The boundary line of an inequality can be determined by replacing the inequality symbol with an equal sign.
Inequality & Boundary Line
y > -2|x+4|+3 & y = -2|x+4|+ 3
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, it is 3. 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+4=0 ⇒ x=-4
The vertex of this equation is (-4,3). 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=-3 and x=-5.
x
y=-2|x+4|+3
Simplify
y
-3
y=-2| -3+4|+3
y=-2|1|+3
1
-5
y=-2| -5+4|+3
y=-2|-1|+3
1
Connecting these points, we are able to graph our boundary line. Please remember the graph of an 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.
