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

Question

GMAT CLUB'S FRESH QUESTION:

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

M36-75

chetan2u · Accepted Answer

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     
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
eq{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
eq{z} 
But if y=z, value of x cannot be determined
insuff

combined
 y
eq{z} 
therefore x=2
suf

Bunuel · Answer

Official Solution:

If  |x - y| = |x - z| , what is the value of  x ? 
 
 |x - y| = |x - z|  means that  the distance between  x  and  y , on the number line, equals to the distance between  x  and  z  . This can occur in two cases:
 
(i) when two points  y  and  z  coincide, so when  z=y . Notice that in this case  x  can be any number (any  x  will be equidistant from  y  and  z  if  z=y ).
 
(ii) when  x  is exactly in the middle between two points  y  and  z , so when  x=\frac{y+z}{2} . 
 
(1)  y < z 
 
This statement rules out the first case from above ( z=y ), so we are left with the second case:  x=\frac{y+z}{2} . But we still don't know the values of  y  and  z  to get  x . Not sufficient. 
 
(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} 
 
The above gives:  y+z=4 . 
 
If we have  z=y  case from above, then  z=y=2  and  x  can be any number;
 
If we have  x=\frac{y+z}{2}  case from above, then  x=\frac{y+z}{2}=\frac{4}{2}=2 
 
Not sufficient.
 
(1)+(2) From (1) we got that we have  x=\frac{y+z}{2} , so from (2) we get that  x=\frac{y+z}{2}=\frac{4}{2}=2 . Sufficient.

Answer: C

gvij2017 · Answer

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.

Bunuel · 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.

vaibhav1221 · Answer

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

GyMrAT · Answer

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

akhiparth · Answer

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

chetan2u · Answer

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

Raksat · Answer

|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

piyushsahay1001 · Answer

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

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

Ypsychotic · Answer

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.

The correct answer is C.

rahul16singh28 · Answer

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.

Jitu20 · Answer

Long Method aka Brute force:

1. |x-y| = |x-z| =

case 1: x-y = x-z => y = z 
case 2: -(x-y) = x-z => x = y+z/2
case 3: x-y = -(x-z) => x = y+z/2

we know y<z so x = y+z/2 but no possibility of finding values. Not Sufficient AD are out..

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

y+z/2 = y+z+2/3 => y+z =4, no possibility to find x as we dont know the relation. B is out.

Combining 1,2 => x=y+z/2 = 2. Unique value

GMATGuruNY · Answer

Prompt:
 |x-y| = |x-z|

Case 1: Signs unchanged
 x-y = x-z 
 y=z 
Here, x can be ANY VALUE.

Case 2: Signs changed in ONE of the absolute values
 x-y = -x+z 
 2x = y+z 
 x = \frac{y+z}{2}

Statement 1: y < z
Since only Case 2 is viable,  x = \frac{y+z}{2} .
No way to determine the value of x.
INSUFFICIENT.

Statement 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

The resulting equation implies the following:
Case 1 -->  y=z=2  --> x can be ANY VALUE
Case 2 -->  x=\frac{(y+z)}{2} = \frac{4}{2} = 2 
Since x can be any value in Case 1, INSUFFICIENT.

Statements combined:
Only Case 2 satisfies both statements.
In Statement 2, the result yielded by Case 2 is that x=2.
SUFFICIENT.

C

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

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