The boundary line of an inequality can be determined by replacing the inequality symbol with an equal sign.
Inequality & Boundary Line
y ≤ |x-2|-4 & y = |x-2|-4
The graph of this function is the graph of y=|x| after a few transformations. Let's first figure out which transformations were involved so that we can graph it.
Transformations of y=|x|
Vertical Translations
Translation up k units, k>0
y=|x|+ k
Translation down k units, k>0
y=|x|- k
Horizontal Translations
Translation right h units, h>0
y=|x- h|
Translation left h units, h>0
y=|x+ h|
Vertical Stretch or Compression
Vertical stretch, a>1
y= a|x|
Vertical compression, 0< a<1
y= a|x|
Horizontal Stretch or Compression
Horizontal stretch, 0< b<1
y=| bx|
Horizontal compression, b>1
y=| bx|
Reflections
In the x-axis
y=- |x|
In the y-axis
y=|- x|
Using the table, we can see that there is a horizontal translation 2 units right and a vertical translation down by 4 units. Applying the transformations to y=|x|, we can draw the boundary line. 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.
