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