Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set. \[ y \geq-x+8 \] \[ y \leq \frac{3}{2} x-7 \]

To solve the system of inequalities graphically, we will follow these steps: 1. **Graph the boundary lines** of each inequality. 2. **Determine the regions** that satisfy each inequality. 3. **Identify the intersection** of the regions to find the solution set. 4. **State the coordinates** of a point in the solution set. ### Step 1: Graph the Boundary Lines 1. **For the first inequality** \( y \geq -x + 8 \): - The boundary line is \( y = -x + 8 \). - To graph this line, we can find two points: - When \( x = 0 \): \( y = 8 \) (point (0, 8)) - When \( y = 0 \): \( 0 = -x + 8 \) → \( x = 8 \) (point (8, 0)) - This line has a negative slope and will be solid since the inequality is "greater than or equal to". 2. **For the second inequality** \( y \leq \frac{3}{2} x - 7 \): - The boundary line is \( y = \frac{3}{2} x - 7 \). - To graph this line, we can find two points: - When \( x = 0 \): \( y = -7 \) (point (0, -7)) - When \( y = 0 \): \( 0 = \frac{3}{2} x - 7 \) → \( \frac{3}{2} x = 7 \) → \( x = \frac{14}{3} \approx 4.67 \) (point \((\frac{14}{3}, 0)\)) - This line has a positive slope and will also be solid since the inequality is "less than or equal to". ### Step 2: Determine the Regions - For \( y \geq -x + 8 \), the region above the line is included. - For \( y \leq \frac{3}{2} x - 7 \), the region below the line is included. ### Step 3: Identify the Intersection The solution set is where the shaded regions of both inequalities overlap. ### Step 4: State the Coordinates of a Point in the Solution Set To find a point in the solution set, we can check the intersection of the two lines: 1. Set the equations equal to each other to find the intersection point: \[ -x + 8 = \frac{3}{2} x - 7 \] 2. Solve for \( x \): \[ 8 + 7 = \frac{3}{2} x + x \] \[ 15 = \frac{5}{2} x \] \[ x = \frac{15 \cdot 2}{5} = 6 \] 3. Substitute \( x = 6 \) back into one of the equations to find \( y \): \[ y = -6 + 8 = 2 \] Thus, the intersection point is \( (6, 2) \). ### Conclusion The coordinates of a point in the solution set are \( (6, 2) \).