Determine which portion of the plane we should shade to show the solution set.
Boundary Line
The boundary line of an inequality can be determined by replacing the inequality symbol with an equal sign.
Inequality & Boundary Line
y ≤ |x-3|+4 & y = |x-3| + 4To graph this absolute value equation, let's first identify the vertex. The y-value of the vertex is equal to the term that is not connected to the absolute value. In this case, it is 4. The x-value of the vertex is the zero of the expression inside the absolute value. To find this, we need to set the expression equal to 0 and solve.
x-3=0 ⇒ x=3
The vertex of this equation is (3,4). To draw the graph, we will need two more points. We can find one on the left side of the vertex and one on the right side. Let's find the corresponding y-values for x=2 and x=4.
x
y=|x-3|+4
Simplify
y
2
y=| 2-3|+4
y=|-1|+4
5
4
y=| 4-3|+4
y=|1|+4
5
Connecting these points, we are able to graph our boundary line. Please remember that the graph of an absolute value equation is V-shaped. Because the inequality is not strict, the boundary line will be solid.
Shading the Solution Set
In order to decide which part of the plane to shade, we can test a point which is not on the boundary line. Let's test the point ( 0, 0).
If the point satisfies the inequality, we shade the region that contains the point. Otherwise, we shade the region that does not contain the point.
