. For me is B. Please respond and I'll provide official explanation! Thanks Let's solve this question differently Here following is given I x I >3 So, Either x>3 OR x<-3 If we plot the values in line we get following - ----(-4)----(-3)-----(-2)-----(-1)-----(0)------(1)-------(2)-------(3)------(4) <------------I-----------------------------------------------------------I------------> (x<-3)...............................................................................(x>3) Now, let us take some value (-4 and 4) and plug into the options. Option I with x=4 : $$4> 3$$ Correct Option I with x=-4 $$-4>3$$ Not correct. So, options I is out. Option II with x=4 : $$4^2> 9$$ Correct Option II with x=-4 $$(-4)^2>3$$ Correct. So, options II is right Option III with x=4 : $$I 4-1I > 2$$ Correct Option III with x=-4 $$I (-4-1) I >2$$ Correct. So, options III is right Hence, correct answer is D _________________ My mantra for cracking GMAT: Everyone has inborn talent, however those who complement it with hard work we call them 'talented'. +1 Kudos = Thank You Dear Are you saying thank you? Kudos [?]: 155 [4], given: 5 Verbal Forum Moderator Joined: 10 Oct 2012 Posts: 627 Kudos [?]: 1387 [4], given: 136 Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 05 Apr 2013, 01:42 4 This post received KUDOS 1 This post was BOOKMARKED payals66 wrote: i am clearly unclear how option 3 is true, since if i take the value of x as -2. It holds true for |x-1|>2. However the value of -2 does not hold true for |x| > 3. Please clarify. The question reads "If |x| > 3, which of the following must be true?". The value of x=-2 doesn't subscribe to this condition in the first place. You have to filter all the possible values for x, BASED on the condition given in the problem. _________________ Kudos [?]: 1387 [4], given: 136 Intern Joined: 26 May 2012 Posts: 5 Kudos [?]: 81 [2], given: 1 Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 10 Sep 2012, 05:25 2 This post received KUDOS carcass wrote: corvinis wrote: carcass wrote: I'm ok untill B and D But the 3 option says: |x - 1| > 2. Then x > 3 AND x< -1. Our statement says: x > 3 AND x < -3. So why is always TRUE?? Please clarify From|x|>3, x will always have to be >3 or <-3, so "our" x will never be between -3 and 3, making the III statement always true. I agree but on one hand if "our" X is > 3 could be 3.1 or 200 but if "our" X is < -3 could be -3.1 or -200 so X < -1 could be -2 or -2.5 or -300. This is what I meant. So where is the flaw in my reasoning ?? thanks lol. you're doing the same error in reasoning I was doing. You have to prove statement III as true an not the other way around (prove |x|>3 true). If you know that your x will always be more than 3 OR always less than -3 (from |x|>3) then you are restricted to these values when looking to prove statement III as true. Hope you understand. Edit: Upppss..You got it. Cheers. Kudos [?]: 81 [2], given: 1 Current Student Joined: 26 Jul 2012 Posts: 63 Kudos [?]: 14 [2], given: 8 Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 11 Nov 2012, 17:43 2 This post received KUDOS 1 This post was BOOKMARKED I just came across this one and it is clear that this is a tough cookie. Given that |x| > 3, so that means x > 3 or x < -3. We can also test values here, only 3,4,5,6 or -3, -4, -5 etc work. Now, let's tackle the statements: I. x > 3 We know that x > 3 or x < -3, but it is not ALWAYS the case that x > 3. We do have values of x that are less than 3, i.e., when x is -4 or -5. So, I is false. II. x^2 > 9. This means x>3 or x < -3. That is what we have above so this is golden. True. III. |x - 1| > 2 When x > 0 x -1 > 2 x > 3 x < 0 -x + 1 > 2 -x > 1 x < -1 So , III says IF x > 3 or x < -3, THEN x > 3 OR x < -1. True. Test values to prove this x > 3 or x < -3 means x can be -4 and that hits x < -1 Kudos [?]: 14 [2], given: 8 Board of Directors Joined: 01 Sep 2010 Posts: 3378 Kudos [?]: 9290 [1], given: 1169 Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 10 Sep 2012, 04:41 1 This post received KUDOS I'm ok untill B and D But the 3 option says: |x - 1| > 2. Then x > 3 AND x< -1. Our statement says: x > 3 AND x < -3. So why is always TRUE?? Please clarify _________________ Kudos [?]: 9290 [1], given: 1169 Intern Joined: 26 May 2012 Posts: 5 Kudos [?]: 81 [1], given: 1 Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 10 Sep 2012, 05:04 1 This post received KUDOS carcass wrote: I'm ok untill B and D But the 3 option says: |x - 1| > 2. Then x > 3 AND x< -1. Our statement says: x > 3 AND x < -3. So why is always TRUE?? Please clarify From|x|>3, x will always have to be >3 or <-3, so "our" x will never be between -3 and 3, making the III statement always true. Kudos [?]: 81 [1], given: 1 Retired Moderator Joined: 05 Jul 2006 Posts: 1749 Kudos [?]: 443 [1], given: 49 Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 27 Apr 2013, 08:57 1 This post received KUDOS if x<-3 then sure x<-1 .... Kudos [?]: 443 [1], given: 49 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7738 Kudos [?]: 17823 [1], given: 235 Location: Pune, India Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 24 Nov 2014, 20:36 1 This post received KUDOS Expert's post 1 This post was BOOKMARKED aj0809 wrote: Thanks WondedTiger, Maybe I didn't present my question correctly. I didn't leave the 3rd option but came to the conclusion that it was wrong and chose my answer as B. I just want to prevent that in timed conditions for difficult questions like these, which have subtle differences that makes an answer choice right. Try using the number line for inequalities and absolute values. |x| > 3 means distance of x from 0 is more than 3. So x is either greater than 3 or less than -3. So on the number line, it looks like this: ___________-3________0________3____________ The red part is the range where x will lie. Is |x-1| > 2? |x-1| > 2 represents that distance of x from 1 is more than 2. So x is either greater than 3 or less than -1. Is x either greater than 3 or less than -1? ___________-3________0________3____________ All points on the red lines satisfy this. They are either greater than 3 or less than -1. Hence option III must be true. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: If |x|>3, which of the following must be true? [#permalink]

