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.
Inequality &
Boundary Line
3x+2 < 0 &
3x+2 = 0
Before we graph the boundary line, let's first simplify it's equation by isolating the variable on one side.
Notice that the equation for this boundary line only has one variable. This equation is telling us that each and every point that lies on the line will have an x-coordinate equal to - 23. This gives us a vertical line. Also, the inequality is strict, so the points on the boundary line are not included in the solution set. We show this by drawing a dashed 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 did not create a true statement, we will shade the region that does not contain the point.
The x-intercept of the graph is at x=- 23. Additionally, we shaded the half-plane to the left of our boundary line. Therefore, every possible (x,y) coordinate pair with an x-value that is less than - 23 lies within the solution set. Let's write our answer as an inequality.
x < -2/3
