Graphing an inequality involves two main steps.

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

Boundary Line

To graph the inequality, we first 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.

Inequality	Boundary Line
$y \leq \frac{1}{2} x - 3$	$y = \frac{1}{2} x - 3$

Fortunately, this equation is already in slope-intercept form, so we can identify the slope

m

and

y -

intercept

(0, b) .

y = \frac{1}{2} x - 3 \Leftrightarrow y = \frac{1}{2} x + (- 3)

We will plot the

y -

intercept

(0, - 3),

then use the slope

m = \frac{1}{2}

to plot another point on the 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.

y \leq \frac{1}{2} x - 3

SubstituteII

$x = 0$ , $y = 0$

0 \leq ? \frac{1}{2} (0) - 3

ZeroPropMult

Zero Property of Multiplication

0 ≰ - 3 \times

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

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