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
2x-3y ≤ 1 &
2x-3y = 1To draw this line, we will first rewrite the equation in slope-intercept form.
Now that the equation is in slope-intercept form, we can identify the slope m and y-intercept b.
y=2/3x-1/3 ⇔ y= 2/3x+( - 1/3)
We will plot the y-intercept b= - 13, and then use the slope m= 23 to plot another point on the line. Notice that the inequality is notstrict, so the points on the boundary line are included in the solution set. We have shown 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 (0,0).
Determining Solutions
Finally, we can check the given points to see which ones are contained in the solution set. We will do this by adding the list of points to our coordinate plane.
Points that lie within the shaded region are solutions to the inequality. Note that points that lie on the solid line are also solutions. This means that three of the given points are part of the solution set.
(0,0), (2,3),and(5,3)
