`x gt 2, y gt -1` the which of the following holds good ?

Options

A. `xy gt -2`
B. `xy lt -2`
C. `x gt -(2)/(y)`
D. None of these

Answer

To solve the problem where we have the inequalities \( x > 2 \) and \( y > -1 \), we need to determine what can be inferred from these inequalities.

### Step-by-Step Solution:

1. **Identify the inequalities**: We are given two inequalities:
\[ x > 2 \quad \text{and} \quad y > -1 \]

2. **Multiply the inequalities**: To find a relationship involving \( x \) and \( y \), we can multiply the left-hand sides and the right-hand sides of the inequalities:
\[ x \cdot y > 2 \cdot (-1) \]
This simplifies to:
\[ x \cdot y > -2 \]

3. **Conclusion**: From the multiplication of the inequalities, we conclude that:
\[ x \cdot y > -2 \]
This means that the product of \( x \) and \( y \) is greater than \(-2\).

4. **Verification with examples**: To verify this, we can choose specific values for \( x \) and \( y \):
- Let \( x = 3 \) (which is greater than 2) and \( y = 0 \) (which is greater than -1).
- Then, \( x \cdot y = 3 \cdot 0 = 0 \), which is indeed greater than \(-2\).
- Another example: Let \( x = 4 \) and \( y = -0.5 \). Then, \( x \cdot y = 4 \cdot (-0.5) = -2 \), which is equal to \(-2\).

Thus, we can conclude that the condition \( x \cdot y > -2 \) holds true.

### Final Answer:
The correct conclusion is that \( x \cdot y > -2 \).

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