Bunuel wrote:

Is \(xy \leq 6\) ?

(1) \(1 \leq x \leq 2\) and \(4 \leq y \leq 6\)

(2) x + y = 7

Is \(xy \leq 6\)? If x and y are opposite signs, then xy is ≤ 6; If they are the same sign, then we must solve for x and y.

(1) \(1 \leq x \leq 2\) and \(4 \leq y \leq 6\). If we multiply both inequalities we have: \(4 \leq xy \leq 12\), then xy could be ≥ 6 or xy ≤ 6, insufficient.

(2) \(x + y = 7\). If x = 3 and y = 4, xy = 12, condition is false. If x = 1 and y = 6, xy = 6, condition is true. Both cases, x and y could be opposite signs or not.. insufficient.

(3) Combining both: (1) x and y are the same signs (positive), and (2) their sum is equal to 7. There are

only 2 solutions for x that meet the inequality: x = 1 or x = 2. And to meet the (2) statement, y has to be y = 6 or y = 5. Thus, xy is either (1)(6) = 6 or (2)(5) = 10, which both are ≥ 6, sufficient.

(C) is the answer.

Based on what you have provided, the answer should be E.

Also small comment. The highlighted is not correct. You assumed that x and y are integers but x could be 1.9 & y could be 5.1. Nothing in the questions says that those are integers.