Linear Inequalities, Systems, and Linear Programming


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1 8.8 Linear Inequalities, Sstems, and Linear Programming Linear Inequalities, Sstems, and Linear Programming Linear Inequalities in Two Variables Linear inequalities with one variable were graphed on the number line in an earlier chapter. In this section linear inequalities in two variables are graphed in a rectangular coordinate sstem. Linear Inequalit in Two Variables An inequalit that can be written as A B v C or A B w C, where A, B, and C are real numbers and A and B are not both, is a linear inequalit in two variables. The smbols and ma replace and in this definition. A line divides the plane into three regions: the line itself and the two halfplanes on either side of the line. Recall that the graphs of linear inequalities in one variable are intervals on the number line that sometimes include an endpoint. The graphs of linear inequalities in two variables are regions in the real number plane and ma include a boundar line. The boundar line for the inequalit A B C or A B C is the graph of the equation A B C. To graph a linear inequalit, we follow these steps. Graphing a Linear Inequalit Step 1: Draw the boundar. Draw the graph of the straight line that is the boundar. Make the line solid if the inequalit involves or ; make the line dashed if the inequalit involves or. Step : Choose a test point. Choose an point not on the line as a test point. Step : Shade the appropriate region. Shade the region that includes the test point if it satisfies the original inequalit; otherwise, shade the region on the other side of the boundar line. EXAMPLE 1 Graph 6. First graph the straight line 6. The graph of this line, the boundar of the graph of the inequalit, is shown in Figure 4 on the net page. The graph of the inequalit 6 includes the points of the line 6, and either the points above the line 6 or the points below that line. To decide which, select an point not on the line 6 as a test point. The origin,,, often
2 48 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities The TI8 Plus allows us to shade the appropriate region for an inequalit. Compare with Figure 4. is a good choice. Substitute the values from the test point, for and in the inequalit 6. 6? 6 False Since the result is false,, does not satisf the inequalit, and so the solution set includes all points on the other side of the line. This region is shaded in Figure 4. + > 6 + = 6 (1, ) (, ) 4 = 4 > 4 FIGURE 4 FIGURE Compare with Figure 44. Can ou tell from the calculator graph whether points on the boundar line are included in the solution set of the inequalit? 1 > 4 EXAMPLE Graph 4. First graph the boundar line, 4. The graph is shown in Figure 44. The points of the boundar line do not belong to the inequalit 4 (since the inequalit smbol is and not ). For this reason, the line is dashed. To decide which side of the line is the graph of the solution set, choose an point that is not on the line, sa 1,. Substitute 1 for and for in the original inequalit. 1 4? 5 4 False Because of this false result, the solution set lies on the side of the boundar line that does not contain the test point 1,. The solution set, graphed in Figure 44, includes onl those points in the shaded region (not those on the line). Sstems of Inequalities Methods of solving sstems of equations were discussed in the previous section. Sstems of inequalities with two variables ma be solved b graphing. A sstem of linear inequalities consists of two or more such inequalities, and the solution set of such a sstem consists of all points that make all the inequalities true at the same time The crosshatched region shows the solution set of the sstem 4. Graphing a Sstem of Linear Inequalities Step 1: Graph each inequalit in the same coordinate sstem. Graph each inequalit in the sstem, using the method described in Eamples 1 and. Step : Find the intersection of the regions of solutions. Indicate the intersection of the regions of solutions of the individual inequalities. This is the solution set of the sstem.
