If `ab gt0`, which of the following must be negative ?

Options

A. `a+b`
B. `|a|+b`
C. `b-a`
D. `-(a)/(b)`

Answer

To solve the problem, we need to analyze the given condition \( ab > 0 \) and determine which of the provided options must be negative.

### Step-by-Step Solution:

1. **Understanding the Condition**:
The condition \( ab > 0 \) indicates that the product of \( a \) and \( b \) is positive. This can occur in two scenarios:
- **Case 1**: Both \( a \) and \( b \) are positive (\( a > 0 \) and \( b > 0 \)).
- **Case 2**: Both \( a \) and \( b \) are negative (\( a < 0 \) and \( b < 0 \)).

2. **Analyzing Each Option**:
We need to check each option to see if it must be negative under both cases.

- **Option A: \( a + b \)**:
- In Case 1 (both positive): \( a + b > 0 \) (not negative).
- In Case 2 (both negative): \( a + b < 0 \) (negative).
- Conclusion: **Not necessarily negative**.

- **Option B: \( |a| + |b| \)**:
- In Case 1: \( |a| + |b| > 0 \) (not negative).
- In Case 2: \( |a| + |b| > 0 \) (not negative).
- Conclusion: **Not necessarily negative**.

- **Option C: \( b - a \)**:
- In Case 1: If \( b < a \), \( b - a < 0 \) (negative), but if \( b > a \), \( b - a > 0 \) (not negative).
- In Case 2: If both are negative, the result can vary based on their values.
- Conclusion: **Not necessarily negative**.

- **Option D: \( -\frac{a}{b} \)**:
- In Case 1: \( -\frac{a}{b} < 0 \) (negative).
- In Case 2: \( -\frac{a}{b} < 0 \) (negative).
- Conclusion: **Must be negative**.

3. **Final Conclusion**:
The only option that must be negative in both cases is **Option D: \( -\frac{a}{b} \)**.

### Final Answer:
**Option D must be negative.**

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