Introduction

A lot of interesting real-world problems can be solved with systems of linear inequalities.

For example, you go to your favorite restaurant and you want to be served by your best friend who happens to work there. However, your friend only waits tables in a certain region of the restaurant. The restaurant is also known for its great views, so you want to sit in a certain area of the restaurant that offers a good view. Solving a system of linear inequalities will allow you to find the area in the restaurant where you can sit to get the best view and be served by your friend.

Often, systems of linear inequalities deal with problems where you are trying to find the best possible situation given a set of constraints. Most of these application problems fall in a category called linear programming problems.

Linear programming is the process of taking various linear inequalities relating to some situation, and finding the best possible value under those conditions. A typical example would be taking the limitations of materials and labor at a factory, then determining the best production levels for maximal profits under those conditions. These kinds of problems are used every day in the organization and allocation of resources. These real-life systems can have dozens or hundreds of variables, or more. In this section, we'll only work with the simple two-variable linear case.

The general process is to:

  • Graph the inequalities (called constraints) to form a bounded area on the coordinate plane (called the feasibility region).
  • Figure out the coordinates of the corners (or vertices) of this feasibility region by solving the system of equations that applies to each of the intersection points.
  • Test these corner points in the formula (called the optimization equation) for which you're trying to find the maximum or minimum value.


Source: cK-12, https://www.ck12.org/algebra/Linear-Programming/lesson/Linear-Programming-ALG-I/
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 License.

Back to '6.8: Applications of Systems of Linear Inequalities'