Is 1/(a-b)<(b-a)?
1. a<b
2. 1<|a-b|
Ans: A
Statement 1: If a is smaller than b, then the left hand side of the inequality in the question will be negative, and the right hand side will be positive (and the answer to the question is definitely yes): When you subtract something bigger (b) from something smaller (a), the result is always negative. And when you subtract something smaller (a) from something bigger, the result is always positive. Sufficient.
Statement 2: |a-b|>1 means a-b is more than one unit away from zero on the number line. There are two ways this can happen: either a-b>1 or a-b<-1
case 1: a-b>1: In this case, the left hand side of the inequality in the question is clearly positive (even though it would be a fraction).
For the right hand side:
a-b>1
factoring out negative 1 on the left hand side:
-1(b-a)>1
dividing both sides by negative one (gotta flip the sign)
b-a<-1
So, under case 1, the left hand side is positive but the right hand side is negative, and the answer to the question is "no".
case 2: a-b<-1
Under case 2, the left hand side of the inequality in the question is clearly negative.
For the right hand side:
a-b<-1
-1(b-a)<-1
b-a>1
So, under case 2, the left hand side of the inequality in the question is negative, and the right hand side is positive, and the answer to the question is "yes".
So, in statement 2, we can get both a "yes" and a "no" answer to the question. Insufficient.
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