Chapter 3: Nature of Graphs
Section 3-3: Graphs of nonlinear
inequalities
Build on what you already know
z You know how to graph linear inequalities from
chapter 2.
z Find the boundary line.
z Determine if it is solid or dashed
z Test a point with the inequality
z Shade the appropriate area.
z With nonlinear inequalities follow the same
format. But instead of a line as a boundary, it will
be some other function.
Example # 1
Graph y ≤ 3 x − 1 + 2 . I know what a cubed parent
function looks like and I know what a square
root parent looks like. This is a cubed root
function. So it will be a cubed function on its
side. If I don’t know this, I can always set up a
table of values and plot points.
X
3 x −1 + 2
Y
1
3 1−1 + 2
2
9
3 9 −1 + 2
4
-7
3 −7 − 1 + 2
0
We also know that the
boundary will be solid
because we have
≤
Now test a point on the graph that is not on the
boundary
Let’s test an easy point (0,0). It works so we shade that area.
0 ≤ 3 0 −1 + 2
0 ≤ −1 + 2
0 ≤ 1 yes
Example #2
Graph y > x − 5 + 4
We know we are working with an absolute
value child that looks like mom and is translated 5 units to the right
and 2 units up. We also know that the boundary is dashed because
the points on the V are not included. We draw the boundary and
then test a point (5,0)
0 > 5−5 +4
0>4
no, so shade other side
Solving absolute value inequalities
When we graph an absolute value inequality, we are showing the
solution. We can also find the solution algebraically.
Solve
3+ x − 4 > 8
There are two cases to be solved here. In one
case (x - 4)<o and the other case (x – 4)>0.
(x – 4)<0
3-(x-4)>8
3-x+4>8
-x+7>8
-x>1
x<-1
(x – 4)>0
3+ (x – 4)>8
x-1>8
x>9
So the solution “area” to this inequality is x<-1 or x>9
HW # 18
zSection 3-3
zPp150-151
z#13,15,17,20,23,24,30,33,49,52