Shading half of the plane to show the solution set
Boundary Line
To graph the inequality, we have to draw the boundary line. The equation of a boundary line is written by replacing the inequality symbol from the inequality with an equals sign.
ccc Inequality&&Boundary Line [0.3em] y < - 2/3x+5 && y = - 2/3x+5
Fortunately, this equation is already in slope-intercept form, so we can identify the slope m and y-intercept (0, b).
y= -2/3x+ 5
We will plot the y-intercept (0, 5), then use the slope m= - 23 to plot another point on the line. Connecting these points with a dashed line will give us the boundary line of our inequality. Note that the boundary line is dashed, not solid, because the inequality is 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.
