1. In your teams, give at least one combination of boomerangs that would meet Phil and Cathy s requirements. How much money would they make?

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1 Algebra 2 Chapter 3: Linear Systems LS 3-3 Systems of inequalities Phil and Cathy make and sell boomerangs for a school event in order to raise money for the local food bank. They plan to make them in two sizes: small and large. Phil will carve them from wood. The small boomerang takes 2 hours to carve and the large one takes 3 hours to carve. Phil has at most 24 hours of free time available for carving. Cathy will decorate them. She only has time to decorate at most 10 boomerangs of either size. The small boomerang will make $8 for charity. The large boomerang will make $10 for charity. They want to make as much money for charity as they can. How many small and large boomerangs should they make? How much money will they then make? Problem adapted from The Mathematics Assessment Project: Boomerangs 1. In your teams, give at least one combination of boomerangs that would meet Phil and Cathy s requirements. How much money would they make? 2. Look Back! What are the different requirements that have to be taken into consideration when finding combinations of boomerangs? 1 P a g e

2 3. One way to completely analyze this problem is to write a system of linear inequalities. Let x = the number of small boomerangs and y = the number of large boomerangs. Write several inequalities to represent the different requirements that need to be taken into account. 4. Graph these inequalities in the grid below. IMPORTANT!! Use two different crayon colors to shade in your inequalities. y x Since the problem requires that we meet both conditions at the same time, then we look at where the shading overlaps. This is the solution to a system of inequalities. Use a marker to outline the region of the graph where the two shadings overlap. 5. The other factor that we need to consider is how much money is made from the sale of the boomerangs. There is a theorem in mathematics that says the maximum/minimum in a situation like this will occur at one of the vertices of the outlined region. Find the intersection point of the two lines and then list all the vertices of the region. Find which of these points give the maximum profit. 2 P a g e

3 At this point complete the foldable to go into our learning log entry then paste in the learning log. 2. Solve the following systems using graphs P a g e

4 Using the graphing calculator to solve the systems. Make sure your INEQUALZ apps is on Press your Y= key to access the y editor and enter in the inequalities. Pressing the green ALPHA key will allow you to change the inequality signs. Pick an appropriate window to view your system (Window key or Zoom Key) then graph the inequalities Initially, the graph will appear with both regions shaded and it is rather difficult to see where the intersection is. To fix this press access the menu as seen at right. Arrow up and choose 1: Ineq Intersection. The calculator will regraph the inequalities showing only the shading that overlaps. It will look like the second figure. to Pressing to access the Po-I Trace has the trace feature move from intersection point to intersection point on the graph. Team Work: Write and solve a system of inequalities to represent the situations given below. 4. Alyssa plays soccer and baseball. She burns 400 calories/h playing soccer and 50 calories/h playing baseball. Each week she is willing to spend at most 20 h exercising and wishes to burn at least 4000 calories. y x 4 P a g e

5 5. A company manufactures solar-powered calculators and battery-powered calculators. In one day, a maximum of 110 solar-powered and 80 batterypowered calculators can be made. Each solar-powered calculator requires 1 work-hour to produce and each battery-powered calculator requires 2 workhours. There are a maximum of 200 work- hours available each day P a g e

6 7. Think about it!!! When we were solving systems of equations, there were two special cases, Inconsistent systems and Dependent systems. a. Review: What were the definitions of these two special cases? Sketch an example of what was special about the graphs of each. b. What would an inconsistent system of inequalities look like? Make up an example of a system that would be inconsistent. c. What would a dependent system of inequalities look like? Make up an example of a system that would be dependent. Assignment: page 136: 12, 20, 24, 30, 32, 34 6 P a g e