7.5. Systems of Inequalities. The Graph of an Inequality. What you should learn. Why you should learn it

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1 0_0705.qd /5/05 9:5 AM Page 5 Section Sstems of Inequalities 5 Sstems of Inequalities What ou should learn Sketch the 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 You can use sstems of inequalities in two variables to model and solve real-life problems. For instance, in Eercise 77 on page 550, ou will use a sstem of inequalities to analze the retail sales of prescription drugs. The Graph of an Inequalit The statements < and are inequalities in two variables. An ordered pair a, b is a solution of an inequalit in and if 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.. 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. Sketching the Graph of an Inequalit in Two Variables. Replace the inequalit sign b an equal sign, and sketch the graph of the resulting equation. (Use a dashed line for < or > and a solid line for or.). Test one point in each of the regions formed b the graph in Step. If 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 To sketch the graph of, begin b graphing the corresponding equation, which is a parabola, as shown in Figure 7.9. B testing a point above the parabola 0, 0 and a point below the parabola 0,, ou can see that the points that satisf the inequalit are those ling above (or on) the parabola. Jon Feingersh/Masterfile = Note that when sketching the graph of an inequalit in two variables, a dashed line means all points on the line or curve are not solutions of the inequalit. A solid line means all points on the line or curve are solutions of the inequalit. (0, 0) Test point above parabola FIGURE 7.9 Now tr Eercise. Test point below parabola (0, )

2 0_0705.qd /5/05 9:5 AM Page 5 5 Chapter 7 Sstems of Equations and Inequalities 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. Eample Sketching the Graph of a Linear Inequalit Technolog A graphing utilit can be used to graph an inequalit or a sstem of inequalities. 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 below. Consult the user s guide for our graphing utilit for specific kestrokes. 0 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 this line, as shown in Figure 7.0. 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 7.. > = = FIGURE 7.0 FIGURE 7. Now tr Eercise. Eample Sketching the Graph of a Linear Inequalit (0, 0) < Sketch the graph of <. The graph of the corresponding equation is a line, as shown in Figure 7.. Because the origin 0, 0 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 ou choose, ou will see that it does not satisf the inequalit.) Now tr Eercise 9. FIGURE 7. = To graph a linear inequalit, it can help to write the inequalit in slope-intercept form. For instance, b writing < in the form > ou can see that the solution points lie above the line or, as shown in Figure 7..

3 0_0705.qd /5/05 9:5 AM Page 5 Sstems of Inequalities Section 7.5 Sstems of Inequalities 5 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, 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. This region represents the solution set of the sstem. For sstems of linear inequalities, it is helpful to find the vertices of the solution region. Eample Solving a Sstem of Inequalities Using different colored pencils to shade the solution of each inequalit in a sstem will make identifing the solution of the sstem of inequalities easier. Sketch the graph (and label the vertices) of the solution set of the sstem. Inequalit Inequalit < > Inequalit The graphs of these inequalities are shown in Figures 7., 7.0, and 7., respectivel, on page 5. The triangular region common to all three graphs can be found b superimposing the graphs on the same coordinate sstem, as shown in Figure 7.. 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. Verte A:, Verte B: 5, Verte C:, = = C = (, ) B = (5, ) 5 5 = set A = (, ) FIGURE 7. Note in Figure 7. 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.

4 0_0705.qd /5/05 9:5 AM Page 5 5 Chapter 7 Sstems of Equations and Inequalities For the triangular region shown in Figure 7., 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 7.. 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 7. Eample 5 Solving a Sstem of Inequalities = + = + (, ) (, 0) FIGURE 7.5 Sketch the region containing all points that satisf the sstem of inequalities. Inequalit Inequalit As shown in Figure 7.5, the points that satisf the inequalit Inequalit are the points ling above (or on) the parabola given b. Parabola The points satisfing the inequalit 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, 0 and,. So, the region containing all points that satisf the sstem is indicated b the shaded region in Figure 7.5. Now tr Eercise 7.

5 0_0705.qd /5/05 9:5 AM Page 55 Section 7.5 Sstems of Inequalities 55 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 Activities. Sketch the graph of the inequalit. Answer: 5. Sketch the graph of the solution of the sstem of inequalities. < > Answer: 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 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 7.. These two half-planes have no points in common. So, the sstem of inequalities has no solution. FIGURE 7. Now tr Eercise 9. + < + > Eample 7 An Unbounded Set + = + = (, 0) FIGURE 7.7 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 7.7. 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, 0. So, the solution set of the sstem of inequalities is unbounded. Now tr Eercise.

6 0_0705.qd /5/05 9:5 AM Page 5 5 Chapter 7 Sstems of Equations and Inequalities Price p Consumer surplus Producer surplus FIGURE 7.8 Demand curve Equilibrium point Suppl curve Number of units Applications Eample 9 in Section 7. discussed the equilibrium point for a sstem of demand and suppl functions. The net eample discusses two related concepts that economists call consumer surplus and producer surplus. As shown in Figure 7.8, the 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. Eample 8 Consumer Surplus and Producer Surplus Price per unit (in dollars) p FIGURE 7.9 Suppl vs. Demand p = Consumer surplus Producer surplus p = 0 p = ,000,000,000,000 Number of units The demand and suppl functions for a new tpe of personal digital assistant are given b p Demand equation p 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 equilibrium point (when suppl and demand are equal) b solving the equation In Eample 9 in Section 7., ou saw that the solution is,000,000 units, 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 p p In Figure 7.9, ou can see that the consumer and producer surpluses are defined as the areas of the shaded triangles. Consumer surplus (base)(height) $5,000,000,000,000 0 Producer surplus (base)(height) $90,000,000,000,000 0 Now tr Eercise 5.

