PROBLEM 1 Whitewater Rafting Chase is an eperienced whitewater rafter who guides groups of adults and children out on the water for amazing adventures. The super-raft he uses can hold 00 pounds of weight. An weight greater than 00 pounds will cause the raft to sink, hit more rocks, and maneuver more slowl. 1. Chase estimates the weight of each adult as approimatel 00 pounds and the weight of each child under age siteen as approimatel 100 pounds. Chase charges adults $5 and children under age siteen $50 to ride down the river with him. His goal is to earn at least $150 each rafting trip. a. Write an inequalit to represent the most weight Chase can carr in terms of rafters. Define our variables. Dœs Chase count when determining the weight and the cost? b. Write an inequalit to represent the least amount of mone Chase wants to collect for each rafting trip. c. Write a sstem of linear inequalities to represent the maimum weight of the raft and the minimum amount of mone Chase wants to earn per trip. In a sstem of linear inequalities, the inequalities are known as constraints because the values of the epressions are constrained to lie within a certain region on the graph.. Let s consider the past two trips that Chase guides. Determine whether each combination of rafters is a solution of the sstem of linear inequalities. Then describe the meaning of the solution in terms of this problem situation. a. First Trip: Chase guides adults and children. 0 Chapter Sstems of Inequalities
b. Second Trip: Chase guides 5 adults. 3. Graph the sstem of linear inequalities on the coordinate plane shown. Shade the half-plane of each inequalit differentl. You can use colored pencils or simpl vertical and horizontal lines. Child Rafters 0 Adult Rafters The solution of a sstem of linear inequalities is the intersection of the solutions to each inequalit. Ever point in the intersection region satisfies the solution.. Analze our graph. a. Describe the possible number of solutions for a sstem of linear inequalities. b. Is the intersection point a solution to this sstem of inequalities? Wh or wh not?. Sstems of Linear Inequalities 1
c. Identif three different solutions of the sstem of linear inequalities ou graphed. What do the solutions represent in terms of the problem situation? d. Determine one combination of adults and children that is not a solution for this sstem of linear inequalities. Eplain our reasoning. 5. Analze the solution set of the sstem of linear inequalities shown. 1. 1 1 # 3 a. Graph the sstem of linear inequalities. Notice the inequalit smbols. How do ou think this will affect our graph? 0 Chapter Sstems of Inequalities
b. Choose a point in each shaded region of the graph. Determine whether each point is a solution of the sstem. Then describe how the shaded region represents the solution. Point 1. 1 1 # 3 Description of location (, ) 1 1 ( ) 3 10 3 The point is not a solution to either inequalit and it is located in the region that is not shaded b either inequalit. c. Alan makes the statement shown. Alan The intersection point is alwas an algebraic solution to a sstem of inequalities because that is where the two lines meet. Eplain wh Alan s statement is incorrect. Use the intersection point of this sstem to eplain our reasoning.. Sstems of Linear Inequalities 3
. Solve each sstem of linear inequalities b graphing the solution set. Then identif two points that are solutions of the sstem. 5 3 a. 5 3 0 b. 1 0 Chapter Sstems of Inequalities
PROBLEM Burning Calories Jackson and a group of friends decide to use the fitness room after school. On the wall, the read the information shown: Eercise Calories Burned per Minute Treadmill light effort. Treadmill vigorous effort 1. Stair Stepper light effort.9 Stair Stepper vigorous effort 10. Stationar Bike light effort 5.5 Stationar Bike vigorous effort 11.1 Jackson decides to use the stair stepper. He has at most 5 minutes to eercise and he wants to burn at least 00 calories. 1. Write a sstem of linear inequalities to represent Jackson s workout. Define our variables.. Sstems of Linear Inequalities 5
Let s graph the sstem ou wrote in Question 1. When choosing the inequalit smbol, think about the half-plane ou must shade. You can use a graphing calculator to graph a sstem of linear inequalities. Step 1: Press Y= and enter the two inequalities as Y 1 and Y. Step : While still in the Y= window, access the inequalit function b moving our cursor r to the left until the \ flashes. Press ENTER to select ect the appropriate inequalit smbol ( or ). Step 3: Press WINDOW and set the bounds. Step : Press GRAPH. Remember to solve for the -value before entering the inequalities. Set the WINDOW for this problem using the bounds [0, 50] X [0, 50].. Graph the sstem of inequalities from Question 1 on the coordinate plane shown. Be sure to label our aes. Chapter Sstems of Inequalities
3. Analze our graph. a. Identif two different solutions of the sstem of inequalities using the value function of our graphing calculator. b. Interpret our solutions in terms of Jackson s workout. c. Algebraicall prove that our solutions satisf the sstem of linear inequalities.. Solve each sstem of linear inequalities using our graphing calculator. Graph each sstem then identif two points that are solutions to the sstem on the grid shown. 3 a. 5 3 3 5 3. Sstems of Linear Inequalities
3 b. 5 3 3 5 3 3 c. 5 3 3 5 3 Chapter Sstems of Inequalities
3 d. 5 3 3 5 3?5. Adele states that since the equations in each sstem for Question are the same, the graphs and solutions should all be identical. Is Adele s statement true? Eplain our reasoning. Be prepared to share our solutions and methods.. Sstems of Linear Inequalities 9