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01 Aug 2017, 02:24
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Expert's post
rma26 wrote:
VeritasPrepKarishma wrote:
corvinis wrote:
If |x| > 3, which of the following must be true?

I. x > 3

II. X^2 > 9

III. |x-1|>2

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

I can't understand how the official answer can be right. For me is B. Please respond and I'll provide official explanation! Thanks

Responding to a pm:

|x| > 3 implies that x is a point whose distance from 0 is more than 3. So x could be greater than 3 or less than -3. Before you move further, think about the values x can take: 3.00001, 3.5, 4.2, 5.7, 67, 1000, -3.45, -4, -8, -100 etc. The only values it cannot take are -3 <= x <= 3

Which of the following must be true?

I. x > 3

For every value that x can take, must x be greater than 3? No. e.g. if x takes -3.45, -4 etc, it will not be greater than 3 so this is not true.

II. X^2 > 9
This is the same as |x| > 3 so it must be true

III. |x-1|>2
This implies that the distance of x from 1 must be greater than 2. So x is either greater than 3 or less than -1. Now, recall all the values that x can take. For each value, can be say that x is either greater than 3 or less than -1? Yes.
3.00001 - x is greater than 3
3.5 : x is greater than 3
4.2 : x is greater than 3
5.7 : x is greater than 3
67 : x is greater than 3
1000 : x is greater than 3
-3.45 : x is less than -1
-4 : x is less than -1
-8 : x is less than -1
-100 : x is less than -1

For every value that x can take, x will be either greater than 3 or less than -1. Note that we are not saying that every value less than -1 must be valid for x. We are saying that every value that is valid for x (found by using |x| > 3) will be either greater than 3 or less than -1. Hence |x-1|>2 must be true for every value that x can take.

So if there were 4th option saying x is integer or that x is real numbers , that would have been true also?

