Question

If $x>2$ and $y>-1$, which is true?

• A. $xy>-2$
• B. $-x<2y$
• C. $xy<-2$
• D. $-x>2y$

Hint:

Test extreme boundary values for inequalities to determine the strict bound.

Answer 1

The Correct Option is A

We are given two conditions:

• \(x > 2\) — meaning \(x\) is a positive number greater than \(2\).
• \(y > -1\) — meaning \(y\) can be any real number greater than \(-1\).

We need to find the range of possible values for the product \(xy\).

Step 1: Understanding the minimum possible value of \(y\)

Since \(y > -1\), the smallest possible value of \(y\) is a number just slightly greater than \(-1\). We can write this as: \[ y \to -1^+ \quad \text{(approaching \(-1\) from above)} \]

Step 2: Finding the minimum value of \(xy\)

The smallest \(y\) occurs when \(y\) is as close as possible to \(-1\). The smallest \(x\) allowed by \(x > 2\) is just slightly greater than \(2\), i.e., \(x \to 2^+\). The product in this extreme case would be: \[ xy > (2) \times (-1) = -2 \] This means that no matter what values we choose (as long as \(x > 2\) and \(y > -1\)), \(xy\) will always be greater than \(-2\).

Step 3: Conclusion

Therefore, the inequality that always holds is: \[ \boxed{xy > -2} \]

Step 4: Interpretation

The value \(-2\) is a boundary for the product \(xy\). While \(xy\) can be as close as we like to \(-2\), it will never be equal to \(-2\) because \(x\) cannot be exactly \(2\) and \(y\) cannot be exactly \(-1\).