BabySmurf wrote:

Is |x + y| < |x| + |y|?

(1) | x | ≠ | y |

(2) | x – y | > | x + y |

Appreciate views on how to solve this. Thank you.

I did this by picking numbers choosing various numbers positive, negative, fractions, 0.

Took make almost four minutes. Any other way?

This question can be done very easily if you are familiar with the properties of absolute value.

For all real x and y, \(|x + y| \leq |x| + |y|\)

\(|x + y| = |x| + |y|\) when x and y have the same sign (e.g. x = 4, y = 6 OR x = -2, y = -3) or at least one of them is 0.

When x and y have opposite signs, \(|x + y| < |x| + |y|\)

So the question asks us whether x and y have opposite signs or not.

(1) | x | ≠ | y |

Doesn't tell us anything about the signs of x and y. Not sufficient.

(2) | x – y | > | x + y |

When will | x – y | be greater than | x + y |? Only when x and y have opposite signs e.g. | x – y | = |3 - (-2)| = 5 but | x + y | = |3 - 2| = 1

If x and y have the same sign or one of them is 0, | x + y | will be greater or equal to | x – y |.

Hence this statement tells us that x and y have opposite signs. So it is sufficient alone.

Answer (B)

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