No. x is an integer will not always hold. We are given that |x| > 3. This means x > 3 or x < -3.
So x could be 4 or 5.6 or -3.87 or -100 etc. It is not necessary that x will be an integer.

But yes, x will be a real number because all these values are real numbers.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17823 [1], given: 235 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7738 Kudos [?]: 17823 [1], given: 235 Location: Pune, India Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 07 Aug 2017, 02:41 1 This post received KUDOS Expert's post ssislam wrote: VeritasPrepKarishma wrote: corvinis wrote: If |x| > 3, which of the following must be true? I. x > 3 II. X^2 > 9 III. |x-1|>2 A. I only B. II only C. I and II only D. II and III only E. I, II, and III I can't understand how the official answer can be right. For me is B. Please respond and I'll provide official explanation! Thanks Responding to a pm: |x| > 3 implies that x is a point whose distance from 0 is more than 3. So x could be greater than 3 or less than -3. Before you move further, think about the values x can take: 3.00001, 3.5, 4.2, 5.7, 67, 1000, -3.45, -4, -8, -100 etc. The only values it cannot take are -3 <= x <= 3 Which of the following must be true? I. x > 3 For every value that x can take, must x be greater than 3? No. e.g. if x takes -3.45, -4 etc, it will not be greater than 3 so this is not true. II. X^2 > 9 This is the same as |x| > 3 so it must be true III. |x-1|>2 This implies that the distance of x from 1 must be greater than 2. So x is either greater than 3 or less than -1. Now, recall all the values that x can take. For each value, can be say that x is either greater than 3 or less than -1? Yes. 3.00001 - x is greater than 3 3.5 : x is greater than 3 4.2 : x is greater than 3 5.7 : x is greater than 3 67 : x is greater than 3 1000 : x is greater than 3 -3.45 : x is less than -1 -4 : x is less than -1 -8 : x is less than -1 -100 : x is less than -1 For every value that x can take, x will be either greater than 3 or less than -1. Note that we are not saying that every value less than -1 must be valid for x. We are saying that every value that is valid for x (found by using |x| > 3) will be either greater than 3 or less than -1. Hence |x-1|>2 must be true for every value that x can take. hi very beautiful explanation ....I must say .. anyway ...I quote you " Note that we are not saying that every value less than -1 must be valid for x. We are saying that every value that is valid for x (found by using |x| > 3) will be either greater than 3 or less than -1. Hence |x-1|>2 must be true for every value that x can take." For option # 3 III. |x-1|>2 we got "-1>x>3" but the question stem says "-3>x>3" So, according to the question stem, if we suppose x = -4, and according to the option #3, if you suppose x = -2, then these two values may not match, but the range provided by the option # 3, that is "-1>x>3" will cover the range provided by the question stem that is "-3>x>3", because -4 is certainly less than -1 and so is x... please correct me if I am missing something you meant... thanks in advance ..... Note that -1>x>3 is not correct. x cannot be less than -1 and greater than 3 at the same time. It should be x < -1 OR x > 3 The question stem tells us that x > 3 OR x < -3. So every valid value of x is either greater than 3 OR less than -3. Now, every valid value that is greater than 3 is of course greater than 3. Every valid value that is less than -3 (e.g. -4, -5.67 etc) is obviously less than -1 too. x cannot be - 2 since it must be less than -3. That is why option III is always true. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: If |x|>3, which of the following must be true? [#permalink]

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10 Sep 2012, 03:54
fameatop wrote:
Option3-|x-1|>2,
which means (x-1)>2 ---> x>3 (if x-1>0) - True
it also means (x-1)<-2---->x<-1 (if x-1<0)

X<-1 satisfies x<-3. Thus true

Both 2 & 3 is true

Hope it helps

Now if I take x=-2 will it satisfy the condition?
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Re: If |x|>3, which of the following must be true? [#permalink]

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10 Sep 2012, 05:16
corvinis wrote:
carcass wrote:
I'm ok untill B and D

But the 3 option says: |x - 1| > 2. Then x > 3 AND x< -1. Our statement says: x > 3 AND x < -3. So why is always TRUE??

