In Chapter 2, you learned how to handle systems of linear equations. However there are many situations in which inequalities appear instead of equalities. In such cases we are often interested in an optimal solution extremizing a particular quantity of interest. Questions like this are a focus of fields such as mathematical optimization and operations research. For the case where the functions involved are linear, these problems go under the title linear programming. Originally these ideas were driven by military applications, but by now are ubiquitous in science and industry. Gigantic computers are dedicated to implementing linear programming methods such as George Dantzig’s simplex algorithm–the topic of this chapter.