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

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

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13 May 2013, 01:17
Archit143 wrote:
I too have the same doubt...can anyone address the query

Archit

The question asks is $$x>3$$ or $$x<-3$$?

III tells us that $$x>3$$ or $$x<-1$$. So is $$x>3$$ or $$x<-3$$? YES

$$x>3$$ from question => $$x>3$$from III: Correct
$$x<-3$$ from question => $$x<-1$$ from III: Correct as well. We are asked if x<-3 and III tells us that x<-1 so for sure it will be <-3 also.

Hope it's clear now!
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Re: If |x|>3, which of the following must be true? [#permalink]

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13 May 2013, 08:44
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danzig 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 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?

If some numbers confuse you, don't fixate on them. Go ahead and take some other easier examples.
Let's keep the wording of the question same but make it simple.

If n < 6, which of the following must be true?

I.

II.

III. n < 8

Can we say that III must be true? Yes!
If n is less than 6 then obviously it is less than 8 too.
If n is less than 6, it will take values such as -20, 2, 5 etc. All of these values will be less than 8 too.

Values 6 and 7 are immaterial because n cannot take these values. You are given that n is less than 6 so you only need to worry about values that n CAN take. Those should satisfy n < 8.

Similarly, your question says that x > 3 or x < -3

Then we can say that x > 3 or x < -1. All values that will be less than -3 will be less than -1 too.

