Quote:
If A and B are non-zero numbers such that AB>0, is |A−B|>|A|−|B|?
(1) \(\frac{1}{A}<\frac{1}{B}\)
(2) 2A+B<0
What an intriguing question! There's a lot going on here. Let's slow down and unpack it bit by bit.
First, we have
AB>0. That means A and B have the same sign: either they're both positive, or they're both negative.
Now, what we want to know:
is |A−B|>|A|−|B|?. I spent some time trying to simplify this, but the simplification took a long time, and was even harder to explain than it was to do! So, I think the right approach on test day would be to leave this how it is, write it down on your paper, and get ready to carefully and thoughtfully test cases.
Let's do that now.
Statement 1: 1/A < 1/B.
I'd like to simplify this, but I'm wary, because I know that you usually can't multiply both sides of an inequality by a variable. That's because if you don't know whether the variable is positive or negative, you don't know whether you're supposed to flip the sign or not. But we have something we can actually use here! Because the problem says that AB > 0, we know that AB is positive. So, we can safely multiply both sides of the inequality by AB.
1/A * AB < 1/B * AB
B < A
This statement really says that B < A.
Now let's try some cases where B < A. We're dealing with absolute values, so we should try both positive and negative cases.
Case 1:
B = 1, A = 10. Is |A−B|>|A|−|B|?
|A - B| = |10 - 1| = |9| = 9
|A| - |B| = |10|-|1| = 10 - 1 = 9
Since they're equal, the answer is "no"
Case 2:
B = -10, A = -1. Is |A−B|>|A|−|B|?
|A - B| = |-1 - (-10)| = |9| = 9
|A| - |B| = |-1| - |-10| = 1 - 10 = -9
The left side is bigger, so the answer is "yes."
We got both types of answers, so this statement is
insufficient.
Statement 2: 2A+B<0
We know, from the original question, that A and B are either both positive or both negative. They definitely can't both be positive anymore, though, given this statement! They must both be negative.
And, if they're both negative... this statement is definitely true! This will be true for
any pair of negative values of A and B.
So really, all this is telling us (in the context of this specific problem and the info we already have) is that A and B are both negative. Let's test some cases.
Case 1: First, reuse 'case 2' from the previous statement. That case gave us a "yes" answer.
Case 2: Now, I'm going to try to find a case that works the other way. It seemed to be important whether A is bigger or whether B is bigger, so this time I'm going to try a case where B is bigger than A. Let's say B = -3, and A = -8. Is |A - B| > |A| - |B|?
|A - B| = |-8 - (-3)| = |-5| = 5
|A| - |B| = |-8| - |-3| = 8 - 3 = 5
They're equal, so the answer is "no."
Since we got both a "yes" and a "no," this statement is also insufficient.
Statements 1 and 2 together:
Let's evaluate what we know at this point.
We know from Statement 2 that A and B are both negative.
We also know from statement 1 that A > B.
Is |A - B| > |A| - |B|?
Since A > B, A - B will be a positive number. Therefore, |A - B| = A - B.
Since A and B are both negative, |A| = -A, and |B| = -B.
The question is really asking: Is A - B > -A - (-B)?
Simplify:
Is A - B > B - A?
Is 2A > 2B?
Is A > B?
We already know that the answer to this question is "yes." So, the statements are sufficient together, and the answer to the problem is
C.
Nice one!
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