Exercise 28 Page 253

Graphing an inequality involves two main steps.

1. Plotting the boundary line.
2. Shading one half of the plane to show the solution set.

Boundary Line

The equation of the boundary line is obtained by replacing the inequality symbol with an equals sign. Inequality & Boundary Line 3x-y ≥ 5 & 3x-y = 5To draw this line, we will first rewrite the equation in slope-intercept form.

Now that the equation is in slope-intercept form, we can identify the slope m and y-intercept (0, b). y= 3x+( -5) We will plot the y-intercept (0, -5), and use the slope m= 3 to plot another point on the line. We will then connect these two points with a straight line. Connecting these points with a solid line will give us the boundary line of our inequality. Note that the boundary line is solid, not dashed, because the inequality is not strict.

Shading the Plane

To decide which side of the boundary line to shade, we will substitute a test point that is not on the boundary line into the given inequality. If the substitution creates a true statement, we shade the region that includes the test point. Otherwise, we shade the opposite region. Let's use (0,0) as our test point.

Since the substitution did not create a true statement, we will shade the region that does not contain the point.