1. Systems of
Inequalities

2. This will be the boundary that divides when is less than three and
We are going to look at inequalities with two variables. Generally their solution cannot be written explicitly, but can be shown graphically as a collection of points that satisfy the inequality.
This is a line
First we find where
This will be the boundary that divides
when is less than three and
when it is greater than 3.
true so shade the side that contains (0, 0)
This line is where
slope
y-intercept
The solutions to the inequality will be on one side of this line. We find out which side by testing a point on one side. If it makes the inequality true, that will be the side to shade. If it is false, shade the other side.
All points in this region satisfy the inequality
We only want less than (not "or equal to") so we won't include the line. We "dash" the line to show it is not included.
(0, 0) is easy to test

3. First we'll graph to determine the boundary. Graph:
This is a parabola. Let's get in standard form.
The parabola opens up, is vertically stretched by a factor of 2 and vertically shifted up 3.
We'll leave a solid line since the inequality has "or equal to"
This is false so we need to shade the region of the graph that does not include (0, 0).
Let's test (0, 0)

4. The solution is where we shaded for both.
Here we have a system. We'll find the solution for each and the solution to the system will be where we shaded for both.
Graph:
The solution is where we shaded for both.
This is a circle.
This is a line.
This is false so shade on the opposite side of the line.
This is true so shade the part of the graph that includes (0, 0).
Let's test (0, 0)
Let's test (0, 0)

5. Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.
Shawna has kindly given permission for this resource to be downloaded from and for it to be modified to suit the Western Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar