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If |x - y| = |x - z|, what is the value of x?

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If |x - y| = |x - z|, what is the value of x?  [#permalink]

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New post 05 Jul 2018, 04:35
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A
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E

Difficulty:

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Question Stats:

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GMAT CLUB'S FRESH QUESTION:



If |x - y| = |x - z|, what is the value of x?

(1) y < z
(2) The average (arithmetic mean) of y and z equal to the average (arithmetic mean) of y, z and 2.

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Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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New post 05 Jul 2018, 08:33
3
If |x - y| = |x - z|, what is the value of x?

Now what does this mean...
This means that x is equidistant from y and z, but two options exist
[b][/b]
a) both y and z are on same side that is y=z
b) both are on either side of x then x will be the mean of y and z

(1) y < z
no numeric value ..
Only that \(y\neq{z}\) and hence x is MEAN of y and z
insuff


(2) The average (arithmetic mean) of y and z equal to the average (arithmetic mean) of y, z and 2.
\(\frac{y+z}{2}=\frac{y+z+2}{3}.........3y+3z=2y+2z+4......y+z=4\)
we can find mean of y and z as 4/2 = 2
so x=2 if \(y\neq{z}\)
But if y=z, value of x cannot be determined
insuff

combined
\(y\neq{z}\)
therefore x=2
suf
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
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Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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New post 05 Jul 2018, 07:32
statement 1
y<z
this statement doesn't tell about X. not sufficient.

Statement 2
(X+Y)/2 = (X+Y+2)/3
X+Y= 4

x can be any number.
Not sufficient.

Statement 1 + statement 2
many values for x.

so E is answer.
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Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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New post 05 Jul 2018, 07:35
gvij2017 wrote:
statement 1
y<z
this statement doesn't tell about X. not sufficient.

Statement 2
(X+Y)/2 = (X+Y+2)/3
X+Y= 4

x can be any number.
Not sufficient.

Statement 1 + statement 2
many values for x.

so E is answer.


Sorry. (2) actually reads:
(2) The average (arithmetic mean) of y and z equal to the average (arithmetic mean) of y, z and 2.

Edited.
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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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New post 05 Jul 2018, 07:43
1
Bunuel wrote:

GMAT CLUB'S FRESH QUESTION:



If |x - y| = |x - z|, what is the value of x?

(1) y < z
(2) The average (arithmetic mean) of x and y equal to the average (arithmetic mean) of x, y and 2.


I am not quite sure about this, BUT imo answer should be B.

Consider only statement B.
|x - y| = |x - z|
either
\(x-y = x-z\) => \(y=z\) (1)
or
\(-x+y = x-z\) => \(2x = y+z\) (2)

Statement 2 says
\((x+y)/2 = (x+y+2)/3\)
\(3x+3y = 2x+2y+4\)
\(x+y=4\) (3)

Substituting (3) in (2),

\(2x = 4-x + z\)
and \(z=y\) from (1)

\(2x = 4-x+4-x\)
\(4x=4\)
\(x=1\)
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Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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New post 05 Jul 2018, 09:02
Bunuel wrote:

GMAT CLUB'S FRESH QUESTION:



If |x - y| = |x - z|, what is the value of x?

(1) y < z
(2) The average (arithmetic mean) of y and z equal to the average (arithmetic mean) of y, z and 2.



Given, |x - y| = |x - z|, which means distance of x from y & distance of x from z are equal.

so we have

(a) if (x - y) > 0 & (x - z) > 0, then y = z, & x can take any value

(b) if (x - y) < 0 & (x - z) < 0, then y = z, & x can take any value

(c) if (x - y) > 0 & (x - z) < 0, then x = (y + z)/2, & x > y , x < z & y < x < z

(d) if (x - y) < 0 & (x - z) > 0, then x = (y + z)/2, & x < y, x > z & z < x < y



Statement 1: y < z , hence case (d) is possible, x lies between y & z, as y < x < z. However not sufficient to find value of x.

Statement 2: \(\frac{(y + z)}{2}\) = \(\frac{(y + z + 2)}{3}\)

Simplifying this we get, y + z = 4, however we cannot say anything about of position of x & hence cannot calculate x.

