Marisa needs to hire at least 10 staff members for an upcoming project. The staff members will be made up of junior directors, who will be paid $640 per week, and senior directors, who will be paid $880 per week. Her budget for paying the staff members is no more than $9,700 per week. She must hire at least 3 junior directors and at least 1 senior director. Which of the following systems of inequalities represents the conditions described if x is the number of junior directors and y is the number of senior directors?

Options

A. `640x + 880y ge 9,700`
`x+y le 10`
`x gt 3`
`y ge 1`
B. `640x + 880y le 9,700`
`x+y ge 10`
`x ge 3`
`y ge 1`
C. `640x + 880y ge 9,700`
`x + y ge 10`
x lt 3
`x le 3`
`y le 1`
D. `640x+880y le 9,700`
`x + y le 10`
`x le 3`
`y le 1`

Explanation

To solve the problem, we need to establish a system of inequalities based on the conditions provided. Let's break it down step by step:

### Step 1: Define the Variables
Let:
- \( x \) = number of junior directors
- \( y \) = number of senior directors

### Step 2: Establish the Budget Constraint
Marisa's budget for paying the staff members is no more than $9,700 per week. The cost for junior directors is $640 per week and for senior directors is $880 per week. Therefore, the total cost can be expressed as:
\[ 640x + 880y \leq 9700 \]

### Step 3: Set the Minimum Number of Junior Directors
Marisa must hire at least 3 junior directors. This can be expressed as:
\[ x \geq 3 \]

### Step 4: Set the Minimum Number of Senior Directors
Marisa must hire at least 1 senior director. This can be expressed as:
\[ y \geq 1 \]

### Step 5: Set the Minimum Total Staff Requirement
Marisa needs to hire at least 10 staff members in total. This can be expressed as:
\[ x + y \geq 10 \]

### Final System of Inequalities
Combining all the inequalities, we have the following system:
1. \( 640x + 880y \leq 9700 \) (Budget constraint)
2. \( x \geq 3 \) (Minimum junior directors)
3. \( y \geq 1 \) (Minimum senior directors)
4. \( x + y \geq 10 \) (Minimum total staff)

### Summary of the System of Inequalities
Thus, the system of inequalities representing the conditions described is:
\[ \begin{align*} 640x + 880y & \leq 9700 \\ x & \geq 3 \\ y & \geq 1 \\ x + y & \geq 10 \\ \end{align*} \]