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-x ≤ 3 & y-x = 3To graph this line, we will first rewrite the equation in slope-intercept form.
y-x = 3 ⇔ y = x + 3
Now that the equation is in slope-intercept form, we can identify the slope m and y-intercept (0, b).
y = x + 3 ⇔ y= 1x+ 3
We will plot the y-intercept (0, 3), then use the slope m= 1 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 non-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.
