1. § 9.4
Linear Inequalities in Two Variables and Systems of Linear Inequalities

2. Linear Inequalities in Two Variables
Linear inequality in two variables
Can be written in on of the forms
Ax + By < C Ax + By  C
Ax + By > C Ax + By  C
A, B, and C are real numbers, A and B are not both 0.
An ordered pair is a solution of the linear inequality if it makes the inequality a true statement.

3. Linear Inequalities in Two Variables
To Graph a Linear Inequality in Two Variables
Graph the boundary line found by replacing the inequality sign with an equal sign. If the inequality sign is > or <, graph a dashed boundary line (indicating that the points on the line are not solutions of the inequality). If the inequality sign is  or , graph a solid boundary line (indicating that the points on the line are solutions of the inequality).
Choose a point, not on the boundary line, as a test point. Substitute into the original inequality.
If a true statement is obtained in Step 2, shade the half-plane that contains the test point. If a false statement is obtained, shade the half-plane that does not contain the test point.

4. Linear Inequalities in Two Variablesx y
Example:
Graph 7x + y > –14
Graph 7x + y = –14 as a dashed line.
Pick a point not on the graph: (0,0)
(0, 0)
Test it in the original inequality.
7(0) + 0 > –14, 0 > –14
True, so shade the side containing (0,0).

5. Linear Inequalities in Two Variablesx y
Example:
Graph 3x + 5y  –2
Graph 3x + 5y = –2 as a solid line.
Pick a point not on the graph: (0,0), but just barely
(0, 0)
Test it in the original inequality.
3(0) + 5(0) > –2, 0 > –2
False, so shade the side that does not contain (0,0).

6. Linear Inequalities in Two Variablesx y
Example:
Graph 3x < 15
Graph 3x = 15 as a dashed line.
(0, 0)
Pick a point not on the graph: (0,0)
Test it in the original inequality.
3(0) < 15, 0 < 15
True, so shade the side containing (0,0).

7. Linear Inequalities in Two Variables
Warning!
Note that although all of our examples allowed us to select (0, 0) as our test point, that will not always be true.
If the boundary line contains (0,0), you must select another point that is not contained on the line as your test point.

8. Systems of Linear Inequalities
Two linear inequalities make a system of linear inequalities.
A solution of a system of linear inequalities is an ordered pair that satisfies each inequality in the system.
Graphing the Solution of a System of Linear Inequalities
Graph each inequality in the system on the same set of axes.
The solutions of the system are the points common to the graphs of all the inequalities in the system.

9. Graphing A System of Linear Inequalities
Example:
Graph the solution of the system:
3x + y < 9
2x + 5y  10 y x 4
The solution will be the set of all points that satisfy both of the inequalities in the system.
The boundary line is 3x + y = 9.
Solution
The boundary line is
2x + 5y = 10

10. Graphing A System of Linear Inequalities
Example:
Graph the solution of the system.
y < – 3x + 9. y x 4
2x + 5y  10
Solution
The boundary line is y = – 3x + 9.
The boundary line is
2x + 5y  10
y  – 3x + 9