Under which of the following conditions is `(x(x-y))/(y)` negative?

Options

A. xltylt0
B. yltxlt0
C. 0ltxlt0
D. xlt0lty

Explanation

To determine under which conditions the expression \(\frac{x(x-y)}{y}\) is negative, we need to analyze the expression step by step.

### Step 1: Analyze the expression
The expression is \(\frac{x(x-y)}{y}\). For this expression to be negative, the numerator and denominator must have opposite signs.

### Step 2: Identify the numerator and denominator
- **Numerator**: \(x(x-y)\)
- **Denominator**: \(y\)

### Step 3: Determine the sign of the denominator
The sign of the denominator \(y\) will influence the overall sign of the expression:
- If \(y > 0\), then the denominator is positive.
- If \(y < 0\), then the denominator is negative.

### Step 4: Analyze the numerator \(x(x-y)\)
The numerator \(x(x-y)\) can be analyzed based on the signs of \(x\) and \(x-y\):
- If \(x > 0\) and \(x - y > 0\), then \(x(x-y)\) is positive.
- If \(x > 0\) and \(x - y < 0\), then \(x(x-y)\) is negative.
- If \(x < 0\) and \(x - y > 0\), then \(x(x-y)\) is negative.
- If \(x < 0\) and \(x - y < 0\), then \(x(x-y)\) is positive.

### Step 5: Combine conditions for negativity
To make the entire expression negative, we can have:
1. **Case 1**: \(y < 0\) and \(x(x-y) > 0\)
2. **Case 2**: \(y > 0\) and \(x(x-y) < 0\)

### Step 6: Evaluate the given options
1. **Option A**: \(x < y < 0\)
- Here, \(y < 0\) (denominator is negative).
- Since \(x < y\), both \(x\) and \(y\) are negative, so \(x-y > 0\) (because subtracting a negative number results in a positive).
- Thus, \(x(x-y) < 0\) (negative times positive is negative).
- Overall, the expression is negative.

2. **Option B**: \(y < x < 0\)
- Here, \(y < 0\) (denominator is negative).
- Since \(y < x\), both are negative, but \(x - y < 0\).
- Thus, \(x(x-y) > 0\) (negative times negative is positive).
- Overall, the expression is positive.

3. **Option C**: \(0 < x < 0\)
- This is not valid as \(x\) cannot be both greater than and less than 0.

4. **Option D**: \(x < 0\) and \(y > 0\)
- Here, \(y > 0\) (denominator is positive).
- Since \(x < 0\), \(x-y < 0\).
- Thus, \(x(x-y) > 0\) (negative times negative is positive).
- Overall, the expression is positive.

### Conclusion
The only condition under which \(\frac{x(x-y)}{y}\) is negative is in **Option A**: \(x < y < 0\).