How do you graph the system of linear inequalities #x-y>7# and #2x+y<8#?

Let's first graph the boundary lines for each graph, then figure out what needs to be shaded. To graph the boundary lines, I'll change the form of the equations to slope-intercept:

graph{x-7[-40,40,-20,20]}

To shade, does the origin fall within the solution?

And so we shade the other side:

graph{y -x+7< 0[-40,40,-20,20]}

graph{2x+8[-40,40,-20,20]}

Is the origin part of this solution?

So we shade that side:

graph{y-2x-8<0[-40,40,-20,20]}

The point of intersection sits at:

To graph the system of linear inequalities (x-y > 7) and (2x + y < 8), follow these steps:

1. Graph the boundary line for each inequality:

   • For (x-y > 7), graph the line (x-y = 7).
   • For (2x + y < 8), graph the line (2x + y = 8).

2. Since the inequalities are strict inequalities ((>) and (<)), the boundary lines should be dashed.

3. Choose a test point not on the boundary lines. For simplicity, the origin (0,0) is often a good choice.

4. Test the chosen point in each inequality:

   • If the test point satisfies the inequality, shade the region containing the test point.
   • If the test point does not satisfy the inequality, shade the region not containing the test point.

5. The region where the shaded areas overlap represents the solution to the system of inequalities.

6. Ensure the shading is consistent with the inequality signs:

   • For (x-y > 7), shade the region above the line (x-y = 7).
   • For (2x + y < 8), shade the region below the line (2x + y = 8).

7. Optionally, label the shaded region as the solution set.

8. Your final graph will show the overlapping shaded region, which represents the solution to the system of inequalities.