Topic Overview: Part 1: Systems of Equations Part 2: Systems of Inequalities

Part 2: Systems of Inequalities

Students can investigate systems of inequalities by solving linear programming problems. These systems can be used to model a number of real world situations.

Explanation

A system of linear inequalities is an extension of a system of linear equations and consists of two (or more) linear inequalities that have the same variables. For example, 2x + 3y < 4 and 3x + 4y < 5 constitute a system of inequalities if x represents the same item in both equations, y represents the same item in both equations, and both equations describe the same context. The solution consists of a region in the coordinate plane that satisfies all of the inequalities in the system.

Systems of inequalities can be useful in determining which combinations of products sold by a business will yield the maximum profit. For instance, one can find the combination of paintings, given a set of constraints, that produces the greatest profit for a painter who sells two different types of artwork.

To solve a system of linear inequalities, one must first determine the boundary lines by graphing each inequality as though it were an equation and then identifying the region where all of the inequalities would be shaded at the same time. In general, if the inequality is "less than," one shades below the line, and if the inequality is "greater than," one shades above the line. The shaded region is called the feasible region because it represents all the possible points that satisfy the system of inequalities.

One (and sometimes more than one) point in the feasible region is considered the optimum point. This is the point where profits are maximized or costs minimized. The optimum point is located on a corner of the feasible region (or the intersection of two of the boundary lines), and its coordinates are usually integer values.

To visualize what this entire process looks like, go to the Role in the Curriculum Section and follow through the example.

Other examples:
• A school system wants to hire a combination of teachers and aides. Given a set of constraints, what combination of teachers and aides would be least costly?

• A pet store sells dogs and cats. Given a set of constraints, what combination of dogs and cats should they sell to maximize their profit?
Mathematical Definition

Two or more linear inequalities with the same variables form a system of inequalities.

Also known as simultaneous inequalities, a system of inequalities consists of two or more inequalities that are conditions imposed simultaneously on all the variables, but may or may not have common solutions.

For example, x2 + y2 < 1 and y > 0, when treated as simultaneous inequalities, have as their solution set the set of all points above the x-intercept and inside the unit circle about the origin. The interior of a convex polygon is the graph (or solution set) of suitable simultaneous linear inequalities - in two variables for the polygon.
(Source: James, Robert C. and Glenn James. Mathematics Dictionary (4th edition). New York: Chapman & Hall, 1978.)

Linear programming is an important element in solving systems of inequalities. A linear programming problem is an optimization problem for which:
1. The function to be maximized or minimized - called the objective function - is a linear function of the decision variables.
2. The values of the decision variables must satisfy a set of constraints. Each constraint must be a linear equation or a linear inequality.
3. The variables must be non-negative.
Another widely accepted definition of linear programming is:
The mathematical theory of the minimization or maximization of a linear function subject to linear constraints. As often formulated, it is the problem of minimizing a linear expression in two or more variables subject to one or more linear constraints.
(Source: James, Robert C. and Glenn James. Mathematics Dictionary (4th edition). New York: Chapman & Hall, 1978.)

Role in the Curriculum

Students should begin to work with systems of linear inequalities and then investigate a broad array of linear programming problems. This is an opportunity to explore more sophisticated mathematical modeling situations. Modeling involves identifying and selecting relevant features of a real-world situation, representing those features symbolically, and analyzing and considering the accuracy and limitations of the model. Linear programming problems require all of the modeling processes listed above. They provide students with a rich opportunity to glimpse important applications that are used in a wide range of business settings.

For example, in Algebra 1, students should be able to recognize situations that use a system of inequalities, write a system of inequalities from a given set of information or constraints, and solve a system of inequalities by graphing each of the inequalities on the same grid and determining the region (if any) which satisfies all of the inequalities. Algebra 1 curricula usually ensure that students have experience graphing systems of linear inequalities by posing a variety of linear programming problems. These problems use systems of linear inequalities as a part of their solution. After students have solved the system of linear inequalities, they can look for an optimum point that produces either a maximum profit or a minimum cost. In linear programming, the solution to a system of linear inequalities is called the feasible region. The following example illustrates these processes.

A manufacturer of skis produces two types: telemark and cross-country. It takes the manufacturer 4 hours to produce each pair of telemark skis and 2 hours to produce each pair of cross-country skis. The maximum time available for production each week is 80 hours. It takes 2 hours to wax and put finishing touches on each pair of telemark skis and it takes 2 hours to complete the same processes for the cross-country skis. The maximum time allowed for waxing and finishing altogether is 64 hours each week. When the skis are sold the manufacturer makes \$140 profit on the telemark skis and \$100 profit on the cross-country skis. How many skis of each type must be produced each week to achieve a maximum profit?

Step 1: Identify the variables.
Let x = the number of pairs of telemark skis manufactured each week.
Let y = the number of pairs of cross-country skis manufactured each week.

Step 2: Write the inequalities based on the given constraints.
Production time constraint:
Finishing time constraint:
Two other constraints that are implied are and because it isn't possible to produce a negative number of skis.

Step 3: Write the profit or cost equation, also known as the objective function.
Profit or Objective equation: P = 140x + 100y

Step 4: Graph the system of inequalities to find the feasible region. To graph the inequalities, solve each inequality for y so that it is written in slope-intercept form. An alternate method is to find the x-intercept and y-intercept of each linear inequality by first substituting y = 0, and then x = 0 into each equation. This is the technique that Patricia's students used in the video for Workshop 3, Part II. As shown by the graphing calculator images below, the feasible region is the region where both inequalities are shaded at the same time. It is the region between the x-intercept, y-axis, and below the two lines. Each point in the feasible region represents a possible combination of telemark skis and cross-country skis that the manufacturer can produce and satisfy both constraints.

Step 5: Identify the corner points in the feasible region. There are four corner points in the feasible region: (0, 0) (0, 32) (20, 0) and (8, 24).

Step 6: Substitute the values of each of the corner points into the objective function, and identify either the maximum profit, or the minimum cost. In this particular example, we are looking for the maximum profit.

 x = # of telemark skis y = # of cross country skis P=140x + 100y Profit (dollars) 0 0 P=140(0)+100(0) \$0 0 32 P=140(0)+100(32) \$3200 20 0 P=140(20)+100(0) \$2800 8 24 P=140(8)+100(24) \$3520

Step 7: State the solution to the problem. To attain the highest possible profit, the manufacturer should produce eight pairs of telemark skis and 24 pairs of cross-country skis each week, for a profit of \$3520.

An understanding of the meaning of and how to solve systems of linear inequalities should be accomplished at the end of Algebra 1. In later study, students will solve systems of non-linear inequalities. The principles students learn solving systems of linear equations help them understand the process of solving more complicated systems of equations. Students might also spend time studying more difficult linear programming problems, including those involving three unknowns.

 Next: Lesson Plans