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

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

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10 Sep 2012, 02:56
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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
[Reveal] Spoiler: OA
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Re: If |x|>3, which of the following must be true? [#permalink]

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10 Sep 2012, 03:17
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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

Original statement |x| > 3, which means either x>3 or x<-3
Now Check the options
Option1- x > 3 - not always true as x can be smaller than -3
Thus option A,C & E is ruled out. Only B & D are left

Option2- X^2 > 9 - Always true for x>3 or x<-3
To check - if x = 4,5,6,7.... or -4,-5,-6,-7
x^2>9

Option3-|x-1|>2,
which means (x-1)>2 ---> x>3 (if x-1>0) - True
it also means (x-1)<-2---->x<-1 (if x-1<0)

X<-1 satisfies x<-3. Thus true

Both 2 & 3 is true
Thus Answer D

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

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10 Sep 2012, 03:24
fameatop 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

Original statement |x| > 3, which means either x>3 or x<-3
Now Check the options
Option1- x > 3 - not always true as x can be smaller than -3
Thus option A,C & E is ruled out. Only B & D are left

Option2- X^2 > 9 - Always true for x>3 or x<-3
To check - if x = 4,5,6,7.... or -4,-5,-6,-7
x^2>9

Option3-|x-1|>2,
which means (x-1)>2 ---> x>3 (if x-1>0) - True
it also means (x-1)<-2---->x<-1 (if x-1<0)

X<-1 satisfies x<-3. Thus true

Both 2 & 3 is true
Thus Answer D

Hope it helps

Isn't x<-3 more restrictive than x<-1??

Edit: Understood...just one of those days. thanks
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Re: If |x|>3, which of the following must be true? [#permalink]

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10 Sep 2012, 03:54
fameatop wrote:
Option3-|x-1|>2,
which means (x-1)>2 ---> x>3 (if x-1>0) - True
it also means (x-1)<-2---->x<-1 (if x-1<0)

X<-1 satisfies x<-3. Thus true

Both 2 & 3 is true
Thus Answer D

Hope it helps

Now if I take x=-2 will it satisfy the condition?
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Re: If |x|>3, which of the following must be true? [#permalink]

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10 Sep 2012, 04:41
Expert's post
I'm ok untill B and D

But the 3 option says: |x - 1| > 2. Then x > 3 AND x< -1. Our statement says: x > 3 AND x < -3. So why is always TRUE??

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

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10 Sep 2012, 05:04
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carcass wrote:
I'm ok untill B and D

But the 3 option says: |x - 1| > 2. Then x > 3 AND x< -1. Our statement says: x > 3 AND x < -3. So why is always TRUE??

Please clarify

From|x|>3, x will always have to be >3 or <-3, so "our" x will never be between -3 and 3, making the III statement always true.
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Re: If |x|>3, which of the following must be true? [#permalink]

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10 Sep 2012, 05:16
Expert's post
corvinis wrote:
carcass wrote:
I'm ok untill B and D

But the 3 option says: |x - 1| > 2. Then x > 3 AND x< -1. Our statement says: x > 3 AND x < -3. So why is always TRUE??

Please clarify

From|x|>3, x will always have to be >3 or <-3, so "our" x will never be between -3 and 3, making the III statement always true.

Ok ok X<- 1 is not possible because our stetement says that X MUST < -3 ...........so X < -1 is not possible. Is This ?????

Silly misunderstood. I apologize

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

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10 Sep 2012, 05:25
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carcass wrote:
corvinis wrote:
carcass wrote:
I'm ok untill B and D

But the 3 option says: |x - 1| > 2. Then x > 3 AND x< -1. Our statement says: x > 3 AND x < -3. So why is always TRUE??

Please clarify

From|x|>3, x will always have to be >3 or <-3, so "our" x will never be between -3 and 3, making the III statement always true.

I agree but on one hand if "our" X is > 3 could be 3.1 or 200 but if "our" X is < -3 could be -3.1 or -200 so X < -1 could be -2 or -2.5 or -300.

This is what I meant. So where is the flaw in my reasoning ??

thanks

lol. you're doing the same error in reasoning I was doing. You have to prove statement III as true an not the other way around (prove |x|>3 true). If you know that your x will always be more than 3 OR always less than -3 (from |x|>3) then you are restricted to these values when looking to prove statement III as true. Hope you understand.

Edit: Upppss..You got it. Cheers.
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Re: If |x|>3, which of the following must be true? [#permalink]

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10 Sep 2012, 08:10
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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

Let's solve this question differently

Here following is given
I x I >3

So, Either x>3 OR x<-3
If we plot the values in line we get following -

----(-4)----(-3)-----(-2)-----(-1)-----(0)------(1)-------(2)-------(3)------(4)
<------------I-----------------------------------------------------------I------------>
(x<-3)...............................................................................(x>3)

Now, let us take some value (-4 and 4) and plug into the options.

