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

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New post 13 Aug 2013, 13:40
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!


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

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New post 13 Aug 2013, 14:27
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
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Re: If |x|>3, which of the following must be true? [#permalink]

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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!


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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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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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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 20 Jun 2015, 02:27
i dont get it...if x=-2, then |x|=2, which does not satisfy the equation, how is it true?
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Re: If |x|>3, which of the following must be true? [#permalink]

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New post 13 Nov 2016, 21:24
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!



Still not clear. It is saying 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 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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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 23 May 2017, 13:39
mau5 wrote:
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.


Thanks mau5

This is the explanation I was looking for.
Re: If |x|>3, which of the following must be true?   [#permalink] 23 May 2017, 13:39

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