Shading half of the plane to show the solution set.
Let's do these two things one at a time.
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 with an equals sign.
Inequality &
Boundary Line
5y+8 ≤ 33 &
5y+8 = 33
Before we graph the boundary line, let's first simplify it's equation by isolating the variable on one side.
Notice that the equation for this boundary line only has one variable. This equation is telling us that each and every point that lies on the line will have a y-coordinate equal to 5. This means that it is a horizontal line. Notice that the inequality is notstrict, so the points on the boundary line are included in the solution set. We show this by drawing a solid line.
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 of the test point created a true statement, we will shade the region that contains the point.
The y-intercept of the graph is at y=5. Additionally, we shaded the half-plane below our boundary line. Therefore, every possible (x,y) coordinate pair with an y-value that is less than or equal to 5 lies within the solution set. Let's write our answer as an inequality.
y ≤ 5
