Official Guide Explanation:
Problem Solving #176

Background

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Solution and Metadata

Question: 176
Page: 177
Difficulty: 5 (Moderate)
Category 1: Algebra > Equations >
Category 2: Algebra > Linear Equations-One Unk >
Category 3: Arithmetic > Fractions >

Explanation: In the first two roman numerals, you can ignore the constants: regardless of what happens to x, -5 and 1 won't change. So, I increases: as x gets larger, so does 2x. II is tricker. As x gets bigger, (1/(x)) gets smaller, but as (1/(x)) gets smaller, -(1/(x)) gets bigger, so II increases. III is also somewhat involved, but we don't have to do it: if I and II increase, the only possible answer choice is (C).

For the sake of completeness, here's how you can tell that III does not increase. First, forget about the fraction and focus on x²-x. As x increases, x² increases, while x does not. But x² increases by a greater amount--166²-165² is greater than 166 - 165. So, x²-x increases. However, since III is the inverse of that, knowing that x²-x increases tells us that (1/(x²-x)) decreases.

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