Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set.
y, is greater than or equal to, minus, start fraction, 4, divided by, 5, end fraction, x, plus, 8
y≥−
5
4
x+8
y, is greater than or equal to, 2, x, minus, 6
To graph the system of inequalities, we can start by graphing each inequality separately on the same set of axes.
The first inequality is y ≥ -4/5x + 8. Let's find two points on this line:
When x = 0, y = -4/5(0) + 8 = 8. So one point is (0, 8).
When x = 5, y = -4/5(5) + 8 = 4. So another point is (5, 4).
Plotting these two points and drawing a solid line through them (since the inequality is "greater than or equal to"), we get:
[There would be a graph here, but I am unable to display it. Please interpret the spoken instructions.]
The second inequality is y ≥ 2x - 6. Let's find two points on this line:
When x = 0, y = 2(0) - 6 = -6. So one point is (0, -6).
When x = 3, y = 2(3) - 6 = 0. So another point is (3, 0).
Plotting these two points and drawing a solid line through them (since the inequality is "greater than or equal to"), we get:
[There would be a graph here, but I am unable to display it. Please interpret the spoken instructions.]
The solution to the system of inequalities is the area where the shaded regions of both inequalities overlap.
[In lieu of a diagram,] the coordinates of a point in the solution set are (3, 0), which is the intersection point of the two lines.