To graph the linear inequality 2x – 3y > 12, you can start by setting the inequality equal to zero and solving for y:
2x – 3y = 12
y = (2/3)x – 4
This represents the line that is the boundary between the solution set of the inequality and the non-solution set. To determine which side of the line represents the solution set, we can test a point that is not on the line. For example, we can try the point (0,0). Substituting these values into the inequality, we get:
2(0) – 3(0) > 12
0 > 12
This inequality is false, so the point (0,0) is not a solution to the inequality. This means that the solution set is the region that is on the opposite side of the line from the point (0,0). To graph the inequality, we can then draw a solid line for the boundary and shade the region that represents the solution set.
This is the graph of the linear inequality 2x – 3y > 12. The shaded region represents the solution set.
[asy]
unitsize(1.5 cm);
pair A, B, C;
A = (0,4);
B = (6,-2);
C = (0,0);
draw((-1,0)--(7,0));
draw((0,-3)--(0,5));
draw(A--B,linewidth(1.5));
draw((-1,4)--(7,4),linewidth(1.5));
draw((-1,-2)--(7,-2),linewidth(1.5));
fill((-1,4)--(7,4)--(7,-2)--(-1,-2)--cycle,gray(0.7));
label("$y = \frac{2}{3}x - 4$",B,E);
[/asy]This is the graph of the linear inequality 2x – 3y > 12. The shaded region represents the solution set.
The shaded region represents the solution set for the inequality. The line is the boundary between the solution set and the non-solution set. Points on the line are not included in the solution set, but points on one side of the line are included. In this case, the solution set includes all points that are on the same side of the line as the shaded region.