Is |x+y|<|x-y|? 1) x+y<0 2) x-y>0

While working with absolute value questions, the smart approach is to always break down the question stem.

When we have a question with absolute values on both sides of an inequality or an equality, we can always square both sides and remove the absolute value. This is because |x| = \sqrt{x^2}

The question here is. 'Is |x + y| < |x - y|'

Squaring both sides and removing the absolute value we get, (x + y)^2 < (x - y)^2 ----> 4xy < 0 ----> xy < 0

The question just asks us if x and y are of different signs.

Statement 1: x + y < 0

Here x and y cab both be the same sign or of different signs. Insufficient.

Statement 2: x - y > 0

Here x and y are both of the same sign or of different sign. Insufficient.

Combining the two statements

The best way to combine two inequality statements is to add the statements up by setting the inequality signs to be the same. Remember we can always add two inequalities.

From 1 we have x + y < 0
From 2 we have -x + y < 0

Adding 1 and 2 we get y to be negative, but x can be either positive or negative. Insufficient.

Answer: E