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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] If |x| > 3, which of the following must be true?   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] If |x| > 3, which of the following must be true?   08 Aug 2017, 01:14
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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] If |x| > 3, which of the following must be true?   07 Mar 2018, 16:47
Expert Reply
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

Show 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

Main Idea: We need to use the normal expressions corresponding to the modulus expression given and solve for x and compare with the choices given for truth of the choices.

Details: By topic specific problem solving aid , which is :

- For converting the modulus to the normal form, one would be just remove the modulus and the other would be to remove the modulus and add a negative sign before the expression. For example |x-3| can correspond to (x-3) or –(x-3).
 , we have

X >3 or x<-3

Checking the choices we see:

(i) Need not be true
(ii) must be true
(iii) implies x-1>2 => x>3 or –(x-1) >2 => x<-1. This is consistent with what is given. Hence true
Hence D.
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Re: If |x| > 3, which of the following must be true? [#permalink] If |x| > 3, which of the following must be true?   15 Aug 2022, 00:39
KarishmaB 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 KarishmaB, I have a few doubts regarding this question and other it's variations.

I understood that for |x| > 3 to be true x <-3 or x> 3 i.e. x = -3.1 , -4 , 4 , 4.1. and so on, but not 3,-3, -2, -1 etc.

Now, in (I) x > 3, why is this not a must be true statement?. How do we differentiate between must be / could be true where we have 2 applicable ranges for the question stem.
Per my understanding, x>3 will always be true for the given question stem because whatever value x>3 gives, it will always satisfy |x| > 3

In (III) we get the range for |x-1| > 2, as x <-1 or x>3,
x>3 satisfies the question stem, but I don't see how x <-1 satisfies it.
What if x takes the value -2 ? in that case |-2| >/ 3.
How is this a must be true scenario. All the values from -1 to -3 inclusive will fail the question stem.

Could you please point out where I'm going wrong? I'm almost always incorrect on questions of this form.
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Re: If |x| > 3, which of the following must be true? [#permalink] If |x| > 3, which of the following must be true?   15 Aug 2022, 23:53
Expert Reply
KartikSingh09 wrote:
KarishmaB 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 KarishmaB, I have a few doubts regarding this question and other it's variations.

I understood that for |x| > 3 to be true x <-3 or x> 3 i.e. x = -3.1 , -4 , 4 , 4.1. and so on, but not 3,-3, -2, -1 etc.

Now, in (I) x > 3, why is this not a must be true statement?. How do we differentiate between must be / could be true where we have 2 applicable ranges for the question stem.
Per my understanding, x>3 will always be true for the given question stem because whatever value x>3 gives, it will always satisfy |x| > 3

In (III) we get the range for |x-1| > 2, as x <-1 or x>3,
x>3 satisfies the question stem, but I don't see how x <-1 satisfies it.
What if x takes the value -2 ? in that case |-2| >/ 3.
How is this a must be true scenario. All the values from -1 to -3 inclusive will fail the question stem.

Could you please point out where I'm going wrong? I'm almost always incorrect on questions of this form.



The relation that is given to us is the one that will provide the valid values of x.
So |x| > 3 is given to us. We know it holds. So we know that x can be -4 or 5 or 100 etc.
The set of VALID values of x is {... -5,.. -4,... -3.1, ,3.05,... 4.90,... 100,... 178...}
.
When we ask "which of the following must be true?" we need to find the option that is true for all VALID values of x.

Now think, "is x always greater than 3 for all valid values of x?"
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Re: If |x| > 3, which of the following must be true? [#permalink] If |x| > 3, which of the following must be true?   25 Aug 2023, 02:48
Asked: If |x| > 3, which of the following must be true?

Either x > 3 or x < -3

I. x > 3
Not true when x < -3

II. x^2 > 9
|x| > 3
(|x|)ˆ2 = xˆ2 > 9
Must be true

III. |x - 1| > 2
Either x-1>2; x> 3
or x-1 < -2; x < -1; If x<-3 then x<-1
Must be true

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

IMO D
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Re: If |x| > 3, which of the following must be true? [#permalink] If |x| > 3, which of the following must be true?   25 Jun 2024, 01:07
­statement third which mention |x - 1| > 2 
x-1>2; x> 3
or x-1 < -2; x < -1; If x<-3 given
but from this statement X=-2 is also possible solution which can't we less than -3 which contradict the what given in the question.
But if in question it was given x<-1 and from statement if we deduce x<-3 then statement third must be true. Kindly explain in details.
Thank You 
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Re: If |x| > 3, which of the following must be true? [#permalink] If |x| > 3, which of the following must be true?   25 Jun 2024, 01:20
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Su_bha wrote:
­statement third which mention |x - 1| > 2 
x-1>2; x> 3
or x-1 < -2; x < -1; If x<-3 given
but from this statement X=-2 is also possible solution which can't we less than -3 which contradict the what given in the question.
But if in question it was given x<-1 and from statement if we deduce x<-3 then statement third must be true. Kindly explain in details.
Thank You 


­Your doubt is addressed several times in the thread. Please review it more carefully. Check similar questions given in this post to understand the concept better:

https://gmatclub.com/forum/if-x-3-which ... l#p3057928
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Re: If |x| > 3, which of the following must be true? [#permalink] If |x| > 3, which of the following must be true?   16 Jul 2024, 08:13
Quote:
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

­let suppose in the III option x=1.1 then it holds true for option 3 but not for the question 
| -1.1 -1 | > 2 but actually x >3 or x < -3, is x actually - 1 no ? its very confusing but what is your take on this one ?
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Re: If |x| > 3, which of the following must be true? [#permalink]
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