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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

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