For all real number a, b and c such that a > b and c < 0. Which of the following inequalities must be true?

Options

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

Answer

To solve the problem, we need to analyze the given inequalities based on the conditions that \( a > b \) and \( c < 0 \).

### Step-by-Step Solution:

1. **Understanding the conditions**: We know that \( a \) is greater than \( b \) (i.e., \( a > b \)) and \( c \) is a negative number (i.e., \( c < 0 \)).

2. **Analyzing the first option**: The first option states that \( \frac{a}{c} < \frac{b}{c} \).
- Since \( c \) is negative, dividing both sides of the inequality \( a > b \) by \( c \) will reverse the inequality sign:
\[ \frac{a}{c} < \frac{b}{c} \]
- This option is **true**.

3. **Analyzing the second option**: The second option states that \( \frac{a}{c} > \frac{b}{c} \).
- As established in the previous step, this is incorrect because we found that \( \frac{a}{c} < \frac{b}{c} \). Thus, this option is **false**.

4. **Analyzing the third option**: The third option states that \( ac > bc \).
- Since \( c < 0 \), multiplying both sides of \( a > b \) by \( c \) will again reverse the inequality:
\[ ac < bc \]
- Therefore, this option is also **false**.

5. **Analyzing the fourth option**: The fourth option states that \( a + c < b + c \).
- Adding \( c \) (which is negative) to both sides of the inequality \( a > b \) does not change the order of the inequality:
\[ a + c > b + c \]
- Thus, this option is **false**.

### Conclusion:
From the analysis, the only inequality that must be true given the conditions is:
- **Option 1: \( \frac{a}{c} < \frac{b}{c} \)**.