Option I with x=4 : $$4> 3$$ Correct
Option I with x=-4 $$-4>3$$ Not correct. So, options I is out.

Option II with x=4 : $$4^2> 9$$ Correct
Option II with x=-4 $$(-4)^2>3$$ Correct. So, options II is right

Option III with x=4 : $$I 4-1I > 2$$ Correct
Option III with x=-4 $$I (-4-1) I >2$$ Correct. So, options III is right

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

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10 Sep 2012, 08:19
Expert's post
premnath 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

Let's solve this question differently

Here following is given
I x I >3

So, Either x>3 OR x<-3
If we plot the values in line we get following -

----(-4)----(-3)-----(-2)-----(-1)-----(0)------(1)-------(2)-------(3)------(4)
<------------I-----------------------------------------------------------I------------>
(x<-3)...............................................................................(x>3)

Now, let us take some value (-4 and 4) and plug into the options.

Option I with x=4 : $$4> 3$$ Correct
Option I with x=-4 $$-4>3$$ Not correct. So, options I is out.

Option II with x=4 : $$4^2> 9$$ Correct
Option II with x=-4 $$(-4)^2>3$$ Correct. So, options II is right

Option III with x=4 : $$I 4-1I > 2$$ Correct
Option III with x=-4 $$I (-4-1) I >2$$ Correct. So, options III is right

Hence, correct answer is D

Good explanation thanks . generally I do not this silly but crucial error but this time I could have sworn tha D was false. Indeed, III was tricky especially if I think under time pressure. The latter changes all things, all prespective.

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

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10 Sep 2012, 09:37
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

You are confusing the question with what is the range of X
We dont have to find the range of x, the question says that distance of x from 0 is more than 3

Satement 1 says x is always positive ... ot always true x can be negative and still the distance from zero will be more than 3

Statement 2 x^2 is more than 9 ofcourse ! is distance of x from 0 is more than 3 than its square will be more than 9 always

Statement 3 distance of x from 1 iss more than 2 .. Of course ! if its distance from 0 is more than three than its distance from 1 will be always more than 3-1 = 2
Weather it is negative or positive
as if it is negative, than distance from 0 is already more than 3, and you are taking distance from 1 so it will be 1 unit more than that from 0 that is 3 + 1 and hence will surely be more than 2

Here these problems are better solved by converting expression to their definition.

Important point is you are given the value of X and you have to see, what all is ture for all those values of X not ! which values of X satisfy the expression mentioned in questions.
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Re: If |x|>3, which of the following must be true? [#permalink]

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11 Nov 2012, 17:43
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I just came across this one and it is clear that this is a tough cookie.

Given that |x| > 3, so that means x > 3 or x < -3. We can also test values here, only 3,4,5,6 or -3, -4, -5 etc work.

Now, let's tackle the statements:
I. x > 3

We know that x > 3 or x < -3, but it is not ALWAYS the case that x > 3. We do have values of x that are less than 3, i.e., when x is -4 or -5. So, I is false.

II. x^2 > 9.

This means x>3 or x < -3. That is what we have above so this is golden. True.

III. |x - 1| > 2

When x > 0
x -1 > 2
x > 3

x < 0
-x + 1 > 2
-x > 1
x < -1

So , III says IF x > 3 or x < -3, THEN x > 3 OR x < -1. True. Test values to prove this x > 3 or x < -3 means x can be -4 and that hits x < -1
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Re: If |x|>3, which of the following must be true? [#permalink]

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20 Nov 2012, 08:46
Please check the trick to solve this in 1 min.

inequalities-trick-91482-40.html#p1145743
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Re: If |x|>3, which of the following must be true? [#permalink]

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03 Feb 2013, 21:36
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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.
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Re: If |x|>3, which of the following must be true? [#permalink]

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05 Apr 2013, 00:24
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.
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Re: If |x|>3, which of the following must be true? [#permalink]

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05 Apr 2013, 01:42
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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.
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If |x| > 3 , which of the following must be true? [#permalink]

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27 Apr 2013, 08:45
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?
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Re: If |x| > 3 , which of the following must be true? [#permalink]

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27 Apr 2013, 08:57
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if x<-3 then sure x<-1 ....
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Re: If |x| > 3 , which of the following must be true? [#permalink]

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13 May 2013, 02:12
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?

I too have the same doubt...can anyone address the query

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

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13 May 2013, 02:17
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] 13 May 2013, 02:17

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