Mathematics 209: Finite Mathematics

Study Guide

Unit 5: Linear Inequalities and Linear Programming

In this unit, you will learn to solve problems using linear programming with two variables. This is a relatively new method, which uses linear inequalities.

Objectives

When you have completed this unit, you should be able to

1. determine the regions given by one and two linear inequalities.
2. solve a system of inequalities graphically.
3. determine whether a solution region is bounded or unbounded.
4. construct the model for an applied linear programming problem.
5. establish whether one or more optimal solutions of a problem exist.
6. apply the geometric solution to solve a linear programming problem with two decision variables.

Inequalities in Two Variables

Indications

1. Read Section 5-1, Inequalities in Two Variables, on pages 261–267 of the textbook.
2. Do odd-numbered exercises 1–37 on page 268 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
3. Solve odd-numbered problems 39–49 on page 269 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
4. Answer the questions listed below, and then compare your answers with those given in the Answers to Study Guide Questions.

Questions

1. Geometrically, what is the difference between the inequalities $ax+by<c$ and $ax+by\le c\text{?}$
2. What is the intersection of the regions $x<1$ and $y\ge 1\text{?}$
3. On page 263, the textbook authors recommend graphing lines using the $x$‑ and $y$‑intercepts. Could you use this recommendation in all cases? If not, what could you do instead?
4. How do you know if the point $\left(3,2\right)$ is in the region given by the inequality $3x+y\ge 6\text{?}$
5. How do you find a point that is not in the region given by the inequality $3x+y\ge 6\text{?}$
6. How do you know if the mathematical model of a problem is a linear inequality?

Systems of Linear Inequalities in Two Variables

Indications

1. Read Section 5-2, Systems of Inequalities in Two Variables, on pages 269–274 of the textbook.
2. Do odd-numbered exercises 1–35 on pages 275–276 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
3. Solve odd-numbered problems 39–43 on pages 276–277 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
4. Answer the questions listed below, and then compare your answers with those given in the Answers to Study Guide Questions.

Questions

7. Is the graphing method an accurate way of solving a system of inequalities with two variables?
8. What is the minimum number of linear equations we need in order to have a bounded region?
9. What is the maximum number of linear equations we can have in order to have an unbounded region?
10. Is it possible that two linear inequalities give an empty region?

Linear Programming in Two Dimensions: A Geometric Approach

Indications

1. Read Section 5-3, Linear Programming in Two Dimensions: A Geometric Approach, on pages 277–287 of the textbook.
2. Do odd-numbered exercises 1–29 on pages 287–288. If you have difficulty, consult your tutor to discuss the problem.
3. Solve odd-numbered problems 31–43 on pages 289–290. If you have difficulty, consult your tutor to discuss the problem.
4. Answer the questions listed below, and then compare your answers with those given in the Answers to Study Guide Questions.

Questions

11. How would you set up the objective function of a problem to be solved using linear programming?
12. What are the constraints in a problem to be solved using linear programming?
13. Why is it important to find the corner points of a feasible region in a problem to be solved using linear programming?
14. When does the linear programming method give a unique solution?

Finishing This Unit

1. Review the objectives of this unit and make sure you are able to meet each of them.
2. Study the section of the Chapter 5 Review titled Important Terms, Symbols and Concepts, on page 291 of the textbook.
3. If there is a concept, definition, example or exercise that is not yet clear to you, go back and re-read it. Contact your tutor if you need help.
4. You might want to do odd-numbered exercises 1–15 from the Review Exercise section on pages 291–292 of the textbook. The questions on the practice examination are taken from this exercise. If you have difficulty, consult your tutor to discuss the problem.
5. Complete the practice examination provided for this unit. Evaluate yourself, first checking your answers against those provided in the Answers section at the end of the textbook, and then comparing your solutions with those provided on pages S-256 to S-263 of the Student Solutions Manual portion of the textbook. The number of points in a question may indicate the number of steps in the solution. Give yourself full credit if your answer is correct and you give a complete solution, even if your solution differs from that shown in the Student Solutions Manual.

Practice Examination

Time: 1 hour
Total points: 20
Passing grade: 50%

Do the following exercises from the Chapter 5 Review Exercise on pages 291–292 of the textbook.

To obtain full credit you must justify all your answers and show your work.

1. Exercise 8  (Marks: 6 pts.)
2. Exercise 10  (Marks: 6 pts.)
3. Exercise 16  (Marks: 8 pts.)