Appendix F: Systems of Inequalities

Transcription

1 Appendi F: Sstems of Inequalities F. Solving Sstems of Inequalities The Graph of an Inequalit What ou should learn The statements < and ⱖ are inequalities in two variables. An ordered pair 共a, b兲 is a solution of an inequalit in and for which the inequalit is true when a and b are substituted for and, respectivel. The graph of an inequalit is the collection of all solutions of the inequalit. To sketch the graph of an inequalit, begin b sketching the graph of the corresponding equation. The graph of the equation will normall separate the plane into two or more regions. In each such region, one of the following must be true.. All points in the region are solutions of the inequalit. Sstems of inequalities in two variables can be used to model and solve real-life problems. For instance, Eercise 85 on page F0 shows how to use a sstem of inequalities to analze the compositions of dietar supplements.. No point in the region is a solution of the inequalit. So, ou can determine whether the points in an entire region satisf the inequalit b simpl testing one point in the region. Sketch graphs of inequalities in two variables. Solve sstems of inequalities. Use sstems of inequalities in two variables to model and solve real-life problems. Wh ou should learn it Sketching the Graph of an Inequalit in Two Variables. Replace the inequalit sign with an equal sign and sketch the graph of the corresponding equation. Use a dashed line for < or > and a solid line for ⱕ or ⱖ. (A dashed line means that all points on the line or curve are not solutions of the inequalit. A solid line means that all points on the line or curve are solutions of the inequalit.). Test one point in each of the regions formed b the graph in Step. When the point satisfies the inequalit, shade the entire region to denote that ever point in the region satisfies the inequalit. Eample Sketching the Graph of an Inequalit Sketch the graph of b hand. Begin b graphing the corresponding equation, which is a parabola, as shown in Figure F.. B testing a point above the parabola 共0, 0兲 and a point below the parabola 共0, 兲, ou can see that 共0, 0兲 satisfies the inequalit because 0 ⱖ 0 and that 共0, 兲 does not satisf the inequalit because >Ⲑ 0. So, the points that satisf the inequalit are those ling above and those ling on the parabola. Now tr Eercise. The inequalit in Eample is a nonlinear inequalit in two variables. Most of the following eamples involve linear inequalities such as a b < c (a and b are not both zero). The graph of a linear inequalit is a half-plane ling on one side of the line a b c. Figure F. F

2 F Appendi F Sstems of Inequalities Eample Sketching the Graphs of Linear Inequalities Technolog Tip Sketch the graph of each linear inequalit. a. > b. a. The graph of the corresponding equation is a vertical line. The points that satisf the inequalit > are those ling to the right of (but not on) this line, as shown in Figure F.. b. The graph of the corresponding equation is a horizontal line. The points that satisf the inequalit are those ling below (or on) this line, as shown in Figure F.. A graphing utilit can be used to graph an inequalit. For instance, to graph, enter and use the shade feature of the graphing utilit to shade the correct part of the graph. You should obtain the graph shown below. > 9 = 9 = Figure F. For instructions on how to use the shade feature, see Appendi A; for specific kestrokes, go to this tetbook s Companion Website. Figure F. Now tr Eercise 9. Eample Sketching the Graph of a Linear Inequalit Sketch the graph of <. The graph of the corresponding equation is a line, as shown in Figure F.. Because the origin 共0, 0兲 satisfies the inequalit, the graph consists of the half-plane ling above the line. (Tr checking a point below the line. Regardless of which point below the line ou choose, ou will see that it does not satisf the inequalit.) < Now tr Eercise. To graph a linear inequalit, it can help to write the inequalit in slope-intercept form. For instance, b writing < in Eample in the form > ou can see that the solution points lie above the line 共or 兲, as shown in Figure F.. Figure F. (0, 0) =

