1. Graphing Systems of Inequalities

2. Objective:
Solve systems of inequalities by graphing.

3. Introduction Consider the system of inequalities shown below:
y ≥ x + 2
y ≤ -2x -1
The solution of this system is the set of all ordered pairs that satisfy both inequalities. This solution can be determined by graphing each inequality in the same coordinate plane as shown below.

4. Introduction
Recall that the graph of each inequality is called a half-plane. The intersection of the two half-planes represents the solution to the system of inequalities. This solution is a region that contains the graphs of an infinite number of ordered pairs. The graphs of
y = x + 2 (y ≥ x + 2)
y = -2x – 1 (y ≤-2x – 1)
are the boundaries of the region and are included in the graph of the system.

5. Ex. 1: Solve each system of inequalities by graphing y > x – 3 and y ≤ -1
The solution is the ordered pairs in the intersection of the graphs of y > x – 3 and y ≤ -1. This region is shaded in green at the right. The graphs of y = -1 and y = x – 3 are the boundaries of the region. The graph of y = x – 3 is a dashed line and is not included in the graph of the system

6. Ex. 2: x – 2y ≤ -4 and 4y < 2x – 4
The graphs of these two are parallel lines. First solve for y.
x – 2y ≤ -4
-2y ≤ -x – 4
y ≥ ½ x + 2
4y < 2x – 4
y < ½ x – 1
NO INTERSECTIONS – NO SOLUTIONS.