Is the product a * b positive? I. a + b is positive. II. a − b is positive.

Introduction / Context:
The goal is to infer the sign of the product a*b using two inequalities involving a and b. This is a classic reasoning pattern that uses simultaneous inequalities to deduce the signs of variables.

Given Data / Assumptions:

• Statement I: a + b > 0.
• Statement II: a − b > 0.
• Numbers are real.

Concept / Approach:
If we can determine that both a and b are positive, then a*b > 0. Conversely, mixed signs may yield a negative product. Use linear combinations of the two inequalities.

Step-by-Step Solution:
Add the inequalities: (a + b) + (a − b) = 2a > 0 ⇒ a > 0.Subtract the second from the first: (a + b) − (a − b) = 2b > 0 ⇒ b > 0.Thus, from I and II together, a > 0 and b > 0 ⇒ a*b > 0.Statement I alone allows cases such as a = 10, b = −1 (sum positive but product negative), so it is not sufficient.Statement II alone allows a = 1, b = 0.5 (product positive) or a = 1, b = −0.5 (product negative); hence it is not sufficient.

Verification / Alternative check:
Testing values confirms that both inequalities together force both variables to be positive.

Why Other Options Are Wrong:
I alone or II alone do not guarantee the product sign; together they do. Claiming insufficiency contradicts the derived positivity.

Common Pitfalls:
Assuming a − b > 0 implies b > 0; it only implies a > b. You need both inequalities to deduce signs.

Final Answer:
Both statements together are necessary and sufficient; the product is positive.