3 Appendi F. Solving Sstems of Inequalities F Sstems of Inequalities Man practical problems in business, science, and engineering involve sstems of linear inequalities. A solution of a sstem of inequalities in and is a point 共, 兲 that satisfies each inequalit in the sstem. To sketch the graph of a sstem of inequalities in two variables, first sketch the graph of each individual inequalit (on the same coordinate sstem) and then find the region that is common to ever graph in the sstem. For sstems of linear inequalities, it is helpful to find the vertices of the solution region. Eample Solving a Sstem of Inequalities Sketch the graph (and label the vertices) of the solution set of the sstem. < > ⱕ Inequalit Inequalit Inequalit The graphs of these inequalities are shown in Figures F., F., and F., respectivel. The triangular region common to all three graphs can be found b superimposing the graphs on the same coordinate sstem, as shown in Figure F.5. To find the vertices of the region, solve the three sstems of corresponding equations obtained b taking pairs of equations representing the boundaries of the individual regions and solving these pairs of equations. Verte A: 共, 兲 Verte B: 共5, 兲 Verte C: 共, 兲 C = (, ) = B = (5, ) Stud Tip = 5 5 set = A = (, ) Figure F.5 Note in Figure F.5 that the vertices of the region are represented b open dots. This means that the vertices are not solutions of the sstem of inequalities. Now tr Eercise 5. Using different colored pencils to shade the solution of each inequalit in a sstem makes identifing the solution of the sstem of inequalities easier. The region common to ever graph in the sstem is where all shaded regions overlap. This region represents the solution set of the sstem.

4 F Appendi F Sstems of Inequalities For the triangular region shown in Figure F.5, each point of intersection of a pair of boundar lines corresponds to a verte. With more complicated regions, two border lines can sometimes intersect at a point that is not a verte of the region, as shown in Figure F.. To keep track of which points of intersection are actuall vertices of the region, ou should sketch the region and refer to our sketch as ou find each point of intersection. Not a verte Figure F. Eample 5 Solving a Sstem of Inequalities Sketch the region containing all points that satisf the sstem of inequalities. Inequalit Inequalit As shown in Figure F.7, the points that satisf the inequalit are the points ling above (or on) the parabola given b. Parabola = The points that satisf the inequalit are the points ling below (or on) the line given b. Line + = (, ) To find the points of intersection of the parabola and the line, solve the sstem of corresponding equations. Using the method of substitution, ou can find the solutions to be 共, 0兲 and 共, 兲. So, the region containing all points that satisf the sstem is indicated b the purple shaded region in Figure F.7. Now tr Eercise 55. (, 0) Figure F.7

5 Appendi F. F5 Solving Sstems of Inequalities When solving a sstem of inequalities, ou should be aware that the sstem might have no solution, or it might be represented b an unbounded region in the plane. These two possibilities are shown in Eamples and 7. Eample A Sstem with No Sketch the solution set of the sstem of inequalities. > < Inequalit Inequalit From the wa the sstem is written, it is clear that the sstem has no solution, because the quantit 共 兲 cannot be both less than and greater than. Graphicall, the inequalit > is represented b the half-plane ling above the line, and the inequalit < is represented b the half-plane ling below the line, as shown in Figure F.8. These two half-planes have no points in common. So the sstem of inequalities has no solution. += + = Figure F.8 No Now tr Eercise 57. Eample 7 An Unbounded Set Sketch the solution set of the sstem of inequalities. < > Inequalit Inequalit += The graph of the inequalit < is the half-plane that lies below the line, as shown in Figure F.9. The graph of the inequalit > is the half-plane that lies above the line. The intersection of these two half-planes is an infinite wedge that has a verte at 共, 0兲. This unbounded region represents the solution set. Now tr Eercise 59. (, 0) + = Figure F.9 Unbounded Region

