If |x+2|=|y+2| , what is the value of x+y ? (1) x≠y (2) x−y=16

Question

If |x + 2| = |y + 2|, what is the value of x + y?

(1) x ≠ y
(2) x − y = 16

chetan2u · Answer

|x + 2| = |y + 2| means x and y are at same distance from -2
so either both x and y are equal or they are on either side of -2..

OR

square both sides..

(|x + 2|)^2 = (|y + 2|)^2

x^2+4x+4=y^2+4y+4

x^2-y^2+4x-4y=0

(x-y)(x+y)+4(x-y)=0

(x-y)(x+y+4)=0

this means x=y or x+y=-4 or both

(1) x ≠ y
colored portion above tells us that x+y=-4
sufficient

(2) x − y = 16
again this tells us that x ≠ y, otherwise x-Y=0
colored portion above tells us that x+y=-4
sufficient

D

SonGoku · Answer

Hello  chetan2u 
I always had this doubt.Even now i am not clear about the OR in the inequalities.could you please explain the functionality of it or posting a thread link explaining that is also fine because it will be time consuming to type it again if explained elsewhere.
Thankyou.

sumit411 · Answer

Hi  SonGoku

Do you want links for modulus functions?

Thank you = Kudos

Mo2men · Answer

Analyzing the stem, we have 2 cases:

x+2 = y+2..........x =y

Or

x+2 = -y-2....... x+y= 4

(1) x ≠ y

This tells as that case 1 is invalid ....we are left with case 2 in that x +y =4

Sufficient

(2) x − y = 16

This tells us that case 1 is invalid because 0 does not equal 16. we are left with case 2 in that x +y =4

Sufficient

Answer: D

chetan2u · Answer

if an equation is (a-b)(a+b-1)=0..
there are two cases 
a-b=0 so a=b or
a+b-1=0.....a+b=1

so if any of the above two cases a=b or a+b=1 then the equation stands. It may also be the case that both are true say a=b=1/2

is there anything else you are wanting to ask

SonGoku · Answer

Thanks for the reply

So, That means if there are two statements and if one of the two statements satisfies any one of above cases.can we consider the statement sufficient? OR Do we need to consider both of them to check whether the statement is sufficient?

QuantMadeEasy · Answer

Step 1: Understanding the question 
Lets understand with cases:
Case 1: Both (x+2) and (y+2) are positive, (x + 2) = (y + 2); x = y
Case 2: One positive and other negative, (x + 2) = -( y + 2); x + y = -4

Step 2: Understanding statement 1 alone 
(1) x ≠ y
When x ≠ y, therefore case 2 is valid, hence x + y = -4
Sufficient

Step 3: Understanding statement 2 alone 
(2) x − y = 16
As difference between x and y is positive, x is greater than y. Hence, x ≠ y, therefore case 2 is valid ie. x + y = -4
Sufficient

D is correct

CEdward · Answer

Hello chetan, I've seen squaring of the mod quite a bit including this post of yours. I am confused about when we should square the mod versus not? What are the indicators for us to do this? How is squaring the mod different from other techniques of dealing with mods, e.g. critical transition points, opening the mod, or graphically?

chetan2u · Answer

Whenever both the sides are positive, you can square them. Here we have modulus on both sides, so we can square. 
But say it was |x+2|=|y-2|+x, you should avoid it as you do not know if RHS is positive.

Rebaz · Answer

Here is how i solved this question within 2min.

If |x + 2| = |y + 2|, what is the value of x + y?

Absolute values has alway two scenario's: Positive and Negative.

Positive scenario:

x + 2=y + 2 Subtract 2 from left and right then we end up with X=Y

Negative scenario:

x + 2=-(y + 2). Expand the minus sign(which is the same as -1) on the right side of the equation by multiplying the minus sign(a.k.a -1) by y and +2, we get -1(y+2)-------becomes -y-2, thus the whole equation in the negative scenario will look like this:

x + 2=-y - 2. Now we just slove the equation by subtracting 2 from both sides of the equation and adding y to both side of the equation. The final reslut will look like this:

X+Y=-4

Rephrasing the question stem, if |x + 2| = |y + 2| , what is the value of x + y?

Either X=Y or X+Y=-4 -------------------------our job is to find out whether based on the statements X=Y.

If X=Y, we cannot find a concrete value for X+Y, on the other hand, once we can determine based on the statements that X ≠ Y, then we can safely conclude that X+Y=-4.

(1) x ≠ y is sufficient, coz now that we know that x ≠ y, then X+Y=-4

(2) x − y = 16 is sufficient as well. by adding y to both side of the equation, we end up with X=16+Y, meaning x ≠ y, meaning X+Y=-4.

I hope it is clear.

bumpbot · Answer

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If |x+2|=|y+2| , what is the value of x+y ? (1) x≠y (2) x−y=16

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If |x+2|=|y+2| , what is the value of x+y ? (1) x≠y (2) x−y=16

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