Since y = z = 2, hence x can take any value

or y = 1, z = 3, then x = 2

Hence statement 2 is not sufficient.

Combining both Statements, we have y < z & y + z = 4. hence we have case (c), y < x < z

Hence x = (y + z)/2 = 4/2 = 2.

Combining both statements is Sufficient.

Answer C.



Thanks,
GyM
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Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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New post 11 Jul 2018, 07:06
chetan2u wrote:
If |x - y| = |x - z|, what is the value of x?

Now what does this mean...
This means that x is equidistant from y and z, but two options exist
[b][/b]
a) both y and z are on same side that is y=z
b) both are on either side of x then x will be the mean of y and z

(1) y < z
no numeric value ..
Only that \(y\neq{z}\) and hence x is MEAN of y and z
insuff


(2) The average (arithmetic mean) of y and z equal to the average (arithmetic mean) of y, z and 2.
\(\frac{y+z}{2}=\frac{y+z+2}{3}.........3y+3z=2y+2z+4......y+z=4\)
we can find mean of y and z as 4/2 = 2
so x=2 if \(y\neq{z}\)
But if y=z, value of x cannot be determined
insuff

combined
\(y\neq{z}\)
therefore x=2
suf


Hi chetan, will you please explain second part explanation, why y=z cant determine value of x ?
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Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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New post 11 Jul 2018, 07:42
akhiparth wrote:
chetan2u wrote:
If |x - y| = |x - z|, what is the value of x?

Now what does this mean...
This means that x is equidistant from y and z, but two options exist
[b][/b]
a) both y and z are on same side that is y=z
b) both are on either side of x then x will be the mean of y and z

(1) y < z
no numeric value ..
Only that \(y\neq{z}\) and hence x is MEAN of y and z
insuff


(2) The average (arithmetic mean) of y and z equal to the average (arithmetic mean) of y, z and 2.
\(\frac{y+z}{2}=\frac{y+z+2}{3}.........3y+3z=2y+2z+4......y+z=4\)
we can find mean of y and z as 4/2 = 2
so x=2 if \(y\neq{z}\)
But if y=z, value of x cannot be determined
insuff

combined
\(y\neq{z}\)
therefore x=2
suf


Hi chetan, will you please explain second part explanation, why y=z cant determine value of x ?



Now we know y+z=4 and that x is equidistant from y and z..
So if the layout is y.....x....z x is in middle of y and z or mean of y and z that is (y+z)/2=4/2=2......
So if y=0, z=4...............0.......x=2......4
Or if y =1 , z=3.............1....x=2.....3
BUT if y=z.... y+z=4, so y=z=2
But the problem lies here..
Every point will be equidistant from y and z now..
X=1....y and z=2.......1.....2&2
Or x=10, y and z=2.......... 2&2........10
So x can be any value
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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New post 11 Jul 2018, 08:25
|x - y| = |x - z|
Squaring both the side
x^2+y^2-2xy= X^2+z^2-2xz
y^2-z^2= 2x(y-z)
(y-z)(y+z)-2x(y-z)=0
(y-z)(y+z-2x)=0
From statement 1
y<z
hence y+z-2x=0
or y+z=2x
we don't know the value of y+z and hence insufficient
Statement 2
(y+z)/2 = (y+z+)2/3
3(y+z)=2(y+z+2)
y+z=4
our equation is (y-z)(y+z-2x)=0
so we don't know if y-z=0 or y+z-2x=0
hence we cannot put y+z = 4 here. Therefore Not sufficient
From both the statements,
we know y<z , hence y+z=2x or x=(y+z)/2 or x= 2 (sufficient)
Hence the answer is C
The explanation looks big but on real time this can be solved under a minute
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Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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New post 07 Aug 2018, 00:04
Hi ,

Its not mentioned that these values are integers. This makes the whole situation different as x and take any value between
1 and 3
Please clarify
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Re: If |x - y| = |x - z|, what is the value of x? &nbs [#permalink] 07 Aug 2018, 00:04
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