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

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

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05 Jul 2018, 05:35
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95% (hard)

Question Stats:

36% (02:21) correct 64% (02:29) wrong based on 295 sessions

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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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05 Jul 2018, 09:33
4
1
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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Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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05 Jul 2018, 08:32
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.

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Posts: 58431
Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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05 Jul 2018, 08:35
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.

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

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05 Jul 2018, 08:43
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.

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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05 Jul 2018, 10:02
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.

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

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11 Jul 2018, 08:06
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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Posts: 8006
Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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11 Jul 2018, 08:42
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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Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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11 Jul 2018, 09:25
|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)
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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07 Aug 2018, 01:04
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
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Posts: 58431
Re: If |x - y| = |x - z|, what is the value of x?  [#permalink]

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24 Dec 2018, 01:35
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.

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

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25 Dec 2018, 23:18
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.

If y=z, x can take any value, there are infinite possibilities for x.

Statement 1) y<z. No information about x. Insufficient.

Statement 2) (y+z)/2=(y+z+2)/3 or y+z=4. Simplifying Expression|x-y|=|x-z| to x=(y+z)/2 assuming x, y and z are distinct numbers. So, as mentioned above, if y=z, x can be anything to satisfy the expression. Insufficient.

(1)+(2),
Our assumption needed to find x is statement 1. i.e. y is not equal to z. So, Sufficient.

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

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25 Dec 2018, 23:55
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 y and z equal to the average (arithmetic mean) of y, z and 2.

By Squaring on both sides, from the question, we have -

(y-z)(y+z-2x) = 0

So either, y =z or y+z = 2x --------Eqn 1

Statement I:

z > y.. Insufficeint.

Statement II:

y + z = 4. This is tempting but from the given statement in the Question we don't know whether y = z or y+z = 2x

Combining I & II:

From I we have , z > y so z cannot be equal to y. Hence. y + z = 2x.

x = 2.
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Re: If |x - y| = |x - z|, what is the value of x?   [#permalink] 25 Dec 2018, 23:55
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