Understanding the Question
We need to determine whether the product xy is positive.
Let's think about when a product is positive:
- Both positive: If \(\mathrm{x > 0}\) and \(\mathrm{y > 0}\), then \(\mathrm{xy > 0}\)
- Both negative: If \(\mathrm{x < 0}\) and \(\mathrm{y < 0}\), then \(\mathrm{xy > 0}\)
- Mixed signs or zero: If the signs differ or either is zero, then \(\mathrm{xy ≤ 0}\)
So we're really asking: Do x and y have the same sign (both positive OR both negative)?
For this yes/no question to be sufficient, we need a definitive answer:
- YES: xy is always positive (same signs guaranteed)
- NO: xy is never positive (opposite signs or zero guaranteed)
Analyzing Statement 1
Statement 1: \(\mathrm{x - y > -2}\)
This tells us that \(\mathrm{x > y - 2}\), meaning x is "not too far below" y (at most 2 units below).
Let's test if this forces x and y to have the same sign:
- Can both be positive? Try \(\mathrm{x = 3, y = 2}\). Check: \(\mathrm{3 - 2 = 1 > -2}\) ✓. Product: \(\mathrm{xy = 6 > 0}\)
- Can one be zero? Try \(\mathrm{x = 0, y = 1}\). Check: \(\mathrm{0 - 1 = -1 > -2}\) ✓. Product: \(\mathrm{xy = 0}\)
Since we found:
- A case where \(\mathrm{xy > 0}\) (both positive)
- A case where \(\mathrm{xy = 0}\) (one is zero)
We cannot determine whether \(\mathrm{xy > 0}\).
Statement 1 is INSUFFICIENT.
Analyzing Statement 2
Statement 2: \(\mathrm{x - 2y < -6}\)
This tells us that \(\mathrm{x < 2y - 6}\), meaning x is "significantly below" twice the value of y.
Let's test different sign combinations:
- Can both be positive? If \(\mathrm{y = 5}\), then \(\mathrm{x < 2(5) - 6 = 4}\). So \(\mathrm{x = 3}\) works ✓. Product: \(\mathrm{xy = 15 > 0}\)
- Can they have opposite signs? If \(\mathrm{y = 2}\), then \(\mathrm{x < 2(2) - 6 = -2}\). So \(\mathrm{x = -3}\) works ✓. Product: \(\mathrm{xy = -6 < 0}\)
Since we found:
- A case where \(\mathrm{xy > 0}\) (both positive)
- A case where \(\mathrm{xy < 0}\) (opposite signs)
We cannot determine whether \(\mathrm{xy > 0}\).
Statement 2 is INSUFFICIENT.
Combining Both Statements
Using both statements together, x must satisfy:
- \(\mathrm{x > y - 2}\) (from Statement 1)
- \(\mathrm{x < 2y - 6}\) (from Statement 2)
For both conditions to work simultaneously, we need:
\(\mathrm{y - 2 < 2y - 6}\)
Solving: \(\mathrm{-2 < y - 6}\), which gives us \(\mathrm{y > 4}\)
Now here's the key insight:
- Since \(\mathrm{y > 4}\) (positive!)
- And \(\mathrm{x > y - 2 > 4 - 2 = 2}\) (also positive!)
- Both variables are forced to be positive
Therefore, \(\mathrm{xy > 0}\) in all cases.
[STOP - Sufficient!]
Combined, the statements are SUFFICIENT.
Answer: C
