Given that x , y and b are real numbers and `x le y, b lt 0 ` then

Options

A. A. `x/b lt y/b`
B. B. `x/b le y/b`
C. C. `x/b gt y/b`
D. D. `x/b ge y/b`

Answer

To solve the problem, we start with the given inequalities:

1. \( x \leq y \)
2. \( b < 0 \)

We need to analyze the implications of these inequalities when we divide by \( b \).

### Step 1: Understand the implications of dividing by a negative number
When we divide both sides of an inequality by a negative number, the direction of the inequality reverses. This is a fundamental property of inequalities.

### Step 2: Divide the inequality by \( b \)
Since \( b < 0 \), we can divide both sides of the inequality \( x \leq y \) by \( b \):

\[ \frac{x}{b} \quad \text{and} \quad \frac{y}{b} \]

### Step 3: Reverse the inequality
Applying the rule of reversing the inequality when dividing by a negative number, we get:

\[ \frac{x}{b} \geq \frac{y}{b} \]

### Conclusion
Thus, the final result of dividing the inequality \( x \leq y \) by \( b \) (where \( b < 0 \)) is:

\[ \frac{x}{b} \geq \frac{y}{b} \]

This means that the correct option is:

**D: \( \frac{x}{b} \geq \frac{y}{b} \)**