1. Solving Systems of Linear Inequalities
Section 4.5
Solving Systems of Linear Inequalities

2. Objectives Solve a system of linear inequalities by graphing
Solve application problems involving systems of linear inequalities

3. Objective 1: Solve a System of Linear Inequalities by Graphing
In Section 4.1, we solved systems of linear equations graphically by finding the point of intersection of two lines.
Now we consider systems of linear inequalities, such as:
To solve systems of linear inequalities, we again find the points of intersection of graphs.
In this case, however, we are not looking for an intersection of two lines, but an intersection of two regions.

4. Objective 1: Solve a System of Linear Inequalities by Graphing
A solution of a system of linear inequalities is an ordered pair that satisfies each inequality.
To solve a system of linear inequalities means to find all of its solutions.
This can be done by graphing each inequality on the same set of axes and finding the points that are common to every graph in the system.

5. Objective 1: Solve a System of Linear Inequalities by Graphing
In general, to solve systems of linear inequalities, we will follow these steps.
1. Graph each inequality on the same rectangular coordinate system.
2. Use shading to highlight the intersection of the graphs (the region where the graphs overlap). The points in this region are the solutions of the system.
3. As an informal check, pick a point from the region where the graphs intersect and verify that its coordinates satisfy each inequality of the original system.

6. EXAMPLE 2
Graph the solutions of the system:

7. Objective 2: Solve Application Problems Involving Systems of Linear Inequalities
Solve Example 5 on the next few slides.

8. EXAMPLE 5
Landscaping
A homeowner budgets from $300 to $600 for trees and bushes to landscape his yard. After shopping around, he finds that good trees cost $150 each and mature bushes cost $75 each. What combinations of trees and bushes can he buy?