Explanation

In this question, y is negative, and we are imagining the situation in which it's traveling to the left down the number line. Roman numeral I definitely also travels down the number line - even faster than y does, in fact.

For II, we can reformat as . Since fractions are involved, we can try fractional y's. I'm trying to disprove this by finding increase. Say y goes from to -10. That's a decrease in y. The expression goes from to , so it decreased as well. You can break it down into the case in which the absolute value of y is less than one, greater than one, or crosses. We just did crossing.Meanwhile, if y goes from -10 to -100, this thing goes from or to . If y goes from to , the expression goes from 5 to 4. At this point, we should stop and conclude that it decreases, but see the note below.

On to III. This thing need not decrease. For example, when y goes from -1 to -10, this thing moves from 2 to 110. I is in, and III is out. The correct answer is (C), I and II.

The ideal situation when evaluating by cases is that you can extrapolate a rule from the cases after a couple. In Roman numeral II here, it boils down to the properties of what we could call . In the coordinate plane, this is a hyperbola - something that does not appear in the official test rubric, but which indirectly appears in some questions. You can play around with such equations on a graphing calculator (if you know someone in high school) or on a site such as Wolfram Alpha. Nevertheless, sometimes when evaluating by cases, it is difficult to know whether we have been exhaustive, and therefore whether we have been conclusive. It is critical not to get stuck in this cases. Make an evaluation and move on, even if it's not totally conclusive.

Again, the correct answer is (C).