Applications of Linear Inequalities

1. Applications of Linear Inequalities

2. Many real-life situations can be dealt with using systems of linear inequalities. • As we set up mathematical models for such real-world problems, we need to be sure that we are using the proper inequality symbols. • We will find it very convenient to remember which inequality symbols the following phrases indicate:

3. At most a certain quantity indicates • That means we can use that quantity or less. • At least a certain quantity indicates • That means we can use that quantity or more. • A certain quantity is neededindicates • That means we should have that quantity or more. • A certain quantity is available indicates • That means we could use that quantity or less.

4. Example #1

5. Let x be the number of units of A produced and y the number of units of B produced. • We will summarize the information provided in the following table. • As we have always done, we arrange the information in the problem with the variables related to columns of the table.

6. x units of product A and y units of product B require 2x + y pounds of steel but only 600 pounds of steel is available. • Hence, 2x + y≤ 600. Similarly, x units of product A and y units of product B require 4x + 5y minutes of labor but only 800 minutes of labor is available. • Therefore, 4x + 5y ≤ 800.

7. Since x and y represent the numbers of units of product A and product B, respectively, • neither x nor y can be negative. • So x ≥ 0 and y ≥ 0. • Collecting all the inequalities found, we obtain the following system of inequalities:

8. The profit from each unit of product A is $50 so the profit from x units of product A will be 50x dollars. Similarly, the profit from y units of product B will be 60y dollars. Thus if P is the total profit, then P = 50x + 60y dollars End of Example #1

9. We can think of the system of inequalities found in the previous example as a set of constraints • It is under these constraints that the manufacturer has to produce a number of units of product A and a number of units of product B. • Of course, we have not yet answered the interesting question: • How many units of product A and how many units of product B will give a profit as big as possible? • We will soon be able to answer such questions.

10. Now what we need is to sharpen our ability to • convert a given real-life problem into a system of inequalities that describe the constraints of the problem

11. Example #2

12. First, we define the variables: • Let x be the number of pills of brand X • Let y be the number of pills of brand Y. • We organize the given information in the following table:

13. Proceeding as we did in Example#1, we obtain the following systems of linear inequalities:

14. The total cost C of x pills of brand X and y pills of brand Y is: C = 6x + 8y cents. Note: In terms of dollars, one can also write C = 0.06x + 0.08y dollars. End of Example #2

15. What did we gain from the two examples considered? • First of all, they show that linear inequalities can be used to described real-life problems • They also prepare us for the study of Linear Programming