Patricia can buy individual songs for \( \$ 1.00 \) to download. Also, an entire album costs \( \$ 10.00 \) to download. She can spend no more than a total of \( \$ 80 \). She wants to buy no more than four albums, and at least 30 individual songs. The following system of inequalities represents this situation, where \( x \) is the number of individual songs and \( y \) is the number of albums. \( \begin{array}{l}x+10 y \leq 80 \\ x \geq 30 \\ y \leq 4\end{array} \) Which blue-shaded region corresponds to Patricia's possible choices?

To determine the blue-shaded region that represents Patricia's possible choices, let's analyze each inequality in the system: 1. **Budget Constraint:** \[ x + 10y \leq 80 \] - **Interpretation:** The total cost of individual songs (\(x\)) plus the cost of albums (\(10y\)) must not exceed \$80. - **Graphical Representation:** This is a line with a negative slope. Points below or on this line satisfy the budget constraint. 2. **Minimum Individual Songs:** \[ x \geq 30 \] - **Interpretation:** Patricia must buy at least 30 individual songs. - **Graphical Representation:** This is a vertical line at \(x = 30\). The region to the right of this line (including the line) satisfies this condition. 3. **Maximum Albums:** \[ y \leq 4 \] - **Interpretation:** Patricia can buy no more than 4 albums. - **Graphical Representation:** This is a horizontal line at \(y = 4\). The region below this line (including the line) satisfies this condition. 4. **Non-Negativity Constraints:** - \(x \geq 0\) - \(y \geq 0\) - **Interpretation:** Number of songs and albums cannot be negative. - **Graphical Representation:** The solution lies in the first quadrant. **Combining All Constraints:** - The feasible region is where all these conditions overlap. - **Boundaries:** - **Left Boundary:** \(x = 30\) - **Top Boundary:** \(y = 4\) - **Diagonal Boundary:** \(x + 10y = 80\) **Vertices of the Feasible Region:** 1. Intersection of \(x = 30\) and \(x + 10y = 80\): \[ 30 + 10y = 80 \Rightarrow y = 5 \] However, since \(y \leq 4\), this point is not included. 2. Intersection of \(x = 30\) and \(y = 4\): \[ (30, 4) \] 3. Intersection of \(y = 4\) and \(x + 10y = 80\): \[ x + 10(4) = 80 \Rightarrow x = 40 \] So, point is \( (40, 4) \). 4. Intersection of \(x + 10y = 80\) and the axes: - When \(y = 0\), \(x = 80\): Point \( (80, 0) \). - When \(x = 0\), \(y = 8\): But since \(y \leq 4\), this point is not in the feasible region. **Final Feasible Region:** - A quadrilateral bounded by: - \(x = 30\) (vertical line) - \(y = 4\) (horizontal line) - \(x + 10y = 80\) (diagonal line) - The x-axis (\(y = 0\)) up to \(x = 80\), but limited by \(y \leq 4\). **Graphically:**  *(Note: Replace with actual graph if available)* **Summary:** The blue-shaded region is where: - \(x \geq 30\) (right of the vertical line \(x = 30\)), - \(y \leq 4\) (below the horizontal line \(y = 4\)), - And \(x + 10y \leq 80\) (below the line \(x + 10y = 80\)). This region represents all possible combinations of individual songs and albums Patricia can purchase within her budget and constraints. **Visualization Example:**  *(Replace with a Desmos graph if needed)*