Graph the solution set of the given system of linear inequaliti \( \begin{array}{r}3 x+y<6 \\ x-y>3\end{array} \)

To graph the solution set of the system of linear inequalities: 1. **Inequality 1:** \(3x + y < 6\) 2. **Inequality 2:** \(x - y > 3\) ### Step 1: Graph the boundary lines First, we will convert the inequalities into equations to find the boundary lines. 1. **For \(3x + y = 6\):** - To find the intercepts: - When \(x = 0\): \(y = 6\) (point (0, 6)) - When \(y = 0\): \(3x = 6 \Rightarrow x = 2\) (point (2, 0)) - Draw a dashed line through the points (0, 6) and (2, 0) because the inequality is strict (<). 2. **For \(x - y = 3\):** - To find the intercepts: - When \(x = 0\): \(-y = 3 \Rightarrow y = -3\) (point (0, -3)) - When \(y = 0\): \(x = 3\) (point (3, 0)) - Draw a dashed line through the points (0, -3) and (3, 0) because the inequality is strict (>). ### Step 2: Determine the shading for each inequality 1. **For \(3x + y < 6\):** - Choose a test point not on the line, such as (0, 0): - \(3(0) + 0 < 6 \Rightarrow 0 < 6\) (True) - Shade the region that includes (0, 0), which is below the line \(3x + y = 6\). 2. **For \(x - y > 3\):** - Choose a test point not on the line, such as (0, 0): - \(0 - 0 > 3 \Rightarrow 0 > 3\) (False) - Shade the region that does not include (0, 0), which is above the line \(x - y = 3\). ### Step 3: Find the intersection of the shaded regions The solution set of the system of inequalities is the region where the shaded areas of both inequalities overlap. ### Final Graph - Draw the dashed lines for both inequalities. - Shade the area below the line \(3x + y = 6\) and above the line \(x - y = 3\). - The overlapping region is the solution set for the system of inequalities. This graphical representation will help visualize the solution set for the given system of linear inequalities.