Solve the system of inequalities by graphing.
y <= - 1/4 * x + 6
y >= - x + 3
To graph the system of inequalities, we will graph each inequality separately and shade the region that satisfies both inequalities.
First, let's graph the inequality y <= -1/4 * x + 6:
1. Plot the y-intercept at y = 6.
2. Use the slope of -1/4. From the y-intercept (0, 6), move 4 units down and 1 unit to the right to get another point.
3. Draw a dashed line through the two points.
Next, let's graph the inequality y >= -x + 3:
1. Plot the y-intercept at y = 3.
2. Use the slope of -1. From the y-intercept (0, 3), move 1 unit down and 1 unit to the right to get another point.
3. Draw a dashed line through the two points.
Once both lines are drawn, we will determine which side of each line to shade by picking a point not on the line and substituting it into the inequality:
For the first inequality, let's pick the point (0, 0):
0 <= -1/4 * 0 + 6
0 <= 6, which is true.
For the second inequality, let's pick the point (0, 0):
0 >= -0 + 3
0 >= 3, which is false.
Since (0, 0) satisfies the first inequality but not the second, we will shade the region below the line representing the first inequality and use a solid line since it is inclusive.
The final graph should show a shaded region below the line y = -1/4 * x + 6, and the dashed line y = -x + 3.