1. 3.3: Graphing and Solving Systems of Linear Inequalities
Intro: Systems of inequalities are similar to systems of equations. A solution is still an ordered pair that is true in both statements.
The graphing is a little more sophisticated - it’s all about the shading.

2. Graphing Method
Example: Graph the inequalities on the same plane: x + y < 6 and 2x - y > 4.
Before we graph them simultaneously, let’s look at them separately.
Graph of x + y < >

3. So what happens when we graph both inequalities simultaneously?
Graphing Method
This is: 2x - y > 4.
So what happens when we graph both inequalities simultaneously?

4. Coolness Discovered! Wow!
The solution to the system is the brown region - where the two shaded areas coincide.
The green region and red regions are outside the solution set.

5. So what were the steps? Graph first inequality Graph second inequality
Shade lightly (or use colored pencils)
Graph second inequality
Shade darkly over the common region of intersection.
That is your solution!

6. Challenge Extended
Graph
y ≥ -3x -1 and
y < x + 2

7. What about THREE inequalities?
Graph x ≥ 0, y ≥ 0, and 4x + 3y ≤ 24
First off, let’s look at x ≥ 0 and y ≥ 0 separately.

8. Graphing THREE inequalities
Now let’s look at x ≥ 0 and y ≥ 0 together.
Clearly, the solution set is the first quadrant.

9. Graphing THREE inequalities
So therefore, after we graph the third inequality, we know the solution region will be trapped inside the first quadrant. So let’s look at 4x + 3y ≤ 24 by itself.

10. Graphing THREE inequalities
Now we can put all of our knowledge together.
The solution region is the right triangle in the first quadrant.