Check out my post on a similar tricky 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 Kudos [?]: 18122 [0], given: 236 Intern Joined: 15 Mar 2013 Posts: 5 Kudos [?]: [0], given: 2 Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 13 Aug 2013, 12:40 What if x = -2 that is < -1 but > than -3 so IIImust be out?no Zarrolou wrote: Archit143 wrote: I too have the same doubt...can anyone address the query Archit The question asks is $$x>3$$ or $$x<-3$$? III tells us that $$x>3$$ or $$x<-1$$. So is $$x>3$$ or $$x<-3$$? YES $$x>3$$ from question => $$x>3$$from III: Correct $$x<-3$$ from question => $$x<-1$$ from III: Correct as well. We are asked if x<-3 and III tells us that x<-1 so for sure it will be <-3 also. Hope it's clear now! Posted from GMAT ToolKit Kudos [?]: [0], given: 2 Senior Manager Joined: 10 Jul 2013 Posts: 326 Kudos [?]: 430 [0], given: 102 Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 13 Aug 2013, 13:27 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 |x| > 3 means x>3 or x<-3 so I is false because x<-3 is true. II. x^2 > 9 satisfies both equations (use x=-4 or x=5 ) III. is also true for x=-4 and x=5. so D _________________ Asif vai..... Kudos [?]: 430 [0], given: 102 Math Expert Joined: 02 Sep 2009 Posts: 42599 Kudos [?]: 135569 [0], given: 12699 Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 13 Aug 2013, 23:56 Expert's post 1 This post was BOOKMARKED rusth1 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 What if x = -2 that is < -1 but > than -3 so IIImust be out?no Zarrolou wrote: Archit143 wrote: I too have the same doubt...can anyone address the query Archit The question asks is $$x>3$$ or $$x<-3$$? III tells us that $$x>3$$ or $$x<-1$$. So is $$x>3$$ or $$x<-3$$? YES $$x>3$$ from question => $$x>3$$from III: Correct $$x<-3$$ from question => $$x<-1$$ from III: Correct as well. We are asked if x<-3 and III tells us that x<-1 so for sure it will be <-3 also. Hope it's clear now! Posted from GMAT ToolKit x cannot be -2 because we are told that |x|>3, and |-2|=2<3. Hope it helps. _________________ Kudos [?]: 135569 [0], given: 12699 Intern Joined: 17 May 2012 Posts: 45 Kudos [?]: 12 [0], given: 126 Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 23 Nov 2014, 22:46 Hi All, After going through the explanations I could understand why option B was not correct. But I am sure under timed conditions, I might make a similar mistake. Does anyone have a way to solve such problems, so that a mistake can be avoided and an important case like the 3rd option be considered while evaluating answer choices? Thanks, AK Kudos [?]: 12 [0], given: 126 Director Joined: 25 Apr 2012 Posts: 721 Kudos [?]: 872 [0], given: 724 Location: India GPA: 3.21 WE: Business Development (Other) Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 23 Nov 2014, 23:29 aj0809 wrote: Hi All, After going through the explanations I could understand why option B was not correct. But I am sure under timed conditions, I might make a similar mistake. Does anyone have a way to solve such problems, so that a mistake can be avoided and an important case like the 3rd option be considered while evaluating answer choices? Thanks, AK Hi AK, Why will you leave an option out...that's a big no...in this question St 2 is true so you can remove answer options which don't have st 2 as one of the option and see if it reduces your work load... You may want to refresh some basics on how to go about solving such questions.. This is why we need to practice and see where we are going wrong...For instance this is an important pointer for you to never to overlook an option.. Check out below link: math-absolute-value-modulus-86462.html _________________ “If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.” Kudos [?]: 872 [0], given: 724 Intern Joined: 17 May 2012 Posts: 45 Kudos [?]: 12 [0], given: 126 Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 23 Nov 2014, 23:43 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. Kudos [?]: 12 [0], given: 126 Director Joined: 25 Apr 2012 Posts: 721 Kudos [?]: 872 [0], given: 724 Location: India GPA: 3.21 WE: Business Development (Other) Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 23 Nov 2014, 23:53 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. hmm.. Did you solve the 3rd option correctly or you made it a mistake. Identify the step where you made the mistake.. Consider making an error log..That will certainly help... _________________ “If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.” Kudos [?]: 872 [0], given: 724 Intern Joined: 18 May 2014 Posts: 35 Kudos [?]: 38 [0], given: 204 GMAT 1: 680 Q49 V35 Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 22 May 2015, 06:26 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. VeritasPrepKarishma Thank you for this! I was doing 'must be true' questions wrong! Kudos [?]: 38 [0], given: 204 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7793 Kudos [?]: 18122 [0], given: 236 Location: Pune, India Re: If |x|>3, which of the following must be true? [#permalink] ### Show Tags 24 Nov 2016, 22:53 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. Responding to a pm: Quote: I couldn't understand the solution for option B. Since |x|>3 we can say that |x|-1>2 ( subtracting 2 from both sides). But how are we saying that |x|-1 is equal to |x-1|. They are not the same: |x - 1| > 2 and |x|-1 > 2 |x - 1| > 2 means x > 3 or x < -1 |x| - 1 > 2 |x| > 3 means x > 3 or x < -3 But not what is given and what is asked. We are GIVEN that |x| > 3 So we KNOW that x is either greater than 3 or it is less than -3. So valid values for x are 3.4, 4, 101, 2398675, -3.6, -5, -78 etc Now the question is: "Is |x - 1| > 2?" "Is x always either greater than 3 or less than -1?" All positive values of x are given to be greater than 3. All negative values of x are given to be less than -3. So obviously they are less than -1 too. Hence, |x - 1| > 2 is true. Helps? _________________ 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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17 May 2017, 20:05
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it is given x<-3 or x>3

so now x can not be -2, but x can be -4

choice 3 doubt is for x<-1
now tell me -4 is <-1 or -4>-1 ?
obviously -4<-1

basically, if x is less than -3 (given in the premise), so x is automatically less than -1 (x<-1)

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

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24 Jun 2017, 23:33
This is an interesting question, lets try to solve.

$$|x| > 3$$

$$-3 > x > 3$$

From this we know that value of x is not in between -3 & 3.

I. $$x > 3$$

x>3 is not always true, take x = 4 ==> True and x = -4 ==> False ======> Hence, I will be FALSE

II. $$x^2 > 9$$

As we are squaring either x > 3 or x < -3 it will give us a value > 9 =====> Hence, II will always be TRUE

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

This a tricky one.

we can solve this as

$$-2 > x - 1 > 2$$

$$-1 > x > 3$$

With the given information of x < -3 & x > 3 if you add any value in this equation x = 4 or x = -4 it will always be either > 3 or < -1.

