Essential Understanding andObjectives• Essential Understanding: you can solve a system of inequalities in more than one way. Graphing the solution is usually the most appropriate method. The solution is the set of all points that are solutions of each inequality in the system• Objectives:• Students will be able to solve systems of linear inequalities
Iowa Core Curriculum• Algebra• A.CED.3 . Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.• A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.• A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
• An inequality and a system of inequalities can have many solutions. A solution of a system of inequalities satisfies all the inequalities in the system.
Solving a System by Using a Table• Assume g and m are whole numbers. What is the solution of the system of inequalities? ìg + m ³ 6 í î5g + 2m £ 20• Since the first inequality has infinitely many solutions, use the second inequality to make a table. g m 0 1 2 3 4• Now, go back and see which numbers satisfy the first inequality and those will be your solution.
Example• Assume x and y are whole numbers. What is the solution of the system of inequalities? ìx + y > 4 í î3x + 7y £ 21
Solving a system by graphing• What do you think the solution is for is for two inequalities? • the solution is the overlap of the two half plane solutions. (the overlap of the shaded region on the graph)• What is the solution of the system of inequalities? ì2x - y ³ -3 ï í 1 ï y ³ - x +1 î 2• Graph each inequality, the overlap is the solution of the system.• Pick a point in the overlap to test in each inequality to check if it is a solution of both systems.
Example• What is the solution of the system of inequalities? ì x + 2y £ 4 í î y ³ -x -1
Example• A pizza parlor charges $1 for each veggie topping and $2 for each meat topping. You want at least five toppings on your pizza. You have $10 to spend on toppings. How many of each topping can you get on your pizza?• Step 1 Relate: • 1(veggie) + 2(meat) <10 • (veggie) + (meat) >5• Step 2 Define: • X = veggie topping • Y = meat topping• Step 3 Write:ì x + 2y £ 10íîx + y ³ 5
Solving a Linear/AbsoluteValue System• What is the solution of the system of inequalities? ìy £ 3 ï í ï y ³ x -1 î ì 1 ï y < - x +1 í 3 ï y > 2 x -1 î