Is xy > 0? x - y > -2 \(\mathrm{x} - 2\mathrm{y}...

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