Lesson 1: Graphing Linear Inequalities in Two Variables Chapter 6.1

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1 MATH 11 FOUNDATIONS Lesson 1: Graphing Linear Inequalities in Two Variables Chapter 6.1 Name: Date: A mathematical inequality must contain one of the following: Examples of linear inequalities in a single variable: The solution to a inequality can be shown on a : Solutions for a two variable inequality can be shown on a Cartesian plane. Example 1: Graph the solution set for 3x + 4y 12

2 Example 2: Graph the solution set for each linear inequality on a Cartesian plane. a) {(x, y) x 2 > 0, x R, y R} b) {(x,y) -3y + 6-6, x I, y I} Example 3: Oliver and Connor are competing in a spelling quiz. Connor gets a point for every word he spells correctly. Oliver is younger than Connor, so he gets 3 points for every word he spells correctly, plus one bonus point. What combinations of correctly spelled words for Oliver and for Connor are possible, if Connor scores more points than Oliver? Choose two combinations that make sense and explain your choices.

3 MATH 11 FOUNDATIONS Lesson 2 & 3: Graphing Systems of Linear Inequalities Chapter 6.2 and 6.3 Review: Solve the linear system using graphing x y 5 2x y 2 Example 1: Using desmos.com, graph the system (similar to that one above). How is it similar? How is it different? x y 5 2x y 2 Example 2: Graph this system of linear inequalities. Determine a solution. Check its validity. {(x, y) x + y 5, x I, y I} {(x, y) x + 3y > 0, x I, y I}

4 Example 3: Graph the solution set for the following system of inequalities. State two possible solutions from the set. ( x, y) 2 x y 6, x W, y W ( x, y) y 3, x W, y W Example 4: To raise funds to buy new instruments, the band committee has 500 t-shirts to sell. The t-shirts come in blue or green. Based on sales of the same t-shirt at a fundraiser five years ago, the committee expects to sell at least twice as many blue t-shirts as red t-shirts a) Define the variables and restrictions. Write a system of linear inequalities that models the situation. b) Graph the system c) Suggest a combination of t-shirt sales that could be made.

5 MATH 11 FOUNDATIONS Lesson 4: Optimization Problems Chapter 6.4 and 6.5 Optimization a technique for finding a maximum or minimum value of a function or several variables subject to a set of constraints. Example 1: Three teams are travelling to a basketball tournament in cars and minivans. Each team has no more than 2 coaches and 14 athletes Each car can take 4 team members, and each minivan can take 6 team members No more than 4 minivans and 12 cars are available The school wants to know the combination of cars and minivans that will require the minimum and maximum number of vehicles. Create a model to represent this situation. STEP 1: Identify the quantity that must be optimized. STEP 2: Define the variables that affect the quantity to be optimized. Identify any restrictions on these variables. STEP 3: Write a system of linear inequalities to describe all the constraints of the problem. Graph the system. Feasible region: the solution region for a system of linear inequalities that is modelling an optimization problem STEP 4: Write an objective function to represent the relationship between the variables and the quantity to be optimized. Objective function: relationship between the two variables in the system of linear inequalities and the quantity to be optimized

6 Example 2: A refinery produces oil and gas. At least 2L of gas is produced for each litre of oil. The refinery can produce up to 9 million litres of oil and 6 million litres of gas each day. Gasoline is projected to sell of $1.10 per litre. Oil is projected to sell for $1.75 per litre. The company needs to determine the daily combination of gas and oil that must be produced to maximize revenue. Create a model to represent the situation. STEP 1: Identify the quantity that must be optimized. STEP 2: Define the variables that affect the quantity to be optimized. Identify any restrictions on these variables. STEP 3: Write a system of linear inequalities to describe all the constraints of the problem. Graph the system. STEP 4: Write an objective function to represent the relationship between the variables and the quantity to be optimized. Optimization Problems: Finding a Solution Consider the situation: Restrictions: x R, y R Constraints: x + 3y 9, x y 3, x 3 Objective function: P = 2x + y a) Draw a graph to model the situation. b) What are the points of intersection? (Vertices of the feasible region) c) What point in the feasible region would result in the maximum value of the objective function? d) What point in the feasible region would result in the minimum value of the objective function?