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Roman numeral questions tend to take some extra time. Look for opportunities to combine work. After you evaluate each Roman numeral, eliminate all the answer choices you can.

The question stem indicates that the absolute value of a times b is greater than just a times b by itself. Try a couple of values for a and b to understand what this means. If a = 1 and b = 2, then it's not true that |2| > 2. So these values aren't possible for a and b. What do you need to change to make this work?
If a = –1 and b = 2, then it's true that |–2| > –2.
What if both variables are negative? If a = –1 and b = –2, then it's not true that |2| > 2.

So exactly one of the two variables has to be negative.
I. a < 0: This could be true, but it doesn't have to be true, since b could be the negative value instead. Eliminate answers (A) and (D).

II. b < 0: This could be true, but it doesn't have to be true, since a could be the negative value instead. Eliminate answers (B) and (E).

And then choose your answer! There's only one left, so it's not necessary to evaluate Roman numeral III. (This one must be true because exactly one of the two variables has to be negative. Since one variable is negative and one is positive, the product of a and b has to be negative.)

The correct answer is (C).