Given that x,y and b real are real numbers and `x ge y , b gt 0` , then

Options

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

Answer

To solve the problem step-by-step, we start with the given information and analyze the options provided.

### Step 1: Understand the Given Information
We know that:
- \( x \geq y \) (which means \( x \) is greater than or equal to \( y \))
- \( b > 0 \) (which means \( b \) is a positive real number)

### Step 2: Analyze the Options
We have four options to evaluate:
- Option A: \( \frac{x}{b} \geq \frac{y}{b} \)
- Option B: \( \frac{x}{b} > \frac{y}{b} \)
- Option C: \( \frac{x}{b} \leq \frac{y}{b} \)
- Option D: \( \frac{x}{b} < \frac{y}{b} \)

### Step 3: Dividing by a Positive Number
Since \( b > 0 \), we can divide both sides of the inequality \( x \geq y \) by \( b \) without changing the direction of the inequality.

So, we divide:
\[ \frac{x}{b} \geq \frac{y}{b} \]

### Step 4: Conclusion
From the above step, we conclude that:
- The statement \( \frac{x}{b} \geq \frac{y}{b} \) holds true.

Thus, the correct option is **Option A**: \( \frac{x}{b} \geq \frac{y}{b} \).

### Step 5: Eliminate Other Options
- **Option B**: \( \frac{x}{b} > \frac{y}{b} \) cannot be concluded since \( x \) could be equal to \( y \).
- **Option C**: \( \frac{x}{b} \leq \frac{y}{b} \) is incorrect as it contradicts our conclusion.
- **Option D**: \( \frac{x}{b} < \frac{y}{b} \) is also incorrect for the same reason.

### Final Answer
The final answer is **Option A**: \( \frac{x}{b} \geq \frac{y}{b} \).

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