https://file.aiquickdraw.com/mathbot/file/1726049993707fy9hwl3o.jpg

To solve the problem from the image:

The charges for Broker A and Broker B are listed as follows:
- **Broker A**: Fixed platform fee of $2.50 and a commission of 0.08% on the amount invested.
- **Broker B**: Fixed platform fee of $1 and a commission of 0.18% on the amount invested.

We need to find the minimum amount of money that Qian Ren would need to invest for **Broker A** to be cheaper than **Broker B**.

Let the amount of money invested be denoted by $ x $.

### Cost formulas for each broker:
- **Broker A's cost**: $ 2.50 + 0.0008 \cdot x $ (where 0.0008 is 0.08% in decimal form).
- **Broker B's cost**: $ 1 + 0.0018 \cdot x $ (where 0.0018 is 0.18% in decimal form).

We need to find the value of $ x $ where **Broker A's cost is less than Broker B's cost**, i.e.,:
\[
2.50 + 0.0008 \cdot x < 1 + 0.0018 \cdot x
\]

Now, let's solve this inequality.

First, subtract $ 0.0008 \cdot x $ from both sides:
\[
2.50 < 1 + 0.001 \cdot x
\]

Next, subtract 1 from both sides:
\[
1.50 < 0.001 \cdot x
\]

Now, divide both sides by 0.001:
\[
x > 1500
\]

### Conclusion:
Qian Ren would need to invest more than **$1500** for Broker A to be cheaper than Broker B.

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Would you like further clarification or details on this solution? Here are five related questions that expand on this:

1. How would the solution change if the platform fees were different?
2. How can the percentage difference in commission fees be calculated?
3. What happens if Qian Ren invests exactly $1500?
4. How does a small change in the investment amount affect the total fees paid to each broker?
5. Can you find the exact cost for both brokers at an investment of $2000?

**Tip:** Always ensure you carefully interpret commission percentages as they can significantly impact the overall costs when investing large amounts.

Qian Ren is deciding between two brokers to invest a sum of dollars. The charges for Broker A and Broker B are illustrated in the table. By forming an inequality, find the minimum amount of money she would need to invest so that Broker A would be cheaper for her to choose.

Find out how much Qian Ren needs to invest for Broker A to be cheaper than Broker B. Using inequalities, this problem compares fixed platform fees and commission rates to determine the optimal choice. Suitable for Grades 9-12.

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