When you're given absolute values with inequalities, you have to establish different cases for POSITIVES v. NEGATIVES.
First, let's interpret the given: |x - a| < |x - b|
Test cases for x, a, and b that fit:
etc. (You wouldn't really need to test all of these, nor is this exhaustive. I'm just giving several examples).
We need more information on whether x, a, and b are positive or negative, and/or information about their relative sizes before we know anything concrete.
Now interpret the question: a < ba < b?
When would the product of two numbers be greater than one number, but smaller than the other?
- Neither of the numbers could be 0 or 1, because then 2 of those values would be equal.
- They can't both be positive integers, because then ab would be greater than a or b.
- a could be a fraction between 0 and 1, and b could be greater than 1. ab then would be in between
- They cannot both be negative, since ab would then be positive
- It could be that a is negative less than -1, and b is a positive fraction btw 0 and 1
There is not one simple way to rephrase this question - there are a variety of scenarios to consider.
1. ab < 0
This tells us that one of the variables is negative, the other is positive. With the given information, several of our examples from the given information fit:
x = 10, a = 3, b = -2
--> a < ab < b ?
--> 3 < -6 < -2
--> NO.
x = 1/8, a = -1/8, b = 7/8
--> a < ab < b ?
--> -1/8 < -7/64 < 7/8
--> YES.
Insufficient.
2. For all x > 0, |x - a| = |x| + |a|
This doesn't tell us anything about b, so we can tell from a glance that it's insufficient.
But, here's how we would interpret: For all x > 0, |x - a| = |x| + |a|
Since x must be positive, a must be either negative or 0.
Together:
(1) a and b have different signs (one negative, one positive)
(2) x is positive, a </= 0
See what scenarios remain from our original analysis:
Two of our original scenarios work.
#1: a = -3, b = 8
--> a < ab < b ?
--> -3 < -24 < 8
--> No.
#2: a = -1/8, b = 7/8
--> a < ab < b ?
--> -1/8 < -7/64 < 7/8
--> Yes.
(1) and (2) together are INSUFFICIENT. The correct answer should be E.
