Final answer:
The solution to the given system of inequalities is (0.5, 2).
Step-by-step explanation:
To find the solutions to the given system of inequalities, we can graph the lines and shade the appropriate regions based on the inequality signs. Let's start with the first inequality:
2x + 3y > 6
First, we graph the line 2x + 3y = 6 by finding two points on the line. For example, when x = 0, y = 2, and when y = 0, x = 3. Plotting these points on a graph and connecting them gives us the line.
Next, we determine which side of the line to shade. Since the inequality sign is >, the shaded region is above the line. Shade everything above the line.
Now let's move on to the second inequality:
3x + 2y < 6
We repeat the same steps as before. Find two points on the line and connect them to graph the line. Then, shade everything below the line because the inequality sign is <.
The region where both shaded areas overlap is the solution to the system of inequalities. Looking at the graph, we can see that the point (0.5, 2) lies in the overlapping region. Therefore, the correct solution is option (b) (0.5, 2).