Question

Evaluate the following expression: \[ \left( \frac{X \ Y}{- X \times Y} \right) = \frac{3}{8}, \] \text{then which of the following must be true?}

• A. \( X = 2, Y = 1 \)
• B. \( X > 0, Y < 0 \)
• C. \( X, Y \text{ both positive} \)
• D. \( X, Y \text{ both negative} \)

Hint:

When solving equations involving operations, always consider the signs of the variables and check the validity of the operations based on those signs.

Answer 1

The Correct Option is B

We are given the equation: \[ \frac{X \ Y}{- X \times Y} = \frac{3}{8}. \] Step 1: Solve for \( X \ Y \) Using the definition of the operation \( \ \), we know: \[ X \ Y = X + Y \quad \text{(if both \( X \) and \( Y \) are positive)}. \] Substitute this into the equation: \[ \frac{X + Y}{- X \times Y} = \frac{3}{8}. \] Step 2: Simplify the equation Now multiply both sides by \( - X \times Y \): \[ X + Y = \frac{3}{8} \times (- X \times Y). \] \[ X + Y = - \frac{3}{8} X \times Y. \] Step 3: Analyze the equation We need to find conditions for \( X \) and \( Y \) that satisfy this equation. For the equation to hold, we analyze the signs of \( X \) and \( Y \). - If \( X > 0 \) and \( Y < 0 \), the product \( - X \times Y \) will be positive, making the equation valid. - Other combinations of signs for \( X \) and \( Y \) will not satisfy this equation. Thus, the correct answer is that \( X > 0 \) and \( Y < 0 \).