From|x|>3, x will always have to be >3 or <-3, so "our" x will never be between -3 and 3, making the III statement always true.

Ok ok X<- 1 is not possible because our stetement says that X MUST < -3 ...........so X < -1 is not possible. Is This ?????

Silly misunderstood. I apologize

Thanks
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Re: If |x|>3, which of the following must be true? [#permalink]

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10 Sep 2012, 08:19
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premnath wrote:
corvinis wrote:
If |x| > 3, which of the following must be true?

I. x > 3

II. X^2 > 9

III. |x-1|>2

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

I can't understand how the official answer can be right. For me is B. Please respond and I'll provide official explanation! Thanks

Let's solve this question differently

Here following is given
I x I >3

So, Either x>3 OR x<-3
If we plot the values in line we get following -

----(-4)----(-3)-----(-2)-----(-1)-----(0)------(1)-------(2)-------(3)------(4)
<------------I-----------------------------------------------------------I------------>
(x<-3)...............................................................................(x>3)

Now, let us take some value (-4 and 4) and plug into the options.

Option I with x=4 : $$4> 3$$ Correct
Option I with x=-4 $$-4>3$$ Not correct. So, options I is out.

Option II with x=4 : $$4^2> 9$$ Correct
Option II with x=-4 $$(-4)^2>3$$ Correct. So, options II is right

Option III with x=4 : $$I 4-1I > 2$$ Correct
Option III with x=-4 $$I (-4-1) I >2$$ Correct. So, options III is right

Good explanation thanks . generally I do not this silly but crucial error but this time I could have sworn tha D was false. Indeed, III was tricky especially if I think under time pressure. The latter changes all things, all prespective.

@Corvinis
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Re: If |x|>3, which of the following must be true? [#permalink]

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10 Sep 2012, 09:37
corvinis wrote:
If |x| > 3, which of the following must be true?

I. x > 3

II. X^2 > 9

III. |x-1|>2

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

I can't understand how the official answer can be right. For me is B. Please respond and I'll provide official explanation! Thanks

You are confusing the question with what is the range of X
We dont have to find the range of x, the question says that distance of x from 0 is more than 3

Satement 1 says x is always positive ... ot always true x can be negative and still the distance from zero will be more than 3

Statement 2 x^2 is more than 9 ofcourse ! is distance of x from 0 is more than 3 than its square will be more than 9 always

Statement 3 distance of x from 1 iss more than 2 .. Of course ! if its distance from 0 is more than three than its distance from 1 will be always more than 3-1 = 2
Weather it is negative or positive
as if it is negative, than distance from 0 is already more than 3, and you are taking distance from 1 so it will be 1 unit more than that from 0 that is 3 + 1 and hence will surely be more than 2

Here these problems are better solved by converting expression to their definition.

Important point is you are given the value of X and you have to see, what all is ture for all those values of X not ! which values of X satisfy the expression mentioned in questions.

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Re: If |x|>3, which of the following must be true? [#permalink]

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20 Nov 2012, 08:46
Please check the trick to solve this in 1 min.

inequalities-trick-91482-40.html#p1145743
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Re: If |x|>3, which of the following must be true? [#permalink]

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05 Apr 2013, 00:24
i am clearly unclear how option 3 is true, since if i take the value of x as -2. It holds true for |x-1|>2. However the value of -2 does not hold true for |x| > 3. Please clarify.

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Re: If |x|>3, which of the following must be true? [#permalink]

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27 Apr 2013, 08:45
If $$|x| > 3$$ , which of the following must be true?

I. $$x > 3$$

II. $$x^2 > 9$$

III. $$|x - 1| > 2$$

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

I don't understand well III. $$|x - 1| > 2$$ is equivalent to $$x > 3$$ or $$x < -1$$. The last inequality ($$x < -1$$ ) includes integers -2 and -3, integers that are not included in one of the original inequalities ( $$x < -3$$ ). How could III be true?

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Re: If |x|>3, which of the following must be true?   [#permalink] 27 Apr 2013, 08:45

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