Under which conditions is `(xy)/(x-y)` negative?

Options

A. `0 lt y lt x`
B. `x lt y lt 0`
C. `x lt 0 lt y`
D. `y lt x lt 0`

Explanation

To determine under which conditions the expression \(\frac{xy}{x-y}\) is negative, we need to analyze the signs of the numerator \(xy\) and the denominator \(x-y\).

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression \(\frac{xy}{x-y}\).

2. **Determine Conditions for Negativity**:
The fraction \(\frac{a}{b}\) is negative if \(a\) and \(b\) have opposite signs. Therefore, we need to analyze the signs of \(xy\) (numerator) and \(x-y\) (denominator).

3. **Analyze the Numerator \(xy\)**:
- \(xy > 0\) when both \(x\) and \(y\) are either positive or both are negative.
- \(xy < 0\) when one of \(x\) or \(y\) is positive and the other is negative.

4. **Analyze the Denominator \(x-y\)**:
- \(x-y > 0\) when \(x > y\).
- \(x-y < 0\) when \(x < y\).

5. **Case Analysis**:
We will consider different cases based on the signs of \(x\) and \(y\):

- **Case 1**: Both \(x > 0\) and \(y > 0\):
- Here, \(xy > 0\) and \(x - y\) can be either positive or negative depending on the specific values of \(x\) and \(y\). If \(x > y\), then the expression is positive; if \(x < y\), it is negative.

- **Case 2**: Both \(x < 0\) and \(y < 0\):
- Here, \(xy > 0\) (since a negative times a negative is positive) and \(x - y < 0\) (since \(x < y\)). Thus, the expression is negative.

- **Case 3**: \(x < 0\) and \(y > 0\):
- Here, \(xy < 0\) (since a negative times a positive is negative) and \(x - y < 0\) (since \(x < y\)). Thus, the expression is positive.

- **Case 4**: \(x > 0\) and \(y < 0\):
- Here, \(xy < 0\) (since a positive times a negative is negative) and \(x - y > 0\) (since \(x > y\)). Thus, the expression is negative.

6. **Conclusion**:
The expression \(\frac{xy}{x-y}\) is negative under the following conditions:
- Both \(x < 0\) and \(y < 0\) (Case 2).
- \(x > 0\) and \(y < 0\) (Case 4).

### Summary:
The expression \(\frac{xy}{x-y}\) is negative when:
- \(x < 0\) and \(y < 0\)
- \(x > 0\) and \(y < 0\)