The graph of the linear inequality 2x - 3y < 12 represents a region in the coordinate plane where the inequality is true. To graph this inequality, you can follow these steps:
1. Start by graphing the boundary line, which is the equation 2x - 3y = 12. To graph this line, you can rewrite it in slope-intercept form: y = (2/3)x - 4. This line represents all the points where the inequality is exactly equal to 12.
2. Since the inequality is less than (<), you need to determine which side of the boundary line to shade. To do this, you can choose a test point not on the boundary line, such as the origin (0,0).
3. Substitute the coordinates of the test point into the inequality: 2(0) - 3(0) < 12. This simplifies to 0 < 12, which is true. Therefore, the region containing the origin is the solution to the inequality.
4. Shade the side of the boundary line where the test point is located. In this case, since the origin is part of the solution, you would shade the region below the line.
5. Lastly, draw a dashed line to represent the inequality 2x - 3y < 12 because the inequality is strict (<) and not inclusive of the line itself.
By following these steps, you can accurately graph the linear inequality 2x - 3y < 12 and visualize the region where the inequality holds true in the coordinate plane.
