When |x+y|

Question

When |x+y| < |x| +|y|, which of the following must be true?

I. xy<0
II. xy^3 < 0
III. x + y < 0

A. I only
B. II only
C. III only
D. I & II only
E. II & III only

Bunuel · Answer

Squaring |x+y| < |x| +|y| yields:
 
x^2 + 2xy + y^2 < x^2 + 2|xy| + y^2

xy < |xy|
 
The above indicates that xy < 0, meaning that they have opposite signs.

I. xy<0. True.

II. xy^3 < 0 True.

III. x + y < 0. Not necessarily true. For example, x = 2 and y = -1.

Answer: D

pierjoejoe · Answer

can anyone confirm my solving strategy? Bunuel

|x+y| < |x| +|y|

we know that in general
|x| +|y| >= |x+y|
and we know that they are equal when the signs are the same
|x| +|y| = |x+y| --> x and y have the same sign
|x| +|y| > |x+y| --> x and y have opposite signs

the problem states that
|x+y| < |x| +|y|

meaning that we are in the condition where the signs are opposite.
only the the I and II ensure the signs are OPPOSITE.
the third does not give us any hint about the sign of x and y.

thus D

thank you so much in advance

BrushMyQuant · Answer

|x+y| < |x| +|y|
Now, this is true only when x and y have different signs.

Let's take some examples to understand this
Ex 1: x = 1, y = 2
=> |x+y| = |1 + 2 | = |3| = 3
=> |x| + |y| = |1| + |2| = 1 + 2 = 3 = |x + y|

Ex 2: x = 1, y = -2
=> |x+y| = |1 + (-2) | = |-1| = 1
=> |x| + |y| = |1| + |-2| = 1 + 2 = 3 > |x + y|

=> x and y have different signs

I. xy<0
This is TRUE ALWAYS as x and y have different signs and both of them cannot be zero

II. xy^3 < 0
=> x * y * y^2 < 0
Now, we know that y^2 will always be > 0
=> x * y < 0
which is same as I, so TRUE ALWAYS

III. x + y < 0
Now, this might not be true always as we just know that x and y have different signs. Let's take some examples
Ex 1: x = 3, y = -2 => x + y = 3 + (-2) = 1 > 0
Ex 2: x = -2, y = 1 => x + y = -2 + 1 = -1 < 0
So, this NEED NOT BE TRUE ALWAYS

So, Answer will be D
Hope it helps!

Watch the following video to MASTER Absolute Values

bumpbot · Answer

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When |x+y| < |x| +|y|, which of the following must be true?

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When |x+y| < |x| +|y|, which of the following must be true?