Name: 

For which inequality is (0, 9) a possible solution?

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) | 5x – 2y 10, x R, y R}?

How would you graph the solution set for the linear inequality 2y – 2x 10?

Which system of linear inequalities has no solution?

Describe the boundary lines for the following system of linear inequalities.
{y – 3x < 12, x + y 0, x   R, y R}

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

Describe the boundary lines for the following system of linear inequalities.
{10y – 5x 0, 4x + 2y > 10, x   I, y   I}

Which test point is in the solution set for the following system of linear inequalities?
{y 2 + x, x + y 0, x   R, y   R}

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.
How would you write the objective function for revenue, R?

Where might you find the maximum solution to the objective function?
Restrictions:
x   R
y R

Constraints:
–2 x 4
–2 y 4

Objective function:
B = 2y + 3x

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

Brent found spiders and grasshoppers in his barn.
• There were at most 12 spiders and at least 10 grasshoppers.
• There were no more than 36 spiders and grasshoppers, in total.
Let s represent the number of spiders and let g represent the number of grasshoppers.
Which inequality represents a restriction of s and g based on the given information?

A butcher shop makes hamburger patties and sausages. Hamburger patties sell for $2 and sausage sell for $1.50. The butcher noticed that they always sell at least twice as many sausages as hamburger patties. Last week they sold 100 hamburger patties.
What is the maximum amount of profit they can make this week?

Graph the system of linear inequalities:
{(x, y) | x + y 2, x > –3, x W, y W}

Complete the graph of the solution set for the following system of inequalities.
{(x, y) | y 3x, 2x + 3y –3}

A publisher makes romance and adventure novels. Romance novels sell for $9 and adventure novels for $7.50. The publishers noticed that each month they sell at least three times as many adventure novels as romance novels, but never more than 1500 novels a month.
Let r represent the number of romance novels sold.
Let a represent the number of adventure novels sold.
Write a system of linear inequalities to describe the constraints. Then, write an objective function that represents the profit made from the sale of novels.

For every calendar that is sold at a fundraising banquet, $8 goes to charity. For every ticket that is sold, $25 goes to charity. The organizers’ goal is to raise at least $6000. The organizers need to know how many calendars 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 calendars and the second is the number of tickets.
i) (400, 100)      ii) (500, 100)      iii) (100, 200)

A magazine sells full-page and half-page advertisements.
• There can be no more than 100 pages worth of advertisements.
• No more than 120 half-page advertisements can be sold.
• At least 25 pages must be full-page advertisement.
• A full-page advertisement costs $300, and a half-page advertisement costs $200.
What combinations of advertisements would maximize the magazine’s revenue?
Create a model of this problem.

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