The boundary line of an inequality can be determined by replacing the inequality symbol with an equal sign.
Inequality & Boundary Line
y > 1/2|4x-5| & y = 1/2|4x-5|
To 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, as we do not have such term, it is 0. 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.
The vertex of this equation is ( 54,0). 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=0 and x=3.
x
y=1/2|4x-5|
Simplify
y
0
y=1/2|4( 0)-5|
y=1/2|-5|
5/2
3
y=1/2|4( 3)-5|
y=1/2|7|
7/2
Connecting these points, we are able to graph our boundary line. Please remember the graph of absolute value equation is V-shaped. Because the inequality is strict, the boundary line will be dashed.
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.
