If |x+1|=|y+1|, what is the value of x+y? 1) xy

Question

If  |x+1|=|y+1| , what is the value of  x+y ?

1)  xy<0 
2)  x>1  and  y<1

Alexey1989x · Answer

opening module we get 2 outcomes:
case 1: x=y
case 2: x+y=-2
So, we need to discover whether x and y are equal or not

(1) xy<0 follows that x and y are not equal, then 2nd case applies only and x+y=-2 Sufficient
(2) x>1; y<1 we see that they are not equal, then 2nd case applies only and x+y=-2 Sufficient

Answer D

amanvermagmat · Answer

When we are given that |a| = |b| (or absolute value of a is equal to absolute value of b), then there could be two cases:
Either a = b 
Or a = -b

So in the given question, either x+1 = y +1 (which gives us x=y) Or x+1 = -y-1 (which gives us x+y = -2)
Now if its the first case or x=y, then we cannot find x+y (multiple values will be possible)
But if its the second case then x+y will be -2.

(1) xy < 0 
If their product is less than 0 then obviously x and y have opposite signs, so they cannot be equal. So definitely we have the second case and x+y = -2. Sufficient.

(2) x > 1 and y < 1
This again tells us that x and y cannot be equal, and thus we have the second case and x+y = -2. Sufficient.

Hence  D answer

MathRevolution · Answer

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question, and then recheck the question

Modifying the original condition gives:

\(|x+1|=|y+1|\)
\(⇔ (x+1)^2=(y+1)^2\)
\(⇔ (x+1)^2=(y+1)^2\)
\(⇔ (x+1)^2-(y+1)2=0\)
\(⇔ (x+1+y+1)(x+1-y-1)=0\)
\(⇔ (x+y+2)(x-y)=0\)
\(⇔ x+y=-2\) or \(x=y\)

As we have 2 variables (x and y) and 1 equation in the original condition, D is most likely to be the answer.

Condition 1)
Since \(xy < 0, x≠y.\)
So, \(x + y = -2\), and condition 1) is sufficient.

Condition 2)
Since \(x>1\) and \(y<1, x≠y.\)
So, \(x + y = -2\).
Condition 2) is sufficient too.

Therefore, the answer is D.

Note: Since conditions 1) and 2) are similar, D is most likely to be the answer by Tip 1).

Answer: D

mysterymanrog · Answer

Let's start by solving the given equation for the two cases:

Case 1:
 x+1=y+1 
 x=y 
 x+y=2y

Case 2:
 -x-1=y+1 
 -x=y 
 x+y=0

So if we can determine that only case 2 holds, we have a value of x+y. Otherwise, if Case 1 holds the value depends on y.

1) 
From this statement, we can deduce that x and y must have opposing signs. If this is the case, then only Case 2 must hold. Suff.

2) Likewise, from this statement we can deduce that x and y have opposing signs (x cannot be a positive fraction, otherwise the equality will never hold -> consider the max value of x (x=0.999999...) and the min value of y(1.000000000001) y+1 will always be bigger, so x must be some negative integer equal to y) Suff.

D is the correct answer choice

gmatophobia · Answer

Sharing an alternative approach to this question.

Alternative approach : Solving the question using a number line and logical reasoning

Given   :

|x+1|=|y+1|

Inference: 
This statement tells us that the distance of the point (x+1) from 0 is the same as the distance of the point (y+1) from zero. This can happen in two scenarios 
1) When x and y lie on the same side of zero →
  In this case, x and y represent the same point and the product of x and y will be non-negative

--- 0 --------------- X ---------
--- 0 --------------- Y ---------

OR

--- X --------------- 0 ---------
--- Y --------------- 0 ---------

OR

Both the points lie at 0

2) When x and y lie on the opposite side of zero →
  In this case, x and y will be equidistant from zero, and the points will lie on the opposite sides of zero. The product of x and y in this case will be negative.

---- X ----------------- 0 ----------------- Y ----

OR

---- Y ----------------- 0 ----------------- X ----

Question   :

what is the value of  x+y

Inference:

If the points X and Y lie on the same side of zero (as depicted in case 1), the value of the sum will be twice the value of x (or y). The value of the sum can be positive, negative, or zero. 
 If the points X and Y lie on the opposite side of zero (as depicted in case 2), the value of the sum will be 0

Target question   : Do x and y lie on the same side of zero ?

Statement 1

1)  xy<0

This statement tells us that x and y have opposite signs, hence x and y lie on opposite sides of 0. The information is sufficient as we know from the pre-analysis that if x and y lie on opposite sides of zero, the sum (x+y) equals 0.

Statement 1 alone is sufficient. Eliminate B, C, and E.

Statement 2

2)  x>1  and  y<1

From the pre-analysis done above, we know that were x and y lie on the same side of zero, x and y would have the same value. However, statement 2 tells us that x and y do not have the same value. Hence, we can conclude that x and y lie on the opposite side of zero.

The information is also sufficient as we know that if x and y lie on opposite sides of zero, the sum (x+y) equals 0.

Hence each statement individually is sufficient to answer the question.

Option D

Note: This approach may seem long because of the sheer amount of text required to explain however we can solve this question without any pen and paper just by visualizing the positions of the points on the number line.

If |x+1|=|y+1|, what is the value of x+y? 1) xy<0

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