ziyuenlau wrote:Is |x|+|y| > |x-y|?
(1) |x| > |y|
(2) |x-y| < |x|
|x|+|y| = nonnegative + nonnegative = nonnegative.
|x-y| = nonnegative.
Since both sides of the question stem are nonnegative, we can SQUARE THE INEQUALITY:
(|x|+|y|)² > |x-y|²
x² + y² + 2|x||y| > x² + y² - 2xy
2|x||y| > -2xy
|x||y| > -(xy).
The inequality in blue will hold true only if -(xy) is negative.
For -(xy) to be negative, xy must be POSITIVE.
Question stem, rephrased:
Do x and y have the SAME SIGN?
Statement 1: |x| > |y|
If x=2 and y=1, then x and y have the same sign.
If x=-2 and y=1, then x and y do NOT have the same sign.
INSUFFICIENT.
|a-b| = the distance between a and b.
|a| = the distance between a and 0.
Statement 2: |x-y| < |x|
In words:
The distance between x and y is less than the distance between x and 0.
Put another way:
x is closer to y than to 0.
For x to be closer to y than to 0, x and y must be TO THE SAME SIDE OF 0, as in the following:
y--x----0------
x--y----0------
------0----y--x
------0----x--y
If x and y are on opposite sides of 0, then x will NOT be closer to y than to 0:
x-----0--y-----
Here, x is closer to 0 than to y.
Since x and y must be to the same side of 0, they have the SAME SIGN.
SUFFICIENT.
The correct answer is
B.
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