Exercise 30 Page 271

Graphing an inequality involves two main steps.

1. Plotting the boundary line.
2. 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 with an equals sign. To draw this line, we will first rewrite the equation in slope-intercept form.

$\text{-}x+4y=\text{-}12$

AddEqn

$\text{LHS}+x=\text{RHS}+x$

$4y=x-12$

DivEqn

$\text{LHS}/4=\text{RHS}/4$

$y=\frac{1}{4}x-3$

Now that the equation is in slope-intercept form, we can identify the slope $m$ and $y\text{-}$intercept $(0,b).$

$$\begin{array}{c}y=\frac{1}{4}x+(\text{-}3)\end{array}$$

We will plot the $y\text{-}$intercept $(0,\text{-}3),$ then use the slope $m=\frac{1}{4}$ 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.

$\text{-}x+4y>\text{-}12$

SubstituteII

$x=0$, $y=0$

$\text{-}(0)+4(0)\stackrel{?}{>}\text{-}12$

ZeroPropMult

Zero Property of Multiplication

$0>\text{-}12✓$

Since the substitution of the test point created a true statement, we will shade the region that contains the point.