6 F Appendi F Sstems of Inequalities p Applications Demand curve Consumer surplus Equilibrium point Price The net eample discusses two concepts that economists call consumer surplus and producer surplus. As shown in Figure F.0, the point of equilibrium is defined b the price p and the number of units that satisf both the demand and suppl equations. Consumer surplus is defined as the area of the region that lies below the demand curve, above the horizontal line passing through the equilibrium point, and to the right of the p-ais. Similarl, the producer surplus is defined as the area of the region that lies above the suppl curve, below the horizontal line passing through the equilibrium point, and to the right of the p-ais. The consumer surplus is a measure of the amount that consumers would have been willing to pa above what the actuall paid, whereas the producer surplus is a measure of the amount that producers would have been willing to receive below what the actuall received. Producer Suppl surplus curve Number of units Figure F.0 Eample 8 Consumer Surplus and Producer Surplus The demand and suppl equations for a new tpe of personal digital assistant are given b p p Demand equation Suppl equation where p is the price (in dollars) and represents the number of units. Find the consumer surplus and producer surplus for these two equations. Begin b finding the point of equilibrium b setting the two equations equal to each other and solving for Set equations equal to each other. Combine like terms.,000,000 Solve for. So, the solution is,000,000, which corresponds to an equilibrium price of p $0. So, the consumer surplus and producer surplus are the areas of the following triangular regions. Consumer Surplus Producer Surplus p p 0 0 p p 0 0 In Figure F., ou can see that the consumer and producer surpluses are defined as the areas of the shaded triangles. Consumer 共base兲共height兲 共,000,000兲共0兲 $5,000,000 surplus Producer 共base兲共height兲 共,000,000兲共0兲 $90,000,000 surplus Now tr Eercise 79. Price per unit (in dollars) Suppl vs. Demand p p = Consumer surplus Producer surplus p = p = ,000,000,000,000 Number of units Figure F.

7 Appendi F. Solving Sstems of Inequalities F7 Eample 9 Nutrition The minimum dail requirements from the liquid portion of a diet are 00 calories, units of vitamin A, and 90 units of vitamin C. A cup of dietar drink X provides 0 calories, units of vitamin A, and 0 units of vitamin C. A cup of dietar drink Y provides 0 calories, units of vitamin A, and 0 units of vitamin C. Set up a sstem of linear inequalities that describes how man cups of each drink should be consumed each da to meet the minimum dail requirements for calories and vitamins. Begin b letting and represent the following. number of cups of dietar drink X number of cups of dietar drink Y To meet the minimum dail requirements, the following inequalities must be satisfied ⱖ 00 ⱖ ⱖ 90 ⱖ 0 ⱖ 0 Calories Vitamin A Vitamin C The last two inequalities are included because and cannot be negative. The graph of this sstem of inequalities is shown in Figure F.. (More is said about this application in Eample in Appendi F..) Liquid Portion of a Diet Cups of drink Y 8 (0, ) (5, 5) (, ) (8, ) (, ) (9, 0) 8 0 Cups of drink X Figure F. From the graph, ou can see that two solutions (other than the vertices) that will meet the minimum dail requirements for calories and vitamins are 共5, 5兲 and 共8, 兲. There are man other solutions. Now tr Eercise 85. Stud Tip When using a sstem of inequalities to represent a real-life application in which the variables cannot be negative, remember to include inequalities for this constraint. For instance, in Eample 9, and cannot be negative, so the inequalities 0 and 0 must be included in the sstem.

8 F8 Appendi F Sstems of Inequalities F. Eercises For instructions on how to use a graphing utilit, see Appendi A. Vocabular and Concept Check In Eercises, fill in the blank(s).. An ordered pair a, b is a of an inequalit in and for which the inequalit is true when a and b are substituted for and, respectivel.. The of an inequalit is the collection of all solutions of the inequalit.. The graph of a inequalit is a half-plane ling on one side of the line a b c.. The of is defined b the price p and the number of units that satisf both the demand and suppl equations. Procedures and Problem Solving Identifing the Graph of an Inequalit In Eercises 5, match the inequalit with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] (a) (c) (e) (g) (b) (d) (f) (h) 5. < < 9 0. > 9. >. Sketching the Graph of an Inequalit In Eercises, sketch the graph of the inequalit.. <. 5.. < <. 5 > > > 0 0. >. < 9. > 9 Using a Graphing Utilit In Eercises, use a graphing utilit to graph the inequalit. Use the shade feature to shade the region representing the solution < > >. < ln. ln 5. >.