Hence, =====> III will always be TRUE

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

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25 Jun 2017, 23:59
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

[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

Given $$|x| > 3$$,

$$x < 3$$ or $$x < -3$$

$$- 3> x < 3$$

Range of $$x$$ could be for positive integers $$4$$ and above and for negative integers $$-4$$ and below.

Checking the options we get;

I. $$x > 3$$ ------- Not always true ($$x$$ could be less than $$-3$$).

II. $$x^2 > 9$$ ------------ Must be True

Let $$x$$ be $$4$$ and $$-4$$. $$x^2$$ will be $$16$$ in both case which is greater than $$9$$. Hence II must be true.

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

$$x - 1 > 2 ==> x > 3$$ --------- (Given it must be true.)

$$-x + 1 > 2 ==> x < -1$$.

(Given all negative values of $$x$$ is less than $$-3$$ which will in turn be less than $$-1$$). Hence III must be true.

II and III. Answer (D)...

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

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29 Jul 2017, 14:15
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?

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

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06 Aug 2017, 10:32
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 .....

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

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07 Aug 2017, 20:32
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 .....

VeritasPrepKarishma
hey

according to the question stem, x can never equal to -2, as it must be less than -3. I got it. I meant according to the option 3 (answer choice 3), x can equal to -2...
It doesn't matter, however, as -2 is always less than -1 and so is -3, and hence the option # 3 holds true always...thanks I got it ...

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

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08 Aug 2017, 00:14
ssislam wrote:
VeritasPrepKarishma
hey

according to the question stem, x can never equal to -2, as it must be less than -3. I got it. I meant according to the option 3 (answer choice 3), x can equal to -2...
It doesn't matter, however, as -2 is always less than -1 and so is -3, and hence the option # 3 holds true always...thanks I got it ...

The point is - what is given and what is asked?

You are GIVEN that x is less than -3. (question stem gives you that)
You are ASKED whether x will always be less than -1 too. (this is point III. You need to find this. You are not given this.)
So x cannot be -2.
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Re: If |x|>3, which of the following must be true? [#permalink]

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08 Aug 2017, 01:23
VeritasPrepKarishma wrote:
ssislam wrote:
VeritasPrepKarishma
hey

according to the question stem, x can never equal to -2, as it must be less than -3. I got it. I meant according to the option 3 (answer choice 3), x can equal to -2...
It doesn't matter, however, as -2 is always less than -1 and so is -3, and hence the option # 3 holds true always...thanks I got it ...

The point is - what is given and what is asked?

You are GIVEN that x is less than -3. (question stem gives you that)
You are ASKED whether x will always be less than -1 too. (this is point III. You need to find this. You are not given this.)
So x cannot be -2.

hi mam

actually -2 is not pertinent here. I used this number just by the way, nothing else...
since x is less than -3 given, any value less than -3, is obviously less than -1 too, no doubt...
anyway, thanks to you, mam for your valued reply at all times...

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

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08 Aug 2017, 19:19
ssislam wrote:
VeritasPrepKarishma wrote:
ssislam wrote:
VeritasPrepKarishma
hey

according to the question stem, x can never equal to -2, as it must be less than -3. I got it. I meant according to the option 3 (answer choice 3), x can equal to -2...
It doesn't matter, however, as -2 is always less than -1 and so is -3, and hence the option # 3 holds true always...thanks I got it ...

The point is - what is given and what is asked?

You are GIVEN that x is less than -3. (question stem gives you that)
You are ASKED whether x will always be less than -1 too. (this is point III. You need to find this. You are not given this.)
So x cannot be -2.

hi mam

actually -2 is not pertinent here. I used this number just by the way, nothing else...
since x is less than -3 given, any value less than -3, is obviously less than -1 too, no doubt...
anyway, thanks to you, mam for your valued reply at all times...

Possible value of x = -5, -6 -4, 4, 5, 6

When x = (-4), |-4 -1| = 5 > 2
When x = 4, |4 - 1| = 3 > 2

III must be TRUE.
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Re: If |x|>3, which of the following must be true?   [#permalink] 08 Aug 2017, 19:19

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