6-6 : Systems Of Inequalities - Teacher.Docx

6-6 : Systems of Inequalities

Students will be able to: solve systems of linear inequalities by graphing.

Systems of Inequalities:

A set of two or more inequalities with the same variables is called a system of inequalities. The solution of a system of inequalities with two variables is the set of ordered pairs that satisfy all of the inequalities in the system. The solution set is represented by the overlap, or intersection, of the graphs of inequalities.

Example 1: Solve the system of inequalities by graphing. y > -2x + 1 y ≤ x + 3

The graph of y = -2x + 1 is dashed and is not included in the graph of the solution. The graph of y = x + 3 is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y > -2x + 1 and y ≤ x + 3.

When graphing more than one region, it is helpful to use two different colored pencils or two different patterns for each region. This will make it easier to see where the regions intersect and find possible solutions.

On Your Own: Practice graphing the system of inequalities: y ≥ -4 3x + y < 2 Example 2: Solve the system of inequalities by graphing.

3x - y ≥ 2 3x - y < -5

The graphs of 3x - y = 2, and 3x - y = -5 are parallel lines. The two regions do not intersect at any point, so the system has no solution.

On Your Own: Practice graphing the system of inequalities: y > 3 y < 1

Applying Systems of Inequalities: When using a system of inequalities to describe constraints on the possible combinations in a real-world problem, sometimes only whole-number solutions will make sense.

Example 3: Monifa is running for student council. The election rules say that for the election to be valid, at least 80% of the 900 students must vote. Monifa knows that she need more than 330 votes to win.

a. Define the variables, and write a system of inequalities to represent this situation. Then graph the system. Let r = the number of votes required by the election rules; 80% of 900 students is 720 students. So r ≥ 720.

Let v = the number of votes that Monifa needs to win. So v > 330.

The system of inequalities is r ≥ 720 and v > 330.

b. Name one viable option. Only whole-number solutions make sense in this problem. One possible solution is (800, 400); 800 students voted and Monifa received 400 votes.

On Your Own: The Theater Club is selling shirts. They only have enough supplies to print 120 shirts. They will sell sweatshirts for $22 and T-shirts for $15, with a goal of at least $2,000 in sales.

a. Define the variables, and write a system of inequalities to represent this situation.

b. Then graph the system.

c. Name one possible solution.

d. Is (45, 30) a solution? Explain.