Exercise 6 Page 269

Graphing an inequality involves two main steps.

1. Drawing the boundary line.
2. 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.

$$\begin{array}{rl}\underline{\text{Inequality}}& \underline{\text{Boundary Line}}\\ x\le \text{-}4& x=\text{-}4\end{array}$$

The equation for this boundary line has only one variable. This equation is telling us that all the points that lie on the line will have an $x\text{-}$coordinate of $\text{-}4.$ This gives us a vertical line. Notice that the inequality is not strict, 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.

$x\le \text{-}4$

Substitute

$x=0$

$0\nleqq \text{-}4\times$

Because $(0,0)$ is not a solution, we should shade the half-plane that does not contain the origin.