Chapter 4 Notes for Math 181
Finite Mathematics, Fifth Edition

In order to tie comments to specific locations in the book, I have used the available page ruler sheet to identify lines in the text. A copy of that page ruler sheet can be printed from The Index Sheet.

Chapter 4: Linear Programming
4.1 Graphing Linear Inequalities
Page	Line #	Notes
267	6	The book takes the unusual position of shading the half-plane that is not in the solution. Their point is that doing so will make things much easier when we start working with systems of linear inequlities, situations where we are drawing 2, 3, 4, and even more inequalities on the same graph. The authors have a good point in taking this position. However, it is fairly non-standard. If you are going to follow the style of the book, then be sure to mark your solutions as such.
269	all	It would seem that we need to have a significant amount of practice in graphing inequalities, and in graphing systems of linear inequalities. To that end, we have a page of practice for such problems.
4.2 Solving Linear Programming Problems Graphically
Page	Line #	Notes
274	all	There is an expanded version of the earlier inequalities practice, this time each problem is also given an objective function (expression) that you are to maximize and minimize over the critical region. See the new page on Linear Programming Graphic solutions for such practice. Please report any problems with this page. An new page to walk through solving a problem is given on Linear Programming by hand.
4.3. The Simplex Method: Solving Standard Maximization Problems
Page	Line #	Notes
290	all	The book does a good job of explaining the beginnings of the Simplex method for solving standard linear programming problems. I have a slightly different system. The first pages of explanation for that system are on the web page for linear programming. I hope to have numerous additional examples in the next few days, but a solid reading of the notes on my pages is more than advisable. The page Simplex Basic provides links to many variations on three simple problems that are solved, step by step, via the simplex method. first second third fourth fifth sixth
4.4 The Simplex Method: Solving General Linear Programming Problems
Page	Line #	Notes
all	all	Note that the method in the book differs slightly from that presented in the web page, now expanded, referenced in the last section. In addition, there is a new web page, Simplex Example, that provides a lengthy walk-through of the Simplex Method, including the use of the SM3 program. In addition, the page lp_demo_index provides links to numerous pages that demonstrate the Simplex Method.
4.5 The Simplex Method and Duality
Page	Line #	Notes
2	36