3 8.8 Linear Inequalities, Sstems, and Linear Programming 48 EXAMPLE Graph the solution set of the linear sstem Begin b graphing 6. To do this, graph 6 as a solid line. Since, makes the inequalit true, shade the region containing,, as shown in Figure 45. Now graph 5 1. The solid line boundar is the graph of 5 1. Since, makes the inequalit false, shade the region that does not contain,, as shown in Figure 46. The solution set of the sstem is given b the intersection (overlap) of the regions of the graphs in Figures 45 and 46. The solution set is the shaded region in Figure 47, and includes portions of the two boundar lines. + < > 1 The shaded region is the solution set of the sstem + < 6 5 > 1. 5 FIGURE 45 FIGURE 46 FIGURE 47 In practice, we usuall do all the work in one coordinate sstem at the same time. In the following eample, onl one graph is shown. EXAMPLE 4 Graph the solution set of the linear sstem The graph is obtained b graphing the four inequalities in one coordinate sstem and shading the region common to all four as shown in Figure 48. As the graph shows, the boundar lines are all solid = = 6 + > > 8 < 6 < 5 FIGURE 48
4 484 CHAPTER 8 Graphs, Functions, and Sstems of Equations and Inequalities Linear Programming A ver important application of mathematics to business and social science is called linear programming. Linear programming is used to find an optimum value, for eample, minimum cost or maimum profit. Procedures for solving linear programming problems were developed in 1947 b George Dantzig, while he was working on a problem of allocating supplies for the Air Force in a wa that minimized total cost. George B. Dantzig of Stanford Universit has been one of the ke people behind operations research (OR). As a management science, OR is not a single discipline, but draws from mathematics, probabilit theor, statistics, and economics. The name given to this multiple shows its historical origins in World War II, when operations of a militar nature called forth the efforts of man scientists to research their fields for applications to the war effort and to solve tactical problems. Operations research is an approach to problem solving and decision making. First of all, the problem has to be clarified. Quantities involved have to be designated as variables, and the objectives as functions. Use of models is an important aspect of OR. (1, 6) (5, 6) (1, 1) (5, 5) Region of feasible solutions FIGURE 49 EXAMPLE 5 The Smartski Compan makes two products, tape decks and amplifiers. Each tape deck gives a profit of $, while each amplifier gives a profit of $7. The compan must manufacture at least 1 tape deck per da to satisf one of its customers, but no more than 5 because of production problems. Also, the number of amplifiers produced cannot eceed 6 per da. As a further requirement, the number of tape decks cannot eceed the number of amplifiers. How man of each should the compan manufacture in order to obtain the maimum profit? We translate the statements of the problem into smbols b letting number of tape decks to be produced dail number of amplifiers to be produced dail. According to the statement of the problem, the compan must produce at least one tape deck (one or more), so 1. No more than 5 tape decks ma be produced: No more than 6 amplifiers ma be made in one da: The number of tape decks ma not eceed the number of amplifiers: The number of tape decks and of amplifiers cannot be negative: and. All restrictions, or constraints, that are placed on production can now be summarized: 1, 5, 6,,,. The maimum possible profit that the compan can make, subject to these constraints, is found b sketching the graph of the solution set of the sstem. See Figure 49. The onl feasible values of and are those that satisf all constraints. These values correspond to points that lie on the boundar or in the shaded region, called the region of feasible solutions. Since each tape deck gives a profit of $, the dail profit from the production of tape decks is dollars. Also, the profit from the production of amplifiers will be 7 dollars per da. The total dail profit is thus given b the following objective function: Profit 7.
5 8.8 Linear Inequalities, Sstems, and Linear Programming 485 It Pas to Do Your Homework George Dantzig, profiled on the previous page, was interviewed in the September 1986 issue of the College Mathematics Journal. In the interview he relates the stor of how he obtained his degree without actuall writing a thesis. One da Dantzig arrived late to one of his classes and on the board were two problems. Assuming that the were homework problems, he worked on them and handed them in a few das later, apologizing to his professor for taking so long to do them. Several weeks later he received an earl morning visit from his professor. The problems had not been intended as homework problems; the were actuall two famous unsolved problems in statistics! Later, when Dantzig began to think about a thesis topic, his professor told him that the two solutions would serve as his thesis. The problem of the Smartski Compan ma now be stated as follows: find values of and in the region of feasible solutions as shown in Figure 49 that will produce the maimum possible value of 7. It can be shown that an optimum value (maimum or minimum) will alwas occur at a verte (or corner point) of the region of feasible solutions. Locate the point, that gives the maimum profit b checking the coordinates of the vertices, shown in Figure 49 on the previous page and listed below. Find the profit that corresponds to each coordinate pair and choose the one that gives the maimum profit. Point 1, 1 1, 6 5, 6 5, 5 Profit k Maimum The maimum profit of $57 is obtained when 5 tape decks and 6 amplifiers are produced each da. To solve a linear programming problem in general, use the following steps. Solving a Linear Programming Problem Step 1: Step : Step : Step 4: Step 5: Write all necessar constraints and the objective function. Graph the region of feasible solutions. Identif all vertices. Find the value of the objective function at each verte. The solution is given b the verte producing the optimum value of the objective function.
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