If x and y are both positive, is x greater than 4?

MANHATTAN GMAT OFFICIAL SOLUTION:

We are told that both x and y are positive, and we are asked whether x is greater than 4. There’s not much rephrasing we can do to this question, so let’s move on to the statements.

Statement (1): INSUFFICIENT. All we know here is that x is larger than y. It is still possible for x to be any positive number (y just has to be a smaller positive number).

Statement (2): INSUFFICIENT. If the point (x, y) lies outside a circle of radius 5 centered at the origin, then that point lies at a distance of more than 5 units away from (0, 0). In the coordinate plane, distance can be computed with the Pythagorean Theorem. We can rephrase this statement to say that x^2 + y^2 > 25. However, x can still be as small or as large a positive number as we wish.

Statement (1) and (2) together: INSUFFICIENT. We can definitely pick a very large value for x to satisfy the statements and answer the question with a “Yes.” Just make y smaller to make the first statement true, and if x is bigger than 5, then we’ll definitely have statement (2) true as well.

The trick is that we can pick a value of x that is not greater than 4 and still satisfy all the conditions. Let’s try picking x equal to 4 (this would give us an answer of “No” to the question). So we need to see whether there are any values of y that satisfy the following 3 conditions:

1. y is positive (from the stem).
2. y is less than 4 (that is, less than x).
3. x^2 + y^2 > 25.

Let’s plug 4 in for x in the last inequality. We get

16 + y^2 > 25

y^2 > 9

y > 3

(since y must be positive, we don’t have to worry about the negative possibilities)

The conditions become these: y is greater than 3 and less than 4. Any number between 3 and 4 satisfies the conditions. Notice that y is not restricted to integer values; nothing in the problem indicates that it should be. Thus, we still cannot definitively answer the question of whether x is greater than 4. There are other values of x less than 4 that will work; one tricky part of this problem is that those values of x are greater than 3.

The correct answer is E.