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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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New post 14 Aug 2013, 00:56
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!


Image Posted from GMAT ToolKit


x cannot be -2 because we are told that |x|>3, and |-2|=2<3.

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

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New post 23 Nov 2014, 23:46
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
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Re: If |x|>3, which of the following must be true?  [#permalink]

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New post 24 Nov 2014, 00:29
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
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Re: If |x|>3, which of the following must be true?  [#permalink]

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New post 24 Nov 2014, 00:43
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.
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Re: If |x|>3, which of the following must be true?  [#permalink]

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New post 24 Nov 2014, 00:53
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...
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Re: If |x|>3, which of the following must be true?  [#permalink]

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New post 24 Nov 2014, 20:36
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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.
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Re: If |x|>3, which of the following must be true?  [#permalink]

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New post 22 May 2015, 07:26
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! :)
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Re: If |x|>3, which of the following must be true?  [#permalink]

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New post 24 Nov 2016, 23:53
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?
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Re: If |x|>3, which of the following must be true?  [#permalink]

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New post 17 May 2017, 21:05
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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New post 25 Jun 2017, 00:33
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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New post 26 Jun 2017, 00:59
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

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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New post 29 Jul 2017, 15:15
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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New post 01 Aug 2017, 02:24
1
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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Re: If |x|>3, which of the following must be true?  [#permalink]

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New post 06 Aug 2017, 11:32
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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New post 07 Aug 2017, 02:41
1
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.
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Re: If |x|>3, which of the following must be true?  [#permalink]

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New post 07 Aug 2017, 21:32
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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New post 08 Aug 2017, 01:14
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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New post 08 Aug 2017, 02:23
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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New post 08 Aug 2017, 20:19
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]

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New post 07 Dec 2017, 17:51
2
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


|x| > 3 is equivalent to x > 3 OR x < -3. Let’s now examine each answer choice.

1) x > 3

Since x > 3 or x < -3, x does not have to be greater than 3. Answer choice 1 is not necessarily true.

2) x^2 > 9

If x > 3, then x^2 > 9, and if x < -3, then x^2 > 9. Answer choice 2 is true.

3) |x - 1| > 2

Recall that |x| > 3 is equivalent to x > 3 or x < -3. Let’s examine each of these two cases for the given inequality.

Case 1: x > 3

If x > 3, then x - 1 > 2. If a quantity is greater than a positive number, its absolute value is also greater than that positive number. Thus, |x - 1| > 2.

Case 2: x < -3

If x < -3, then x - 1 < -4. If a quantity is less than a negative number, its absolute value is actually greater than the absolute value of the negative number.

Thus |x - 1| > |-4|, i.e., |x - 1| > 4. Since |x - 1| > 4, that means |x - 1| > 2.

Therefore, we see that in either case, |x - 1| > 2. Answer choice 3 is also true.

Answer: D
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Re: If |x|>3, which of the following must be true? &nbs [#permalink] 07 Dec 2017, 17:51

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