Is |x + y| |x + y|

Question

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?

KarishmaB · Accepted Answer

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)

cherukuri1011 · Answer

The answer is B.
for |x+y| <|x|+|y|
we have four cases 
1.x +ve ,y+ve
in that case 
|x+y| =|x|+|y|
2.x-ve ,y -ve
in that case 
|x+y| =|x|+|y|
3.x+ve and y -ve
in that case
 |x+y| <|x|+|y|
4.x -ve and Y +ve
in that case
 |x+y| <|x|+|y|
and if x=0 or y =0 (i.e. any one equal to zero)
|x+y| =|x|+|y|

so if x and y are of opposite signs then only |x + y| < |x| + |y|
now 1 gives 
|x|=|y|
with this we cant answer the question as we donot know signs of x and y
2.| x – y | > | x + y |
this implies that x and y are of opposite signs. So with this we can answer the question.
Key point here is you should understand immediately after looking at the inequality that its valid only if x and y are of opposite signs.
Give me kudos if this helps

BabySmurf · Answer

I am unsure with "ve" is... Thanks.

cherukuri1011 · Answer

Hi BabySmurf,
I used a shorter expression X +ve means x is positive number and x -ve means if x is a negative number

Bunuel · Answer

Similar questions to practice: is-x-y-x-y-87926.html is-x-y-x-y-1-x-y-2-x-y-x-101660.html is-x-y-x-y-123108.html is-x-y-x-y-160531.html is-x-y-x-z-1-y-z-2-x-86132.html is-a-b-a-b-105457.html is-x-y-x-y-1-x-y-2-x-y-132654.html is-x-y-146991.html is-x-y-x-y-137050.html is-x-z-y-x-z-y-154245.html Hope this helps.

Avinashs87 · Answer

If I square both sides, I am going to get xy<|x||y|. That would give B as an answer.

I have found squaring methods to help in inequalities (though not in every case)...

mvictor · Answer

i picked B. 1. x has a different absolute value than y. 1st case: x=2, b=3 => |x+y|=|x|+|y| no 2nd case: x=2, b=-3 -> |x+y|<|x|+|y| 2. it must be true that at least one of the numbers is negative, in which case |x + y| < |x| + |y| is always true

bumpbot · Answer

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Is |x + y| < |x| + |y|? (1) |x| ≠ |y| (2) |x – y| > |x + y|

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