Having devoted a great deal of time to the solution of equations and systems of equations, we now turn our attention to solving inequalities and systems of inequalities.

In doing so, we will find it necessary to find all values of the variable or variables that satisfy the inequality or system of inequalities.

There are numerous technical applications of inequalities. For example, in electricity it might be necessary to find the values of a current that are greater than a specified value. In designing a link in a robotic mechanism, it might be necessary to find the forces that are less than a specified value. Computers can be programmed to switch from one part of a program to another, based upon a result that is greater than (or less than) some given value.

Systems of linear equations have been studied for more than 2000 years, but almost no attention was given to systems of linear inequalities until World War II in the 1940s. Problems of deploying personnel and aircraft effectively and allocating supplies efficiently led the U.S. Air Force to have a number of scientists, economists, and mathematicians look for solutions. From this, a procedure for analyzing such problems was devised in 1947 by George Danzig and his colleagues. Their method involved using systems of linear inequalities and is today called linear programming. We introduce the basic method of linear programming in the final section of this chapter.

Here, we see that a mathematical method was developed as a result of a military need. Today, linear programming is widely used in business and industry in order to set production levels for maximizing profits and minimizing costs.