study wrote:
Is |x - y| > |x| - |y|?
(1) y < x
(2) xy < 0
Attachment:
3_DS_Absolute_B.JPG
Target question: Is |x - y| > |x| - |y|? Statement 1: y < x Let's test some values.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 2 and y = 1. In this case, |x - y| = |2 - 1| = 1 and |x| - |y| = |2| - |1| = 1. So, the answer to the target question is
NO, |x - y| is NOT greater than |x| - |y|Case b: x = 2 and y = -1. In this case, |x - y| = |2 - (-1)| = 3 and |x| - |y| = |2| - |-1| = 1. So, the answer to the target question is
YES, |x - y| IS greater than |x| - |y|Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: xy < 0This tells us that one value (x or y) is positive, and the other value is negative. This sets up two possible cases:
Case a: x is positive and y is negative.
So, we're taking a positive value (x) and subtracting a negative value (y). Doing so yields a positive value that is bigger than x.
In other words, we have:
0 < x < |x - y| Now let's examine |x| - |y|
Since x is positive, we know that |x| = x
Since y ≠ 0, we know that 0 < |y|
So, |x| - |y| = x - |y| = some number less than x
In other words,
|x| - |y| < xWhen we combine the inequalities we get:
|x| - |y| < x < |x - y| In this case, the answer to the target question is
YES, |x - y| IS greater than |x| - |y|Case b: x is negative and y is positive.
Here, we're taking a negative value (x) and subtracting a positive value (y). Doing so yields a negative value that is less than x.
In other words, we have: x - y < x < 0 < |x|
Important: since the MAGNITUDE of x - y is greater than the MAGNITUDE of x, we can write:
|x| < |x - y|Now let's examine |x| - |y|
Since x ≠ 0, we know that 0 < |x|
Since |x| is a positive number, we know that subtracting |y| (another positive value) will yield a number that is LESS THAN |x|
In other word,
|x| - |y| < |x|When we combine the inequalities we get:
|x| - |y| < |x| < |x - y|In this case, the answer to the target question is
YES, |x - y| IS greater than |x| - |y|In both possible cases, the answer to the target question is the same:
YES, |x - y| IS greater than |x| - |y|Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: B
Cheers,
Brent