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If |x - 3| > 1 then which of the following must be true?

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If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:00
3
16
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

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If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 19 Jul 2019, 02:31
3
Bunuel wrote:
If \(|x - 3| > 1\) then which of the following must be true?

I. \(|x| > 4\)
II. \(x^2 > 16\)
III. \(x > 4\)

A. I only
B. II only
C. III only
D. I, II and III
E. None


 

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SOLUTION:



\(|x - 3| > 1\) means that:
\(x - 3 > 1\) --> \(x > 4\)
\(-(x - 3) > 1\) --> \(x < 2\)

So, we are given that \(x < 2\) or \(x > 4\).

I. \(|x| > 4\). Not necessarily true. Consider x = 0.
II. \(x^2 > 16\). Not necessarily true. Consider x = 0.
III. \(x > 4\). Not necessarily true. Consider x = 0.

Answer: E.

To understand the underline concept better practice other Trickiest Inequality Questions Type: Confusing Ranges.
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Re: If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:10
1
Answer should be none.
As x=1 satisfies the exuation |x-3|>1
But does not satisfy any of the options.
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If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post Updated on: 04 Jul 2019, 11:06
4
2
1
Refer attached Image.

Ans. E
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WhatsApp Image 2019-07-04 at 11.32.49 PM.jpeg [ 83.22 KiB | Viewed 3330 times ]


Originally posted by MayankSingh on 04 Jul 2019, 08:22.
Last edited by MayankSingh on 04 Jul 2019, 11:06, edited 1 time in total.
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Re: If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:24
|x-3|>1

Then we should consider two cases:
x-3>1 >> x>4
x-3<-1 >> x<2

So the solution is : (-∞;2) to (4;+∞)

I. |x|>4
then x>4; x<-4
Must be true

II. x^2>16
x>4; x<-4
the same as the 1st case
Must be true

III. x>4
Must be true

Answer: D. I, II and III
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Re: If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:25
2
Easy. What |x−3|>1 depicts is that x is at a distance of more than 1 from 3 on the number line, and thus, x>4 OR x<2

For x=1, |x−3|>1, |1-3| = |-2| = 2 and 2>1 and hence, x=1 satisfies the inequality

Now quickly looking at all the options, x=1 does not fit in any of the three, and hence <1 min., you can arrive at your answer, i.e. (E) None
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Re: If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:28
1
2
As explained in the attached picture, x<2 & x>4

All the given choices CAN BE TRUE, but not MUST BE TRUE.

Hence Ans should be (E)
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Re: If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:28
Given |x-3|>1
Which implies x<2 or x>4

Now go thru the given statements:-
I. |x|>4 implies x<-4 or x>4. DISCARD
II. x^2>16 implies |x|>4 which again implies x<-4 or x>4. DISCARD
III. x>4 matches with the interval given in the question stem .KEEP

Ans. (C)
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Re: If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:28
|x-3|>1
i.e.
x-3>1 or -(x-3)<1
x>4 or x <-4
i.e. |x|>4
imo A.

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Re: If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:31
Lets solve the equation

x-3>1 ,gives : x>4 and x-3<-1 gives : x<2

all three equation satisfies.

Option D
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If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post Updated on: 22 Jul 2019, 23:35
2
If |x−3|>1
then which of the following must be true?

I. |x|>4
II. x^2>16
III. x>4

A. I only
B. II only
C. III only
D. I, II and III
E. None

|x-3|>1
x>4 or x<2

I. |x|>4
Take for example x =1 => |x-3| = |1-3| = 2>1 satisfies the equation. But |1| is not >4. NOT NECESSARILY TRUE.
II. x^2>16
Take for example x =1 => |x-3| = |1-3| = 2>1 satisfies the equation. But 1^2 is not >16. NOT NECESSARILY TRUE.
III. x>4
Take for example x =1 => |x-3| = |1-3| = 2>1 satisfies the equation. But 1 is not > 4. NOT NECESSARILY TRUE.

IMO E
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Originally posted by Kinshook on 04 Jul 2019, 08:33.
Last edited by Kinshook on 22 Jul 2019, 23:35, edited 1 time in total.
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Re: If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:33
If |x−3|>1 then which of the following must be true?

