Roberto is an insurance agent who sells two types of policies: a $50,000 policy and a $100,000 policy. Last month, his goal was to sell at least 57 insurance policies. While he did not meet his goal, the total value of the policies he sold was over $3,000,000. Which of the following systems of inequalities describes x, the possible number of $50,000 policies, and y, the possible number of $100,000 policies, that Roberto sold last month?

Options

A. x+y lt 57
50,000x + 100,000y lt 3,000,000
B. x+y gt 57
50,000x +100,000y gt 3,000,000
C. x+ylt57
50,000x + 100,000 y gt 3,000,000
D. x+ygt57
50,000x + 100,000y lt 3,000,000

Explanation

To solve the problem, we need to set up a system of inequalities based on the information provided about Roberto's insurance policies.

1. **Identify Variables**:
- Let \( x \) be the number of $50,000 policies sold.
- Let \( y \) be the number of $100,000 policies sold.

2. **Total Policies Sold**:
- Roberto's goal was to sell at least 57 policies, but he did not meet this goal. This means:
\[ x + y < 57 \]

3. **Total Value of Policies Sold**:
- The total value of the policies sold was over $3,000,000. The value from $50,000 policies is \( 50,000x \) and from $100,000 policies is \( 100,000y \). Therefore, we can express this as:
\[ 50,000x + 100,000y > 3,000,000 \]

4. **Final System of Inequalities**:
- Combining the two inequalities, we have:
\[ \begin{cases} x + y < 57 \\ 50,000x + 100,000y > 3,000,000 \end{cases} \]

5. **Identify the Correct Option**:
- From the options provided, we see that option 3 matches our derived inequalities:
- \( x + y < 57 \)
- \( 50,000x + 100,000y > 3,000,000 \)

Thus, the correct system of inequalities that describes the possible number of $50,000 policies (x) and $100,000 policies (y) that Roberto sold last month is:
\[ \begin{cases} x + y < 57 \\ 50,000x + 100,000y > 3,000,000 \end{cases} \]