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 less than, one half, x, plus, 3
y<
2
1
x+3
y, is less than, minus, 2, x, minus, 7
y<−2x−7
x
y
x
y
y, is less than, one half, x, plus, 3
y<
2
1
x+3
y, is less than, minus, 2, x, minus, 7
y<−2x−7
To graphically solve the system of inequalities, we will plot the graphs of each inequality on the same set of axes and find the region where they overlap or intersect.
First, let's graph the inequality y < (1/2)x + 3:
Plot the y-intercept (0,3) by placing a point on the y-axis at y = 3.
Using the slope of 1/2, plot a second point by moving 1 unit to the right and 1/2 unit up from the y-intercept.
Draw a dashed line to connect the two points.
Next, let's graph the inequality y < -2x - 7:
Plot the y-intercept (0, -7) by placing a point on the y-axis at y = -7.
Using the slope of -2, plot a second point by moving 1 unit to the right and 2 units down from the y-intercept.
Draw a dashed line to connect the two points.
The solution to the system of inequalities is the region where the shaded areas of the two inequalities overlap:
The coordinates of a point in the solution set are (-2, -11).
(Note: The point (-2, -11) is just an example of a point in the solution set. Any point within the shaded region will also be in the solution set.)