Name: 

What is the boundary line for the linear inequality 4x + 2y < 18?

What is the boundary line for the linear inequality 3x – 6y < 18?

Which test point is in the solution set for the linear inequality
{(x, y) | 7x – 5y < 0, x R, y R}?

Which test point is in the solution set for the linear inequality
{(x, y) | 7x + 5y 0, x I, y I}?

How would you graph the solution set for the linear inequality y + 5x 2?

Which system of linear inequalities has no solution?

Describe the boundary lines for the following system of linear inequalities.
{y 2 + x, x + y 0, x R, y   R}

What system of linear inequalities is shown here?

A vending machine sells juice and pop.
• The machine holds, at most, 200 cans of drinks.
• Sales from the vending machine show that at least 3 cans of juice are sold for each can of pop.
• Each can of juice sells for $1.50, and each can of pop sells for $1.00.
Let x represent the number of cans of pop.
Let y represent the number of cans of juice.
What are the restrictions on x and y?

Jan volunteers to fold origami frogs and swans for a display.
• She has 8 squares of green paper for the frogs and 12 squares of white paper for the swans.
• It takes her 4 min to fold an origami frog and 3 min to fold an origami swan.
• There must be two swans for every frog.
Let f represent the number of frogs.
Let s represent the number of swans.
What are the restrictions on x and y?

A football stadium has 60 000 seats.
• 70% of the seats are in the lower deck.
• 30% of the seats are in the upper deck.
• At least 40 000 tickets are sold per game.
• A lower deck ticket costs $100, and an upper deck ticket costs $60.
Let x represent the number of lower deck tickets.
Let y represent the number of upper deck tickets.
How would you write the objective function for revenue, R?

Jan volunteers to fold origami frogs and swans for a display.
• She has 8 squares of green paper for the frogs and 12 squares of white paper for the swans.
• It takes her 4 min to fold an origami frog and 3 min to fold an origami swan.
• There must be two swans for every frog.
Let f represent the number of frogs.
Let s represent the number of swans.
Which of the following is a constraint for this situation?

Which location best describes where would you find the optimal solutions to an objective function?

The following model represents an optimization problem. Determine the maximum solution.
Restrictions:
x R
y R

Constraints:
x 4
x – y 12
x + 3y 24

Objective function:
G = x – 2y

Audrey notices the number of people and dogs in a dog park.
• There are more people than dogs.
• There are at least 12 dogs.
• There are no more than 40 people and dogs, in total.
• All the dogs have four legs and all the people have two legs.
What is the maximum number of legs at the park?

Determine two valid solutions for the following system of linear inequalities.
{3y – 8x 0, y > 2, x > 5, x I, y I}

A Saskatchewan farmer is planting corn and barley.
• He wants to plant no more than 800 ha altogether.
• The farmer wants at least four times as many hectares of barley as corn.
• The yield per hectare of corn averages 60 bushels, and the yield per hectare of barley averages 30 bushels.
• Corn pays the farmer $7 per bushel, and barley pays $4 per bushel.
Let b represent the number of hectares of barley.
Let c represent the number of hectares of corn.
Describe any restrictions on the variables.

Baskets of fruit are being prepared to sell.
• Each basket contains at least 8 apples and more than 4 oranges.
• Apples cost 25¢ each, and oranges cost 40¢ each.
• The budget allows no more than $6, in total, for the fruit in each basket.
Let x represent the number of apples.
Let y represent the number of oranges.
Write the objective function to determine the combination of apples and oranges that will result in the maximum number of pieces of fruit in a basket.

For every quilt that is sold at a fundraising banquet, $90 goes to charity. For every ticket that is sold, $65 goes to charity. The organizers’ goal is to raise at least $7000. The organizers need to know how many quilts and tickets must be sold to meet their goal.
a) Define the variables and write a linear inequality to represent the situation.
b) Graph the linear inequality to help you determine whether each of the following points is in the solution set. The first coordinate is the number of quilts and the second is the number of tickets.
i) (40, 50)      ii) (10, 100)      iii) (20, 75)

The staff in a cafeteria are making two kinds of sandwiches: salami and cheese.
• A maximum of 820 sandwiches are needed.
• Based on previous demand, there should be at least two cheese sandwiches for every three salami sandwiches.
a) Define the variables and write a system of inequalities that models this situation.
b) Suggest two combinations of numbers of sandwiches that the cafeteria staff could make.

A refinery produces oil and gas.
• At least 3.5 L of gasoline are produced for each litre of heating oil.
• The refinery can produce up to 10 million litres of heating oil and 8 million litres of gasoline each day.
• Gasoline is projected to sell for $1.15 per litre. Heating oil is projected to sell for $1.85 per litre.
The company needs to determine the daily combination of gas and heating oil that must be produced to maximize revenue. Create a model to determine this combination. What would the revenue be?
Optimization Model
Let g represent the number of millions of litres of gasoline.
Let h represent the number of millions of litres of heating oil.
Let R represent the total revenue from sales in millions of dollars.
Restrictions:
g Î R, h Î R
Constraints:
g 0
h 0
g 3.5h
g 8
h 10
Objective function to maximize:
R = 1.15g + 1.85h