If y = –x, then is y > –1 ? (1) x^4 > x^2 (2) x^3 < x^2

The variable y only appears once in the problem. It looks like it's just intended to distract us, so let's try getting rid of it first. The question asks whether y > -1, and we also know that y = -x. So, it's asking whether -x > -1. Multiply both sides of this inequality by -1, and flip the inequality symbol. The question translates to "Is x < 1", and now we can ignore y for the rest of the problem!

Statement 1: x^4 > x^2. If this is true, then can we be certain that x < 1?

There are a few different ways to approach this one. A mathematical way to approach it would be to start by dividing both sides of the inequality by x^2. You normally aren't allowed to divide an inequality by a variable like that, but it's actually okay here, since x^2 is always positive (and x can't be 0, because that wouldn't fit the statement.) So, let's divide:

x^4 > x^2
x^2 > 1

This statement really just says that x^2 is greater than 1. What does that tell us? Well, x could be a large positive number, like 100. But x could also be a negative number, like -100. Therefore, x might be greater than 1, or less than 1. So, the statement is insufficient.

Another way to approach this is to test cases. But, you'd have to be very careful to only test cases that fit the statement, and cross off any cases you accidentally test that don't match the statement.

For instance, if we test x = 1/2, then x^4 = 1/16 and x^2 = 1/4. But, since x^4 isn't greater than x^2, we can't look at that case at all and we have to cross it off.

But, we could test x = 2, and x = -2. Both of those match the statement, but one of them is greater than 1 and the other is less than 1. Therefore, the statement is insufficient.

Statement 2: You can do either of the same things with this one. Mathematically, if you divide by x^2, it translates to x < 1. Therefore, the statement is sufficient.

Testing cases, the only cases that work are either positive fractions, or negative numbers (since a negative number cubed will be negative, but a negative number squared will become positive.) Both of these are less than 1.

Since statement 2 is sufficient, the answer is B.