Lesson 4-3 System of Inequalities Objective: Use prior knowledge of graphing Solve a system of linear inequalities

Linear Inequalities

Vocabulary A system of linear inequalities is a set of two or more linear inequalities containing two or more variables. The solutions of a system of linear inequalities are all the ordered pairs that satisfy all the linear inequalities in the system.

System of Inequalities Tell whether the ordered pair is a solution of the given system. y ≤ –3x + 1 y < –3x + 2 (–1, –3); (0, 1); y ≥ x – 1 y < 2x + 2 y < –2x – 1 y > –x + 1 (–1, 5); (0, 0); y ≥ x + 3 y > x – 1

System of Inequalities The solutions of the system are represented by the overlapping shaded regions.

Solving a system of inequalities Solve the following system. Give an order pair that is a solution an one that is not a solution. y ≤ 3 y > –x + 5

Solving a system of inequalities Solve the following system. Give an order pair that is a solution an one that is not a solution. y ≤ x + 1 y > 2

Solving a system of inequalities Solve the following system. Give an order pair that is a solution an one that is not a solution. –3x + 2y ≥ 2 y < 4x + 3

Solving a system of inequalities Solve the following system. Give an order pair that is a solution an one that is not a solution. y > x – 7 3x + 6y ≤ 12

Solving a system of inequalities Solve the following system. Give an order pair that is a solution an one that is not a solution. y ≤ –2x – 4 y > –2x + 5

Solving a system of inequalities Solve the following system. Give an order pair that is a solution an one that is not a solution. y ≥ 4x + 6 y ≥ 4x – 5

Solving a system of inequalities Solve the following system. Give an order pair that is a solution an one that is not a solution. y > 3x – 2 y < 3x + 6

Solving a system of inequalities Solve the following system. Give an order pair that is a solution an one that is not a solution. y > –2x + 3 y > –2x

Solving a system of inequalities Solve the following system. Give an order pair that is a solution an one that is not a solution. y ≥ 4x – 2 y ≤ 4x + 2

A furniture manufacturer can make from 30 to 60 tables a day and from 40 to 100 chairs a day. It can make at most 120 units in one day. The profit on a table is $150, and the profit on a chair is $65. How many tables and chairs should they make per day to maximize profit? How much is the maximum profit?

Your school has contracted with a professional magician to perform at the school. The school has guaranteed an audience of at least 1000 and total ticket sales of at least $4800. The tickets are $4 for students and $6 for non-students, of which the magician receives $2.50 and $4.50 profit respectively. How may students and non-students tickets are needed to determine a minimum amount of money for the magician?

An office manager is purchasing file cabinets and wants to maximize storage space. The office has 60 square feet of floor space for the cabinets and $600 in the budget to purchase them. Cabinet A requires 3 square feet of floor space, has storage capacity of 12 cubic feet, and costs $75. Cabinet B requires 6 square feet of floor space, has a storage capacity of 18 cubic feet, and costs $50. How many of each cabinet should the office manager buy to maximize storage space?

A t-shirt company makes t-shirts and hoodies A t-shirt company makes t-shirts and hoodies. They can make between 80 and 100 t-shirts in one day. They can produce between 50 and 80 hoodies in one day. They can make, at most, 160 total units in one day. If the profit on each t-shirt is $6 and the profit on each hoodie is $10, how many of each kind do they need to make a maximum profit? What will this maximum profit be?

An appliance store manager is ordering chest and upright freezers An appliance store manager is ordering chest and upright freezers. One chest freezer costs $250 and delivers a $40 profit. One upright freezer costs $400 and delivers $60 profit. Based on previous sales, the manager expects to sell at least 100 freezers. Total profit must be at least $4800. Find the least number of each type of freezer the manager should order to minimize costs.

Friends from the math department often pick up lunch for each other Friends from the math department often pick up lunch for each other. When it was Mr. Jones’ turn to make the food run, he bought 5 sandwiches and 3 bags of chips. He spent $29.50. When Mr. Smith went, he got 4 sandwiches and 4 bags of chips for $26. How much does a sandwich cost? How about a bag of chips?