9 Appendi F. Solving Sstems of Inequalities F9 Writing an Inequalit In Eercises 5 8, write an inequalit for the shaded region shown in the graph Checking s In Eercises 9 and 50, determine whether each ordered pair is a solution of the sstem of inequalities Solving a Sstem of Inequalities In Eercises 5 8, sketch the graph of the solution of the sstem of inequalities < > < < 58. > < 0. > < < (a), 7 (c), 0 8 (a) 0, (c) 8, < > 0 > 0 7 > 5 > 5 5 > > 0 < > < > < > 8 (b), (d), (b) 5, (d), < > e 0 0 < Writing a Sstem of Inequalities In Eercises 9 78, find a set of inequalities to describe the region Rectangle: Vertices at,, 5,, 5, 7,, 7 7. Parallelogram: Vertices at 0, 0,, 0,,, 5, 77. Triangle: Vertices at 0, 0, 5, 0,, 78. Triangle: Vertices at, 0,, 0, 0, Consumer Surplus and Producer Surplus In Eercises 79 8, (a) graph the regions representing the consumer surplus and producer surplus for the demand and suppl equations, and (b) find the consumer surplus and the producer surplus. Demand Suppl 79. p p p p p p p p

10 F0 Appendi F Sstems of Inequalities Solving a Sstem of Inequalities In Eercises 8 8, (a) find a sstem of inequalities that models the problem and (b) graph the sstem, shading the region that represents the solution of the sstem. 8. Finance A person plans to invest some or all of $0,000 in two different interest-bearing accounts. Each account is to contain at least $7500, and one account should have at least twice the amount that is in the other account. 8. Arts Management For a summer concert event, one tpe of ticket costs $0 and another costs $5. The promoter of the concert must sell at least 0,000 tickets, including at least 0,000 of the $0 tickets and at least 5000 of the $5 tickets, and the gross receipts must total at least $00,000 in order for the concert to be held. 85. (p. F) A dietitian is asked to design a special dietar supplement using two different foods. The minimum dail requirements of the new supplement are 80 units of calcium, 0 units of iron, and 80 units of vitamin B. Each ounce of food X contains 0 units of calcium, 5 units of iron, and 0 units of vitamin B. Each ounce of food Y contains 0 units of calcium, 0 units of iron, and 0 units of vitamin B. 8. Retail Management A store sells two models of computers. Because of the demand, the store stocks at least twice as man units of model A as units of model B. The costs to the store for models A and B are $800 and $00, respectivel. The management does not want more than $0,000 in computer inventor at an one time, and it wants at least four model A computers and two model B computers in inventor at all times. 87. Architectural Design You design an eercise facilit that has an indoor running track with an eercise floor inside the track (see figure). The track must be at least 5 meters long, and the eercise floor must have an area of at least 500 square meters. E ercise floor (a) Find a sstem of inequalities describing the requirements of the facilit. (b) Sketch the graph of the sstem in part (a). 88. Geometr Two concentric circles have radii of and meters, where > (see figure). The area of the region between the circles must be at least 0 square meters. (a) Find a sstem of inequalities describing the constraints on the circles. (b) Graph the inequalit in part (a). (c) Identif the graph of the line in relation to the boundar of the inequalit. Eplain its meaning in the contet of the problem. Conclusions True or False? In Eercises 89 and 90, determine whether the statement is true or false. Justif our answer. 89. The area of the region defined b the sstem below is 99 square units. 90. The graph below shows the solution of the sstem 5 9 >. 9. Think About It After graphing the boundar of an inequalit in and, how do ou decide on which side of the boundar the solution set of the inequalit lies? 9. Writing Describe the difference between the solution set of a sstem of equations and the solution set of a sstem of inequalities