=>|x−3|>1
=> x−3>1 & x−3 < -1
=> x > 4 & x < 2 --> so x is in this range, the it must be true

<---------------o----------o--------------->
<---------------2----------4--------------->



I. |x|>4 --> must be true : x>4 & x < -4 : <---------------4----------4---------------> is subset of <---------------2----------4--------------->
II. x^2>16 --> must be true: x^2>16 => |x|>4 same as I
III. x>4 --> must be true:4---------------> is subset of <---------------2----------4--------------->

A. I only
B. II only
C. III only
D. I, II and III --> correct
E. None
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If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post Updated on: 21 Jul 2019, 05:51
|x−3|>1

if x>3
x-3>1
x>4

If x<3
x-3<-1 (multiply by -1)
x<2

So we have x in the range (x<2) and (x>4)

Now lets check the options

1)|x| > 4
x>4 for x>0
x<-4 for x<0
AND for x > 0
x>4

So this falls in the range.

2)x^2 > 16
sqrt of both side gives
|x| > 4
This is same as 1) above so again this is also fine.

3)x>4 again falls in the range.

So all the above falls in the range.

I would chose D as OA.

Originally posted by prabsahi on 04 Jul 2019, 08:37.
Last edited by prabsahi on 21 Jul 2019, 05:51, edited 1 time in total.
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If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post Updated on: 05 Jul 2019, 08:34
solving

lx-3l>1
gives x>4 or x<2
so out of given options IMO E is only valid


If |x−3|>1|x−3|>1 then which of the following must be true?

I. |x|>4|x|>4
II. x2>16x2>16
III. x>4x>4

A. I only
B. II only
C. III only
D. I, II and III
E. None

Originally posted by Archit3110 on 04 Jul 2019, 08:38.
Last edited by Archit3110 on 05 Jul 2019, 08:34, edited 1 time in total.
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Re: If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:38
1
|x-3|>1

Either (x-3)>1 or (x-3)<-1

Either x>4 or x<2


IMO, Answer is (E)
All three statements only take x>4 into consideration but we also have x<2

So none of the statements MUST BE true
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Re: If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:44
1
If |x−3|>1 then which of the following must be true?

I. |x|>4
II. \(x^2>16\)
III. \(x>4\)

Given
|x−3|>1
thus either x-3>1 hence x>4
or x-3<-1 or x<2

thus
take x =1
I. |1|>4 not true
II. \(1^2>16\) not true
III. \(1>4\) not true

Thus None E
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Re: If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:44
2
Two ways to solve :

Method 1) Find a counter example (easiest), X=-1 will satisfy the stem

I. |x|>4 - since x = -1 will satisfy this is not true
II. x^2>16 - since x = -1 will satisfy this is not true
III. x>4 - since x = -1 will satisfy this is not true

IMO E None

Method 2: solve using modulus properties to arrive at X cannot lie from 2 to 4. it can take any other value.
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If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post Updated on: 04 Jul 2019, 22:40
|x-3|>1
=> x-3>1 or 3-x > 1
=> x>4 or x<2

I. |x| > 4 implies x > 4 or x < -4 =>definitely true based on the solution above
II. x^2 > 16 implies x>4 or x<-4 => same as above and is true
III. x>4 => true

All statements are true.
Option D should be the answer.


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Originally posted by prashanths on 04 Jul 2019, 08:44.
Last edited by prashanths on 04 Jul 2019, 22:40, edited 1 time in total.
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Re: If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:56
Given |x - 3| > 1
Solving for x;
x - 3 > 1
x > 1 + 3
x > 4
Also, negating the right side of the original equation
x - 3 > -1
x > -1 + 3
x > 2
From the options available, only III fulfilled the condition, hence answer choice "C"

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Re: If |x - 3| > 1 then which of the following must be true?  [#permalink]

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New post 04 Jul 2019, 08:56
Hi,
Given |x-3| > 1
=> (x-3) < -1 or (x-3) > 1
=> x < -4 or x > 4
so possible range or values are => (-infinte , -4) or (4, +infinite)

Now the question:
1. |x| > 4
for all the possible values this will always be true. take x = -5 or x = +5

2. \(x^2\)> 16
for all possible values, this will always be true.

3. x > 4
this comes by solving the inequality.

So Answer is D.

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Re: If |x - 3| > 1 then which of the following must be true?   [#permalink] 04 Jul 2019, 08:56

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