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# |x-2| > 3

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|x-2| > 3 [#permalink]  17 Oct 2017, 14:00
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38% (01:11) correct 61% (01:16) wrong based on 121 sessions

$$|x-2| > 3$$

 Quantity A Quantity B The minimum possible value of |x - 3.5| The minimum possible value of | x - 1.5 |

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
[Reveal] Spoiler: OA

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Re: |x-2| > 3 [#permalink]  18 Oct 2017, 05:01
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Solving the disequality in the header, $$|x-2|>3$$, we get $$x<-1, x>5$$.

Then, the smallest value for $$|x-3.5|$$ is when x = 5, i.e. $$|5-3.5| = 1.5$$. The smallest value for $$|x-1.5|$$ is when x = -1, i.e. $$|-1-1.5| = 2.5$$.

Thus, since 2.5>1.5, quantity B is greater!
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Re: |x-2| > 3 [#permalink]  28 Feb 2018, 14:10
IlCreatore wrote:
Solving the disequality in the header, $$|x-2|>3$$, we get $$x<-1, x>5$$.

Then, the smallest value for $$|x-3.5|$$ is when x = 5, i.e. $$|5-3.5| = 1.5$$. The smallest value for $$|x-1.5|$$ is when x = -1, i.e. $$|-1-1.5| = 2.5$$.

Thus, since 2.5>1.5, quantity B is greater!

Correct Explanation.
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Manager
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Re: |x-2| > 3 [#permalink]  04 Mar 2018, 02:29
good question
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Last edited by boxing506 on 15 Mar 2018, 23:26, edited 1 time in total.
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Re: |x-2| > 3 [#permalink]  05 Mar 2018, 18:34
correct : C
|X -2 | > 3 means either x-2 > 3 or x-2< -3
so
x - 2 > 3 -> x > 5 or
x -2 < -3 -> x < -1
if the numbers are integers then we can't choose x =5 and x = -1 :
the minimum possible value of |x - 3.5 | is when x = 6, in this case |x-3.5| equals 2.5
the minimum possible value of |x - 1.5 | is when x = -1, in this case |x-3.5| equals 2.5 again
so considering numbers as integers these are equal.

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Re: |x-2| > 3 [#permalink]  21 Oct 2018, 03:17
Expert's post
tigran wrote:
Why can't we put o as x at that time it will be the minimum value of x

0 is not a valid value for |x-2|>3....
Substitute x as 0... |0-2|=2 which is NOT greater than 3
_________________

Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

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Re: |x-2| > 3 [#permalink]  02 Nov 2018, 10:49
Expert's post
correct : C
|X -2 | > 3 means either x-2 > 3 or x-2< -3
so
x - 2 > 3 -> x > 5 or
x -2 < -3 -> x < -1
if the numbers are integers then we can't choose x =5 and x = -1 :
the minimum possible value of |x - 3.5 | is when x = 6, in this case |x-3.5| equals 2.5
the minimum possible value of |x - 1.5 | is when x = -1, in this case |x-3.5| equals 2.5 again
so considering numbers as integers these are equal.

nope.
"the minimum possible value of |x - 1.5 | is when x = -1, in this case |x-3.5| equals 2.5 again"

if x = -1, then |-1-3.5| = 4.5
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Re: |x-2| > 3 [#permalink]  07 Nov 2018, 22:15
Expert's post
Carcass wrote:

$$|x-2| > 3$$

 Quantity A Quantity B The minimum possible value of |x - 3.5| The minimum possible value of | x - 1.5 |

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.

$$|x-2| > 3$$

we can find the values of x in three ways...

(I) Since both sides are positive, square both sides
$$|x-2|^2 > 3^2.......x^2-4x+4>9.........x^2-4x-5>0.....x^2-5x+x-5>0........(x+1)(x-5)>0$$
a) either both are positive.. x+1>0 and x-5>0, so x>-1 and x>5...x>5
b) or both are negative .. x+1<0 and x-5<0, so x<-1 and x<5 ... so x<-1
x<-1 and x>5

(II) critical point..
x<=2..... -(x-2)>3.......x-2<-3....x<-1
x>2...... (x-2)>3......x>5

(III) logical approach via number line..
$$|x-2| > 3$$ means the distance of x from 2 is greater than 3 units..
so if x is on the left side of 2, it will be < (2-3) or <-1
if x is on the right side of 2, it will be > (2+30 or >5

https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

so we have range of x as x<-1 and x>5
The minimum possible value of |x - 3.5| ... again which value in range is closer to 3.5, it is 5, so value is just > |5-3.5| or just > 1.5
The minimum possible value of | x - 1.5 |... again which value in range is closer to 1.5, it is -1, so value is just > |-1-1.5| or just > 2.5
so B>A
_________________

Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

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Re: |x-2| > 3 [#permalink]  27 Nov 2018, 11:40
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3 < |x-2|

** If a < |b+c| Then a < (b+c) or (b+c) < -a
So we can conduct that:
3 < x-2 or (x-2) < -3
x > 5 or x < -1

A: The minimum possible value of |x-3.5|
First we should consider that an absolute value can’t be negative. At least it equals to 0.
We have x > 5 or x < -1
For x>5, |x-3.5| will be bigger than 1.5
For x<-1, |x-3.5| will be bigger than 4.5
So the minimum amount for A is 1.5

B: The minimum possible value of |x-1.5|
We have x > 5 or x < -1
For x>5, |x-1.5| will be bigger than 3
For x<-1, |x-1.5| will be bigger than 2.5
So the minimum amount for A is 2.5

So B is bigger than A.
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Re: |x-2| > 3 [#permalink]  16 Dec 2018, 08:11
Can't we take any decimal value in the place of x? As it is not said that x is an integer?

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Re: |x-2| > 3   [#permalink] 16 Dec 2018, 08:11
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