If a, b and c are different integers such that `a lt b lt c lt 0`, then which of the following statements is true ?

Options

A. `a + c lt b`
B. `ab lt c`
C. `a + b gt c`
D. `ac gt ab`

Answer

To solve the problem, we need to analyze the given conditions and the options provided. The integers \( a, b, \) and \( c \) are different integers such that \( a < b < c < 0 \). This means all three integers are negative.

### Step-by-Step Solution:

1. **Understanding the Range of Values**:
Since \( a, b, \) and \( c \) are negative integers and \( a < b < c < 0 \), we can assign values to \( a, b, \) and \( c \). Let's assume:
- \( a = -3 \)
- \( b = -2 \)
- \( c = -1 \)

2. **Evaluating Option A**: \( a + c < b \)
- Calculate \( a + c \):
\[ a + c = -3 + (-1) = -4 \]
- Check if \( -4 < -2 \):
\[ -4 < -2 \quad \text{(True)} \]
- Therefore, Option A is true.

3. **Evaluating Option B**: \( ab < c \)
- Calculate \( ab \):
\[ ab = (-3) \times (-2) = 6 \]
- Check if \( 6 < -1 \):
\[ 6 < -1 \quad \text{(False)} \]
- Therefore, Option B is false.

4. **Evaluating Option C**: \( a + b > c \)
- Calculate \( a + b \):
\[ a + b = -3 + (-2) = -5 \]
- Check if \( -5 > -1 \):
\[ -5 > -1 \quad \text{(False)} \]
- Therefore, Option C is false.

5. **Evaluating Option D**: \( ac > ab \)
- Calculate \( ac \):
\[ ac = (-3) \times (-1) = 3 \]
- Calculate \( ab \) (already calculated):
\[ ab = 6 \]
- Check if \( 3 > 6 \):
\[ 3 > 6 \quad \text{(False)} \]
- Therefore, Option D is false.

### Conclusion:
The only true statement among the options is **Option A**: \( a + c < b \).