7 0_0705.qd /5/05 9:5 AM Page 57 Section 7.5 Sstems of Inequalities 57 Eample 9 Nutrition The liquid portion of a diet is to provide at least 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 or eceed 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 or eceed the minimum dail requirements, the following inequalities must be satisfied The last two inequalities are included because and cannot be negative. The graph of this sstem of inequalities is shown in Figure 7.0. (More is said about this application in Eample in Section 7..) Calories Vitamin A Vitamin C 8 (0, ) If ou use graphing utilities in our course, ou ma want our students to work together on the following activit. Consider grouping students with the same model of graphing calculator together so that the can help one another with this activit. Activit Using the user s guide for our graphing utilit, find how to graph a sstem of inequalities. Then use the graphing utilit to graph each sstem. (, ) FIGURE 7.0 (, ) Now tr Eercise 9. (9, 0) 8 0 W RITING ABOUT MATHEMATICS Creating a Sstem of Inequalities Plot the points 0, 0,, 0,,, and 0, in a coordinate plane. Draw the quadrilateral that has these four points as its vertices. Write a sstem of linear inequalities that has the quadrilateral as its solution. Eplain how ou found the sstem of inequalities.

8 0_0705.qd /5/05 9:5 AM Page Chapter 7 Sstems of Equations and Inequalities 7.5 Eercises Eercise containing a sstem with no solution: 9 VOCABULARY CHECK: Fill in the blanks. Eercises containing unbounded solutions: 0,,,. An ordered pair a, b is a of an inequalit in and if 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.. A of a sstem of inequalities in and is a point, that satisfies each inequalit in the sstem. 5. The area of the region that lies below the demand curve, above the horizontal line passing through the equilibrium point, to the right of the p-ais is called the. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Eercises, sketch the graph of the inequalit.. <. < < 8. > < 9 > 9 5. > In Eercises 5, use a graphing utilit to graph the inequalit. Shade the region representing the solution. 5. < ln. ln 5 7. < < > < In Eercises 7 0, write an inequalit for the shaded region shown in the figure In Eercises, determine whether each ordered pair is a solution of the sstem of linear inequalities.. (a) 0, 0 (b), > (c), 0 (d), 8. 5 (a) 0, (b), < (c) 8, (d), < 7.. > 5 > In Eercises 5 8, sketch the graph and label the vertices of the solution set of the sstem of inequalities (a) 0, 0 (c), 9 (a), 7 (c), 0 < > 0 > 0 (b) 0, (d), (b) 5, (d), 8

9 0_0705.qd /5/05 9:5 AM Page 59 Section 7.5 Sstems of Inequalities > < 0... < > <.. > < In Eercises 9 5, use a graphing utilit to graph the inequalities. Shade the region representing the solution set of the sstem. In Eercises 55, derive a set of inequalities to describe the region < < > < > 5 > 5 > > 0 > 5 0 < 0 < < > e 0 5 < 5 > Rectangle: vertices at,, 5,, 5, 7,, 7. Parallelogram: vertices at 0, 0,, 0,,, 5,. Triangle: vertices at 0, 0, 5, 0,,. Triangle: vertices at, 0,, 0, 0, Suppl and Demand In Eercises 5 8, (a) graph the sstems representing the consumer surplus and producer surplus for the suppl and demand equations and (b) find the consumer surplus and producer surplus. Demand 5. p p p p Suppl p 0.5 p 5 0. p p Production A furniture compan can sell all the tables and chairs it produces. Each table requires hour in the assembl center and hours in the finishing center. Each chair requires hours in the assembl center and hours in the finishing center. The compan s assembl center is available hours per da, and its finishing center is available 5 hours per da. Find and graph a sstem of inequalities describing all possible production levels. 70. Inventor A store sells two models of computers. Because of the demand, the store stocks at least twice as man units of model A as of model B. The costs to the store for the two models 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. Find and graph a sstem of inequalities describing all possible inventor levels. 7. Investment Analsis A person plans to invest up to $0,000 in two different interest-bearing accounts. Each account is to contain at least $5000. Moreover, the amount in one account should be at least twice the amount in the other account. Find and graph a sstem of inequalities to describe the various amounts that can be deposited in each account. ( 8, 8)

10 0_0705.qd /5/05 9:5 AM Page Chapter 7 Sstems of Equations and Inequalities 7. Ticket Sales For a concert event, there are $0 reserved seat tickets and $0 general admission tickets. There are 000 reserved seats available, and fire regulations limit the number of paid ticket holders to 000. The promoter must take in at least $75,000 in ticket sales. Find and graph a sstem of inequalities describing the different numbers of tickets that can be sold. 7. Shipping A warehouse supervisor is told to ship at least 50 packages of gravel that weigh 55 pounds each and at least 0 bags of stone that weigh 70 pounds each. The maimum weight capacit in the truck he is loading is 7500 pounds. Find and graph a sstem of inequalities describing the numbers of bags of stone and gravel that he can send. 7. Truck Scheduling A small compan that manufactures two models of eercise machines has an order for 5 units of the standard model and units of the delue model. The compan has trucks of two different sizes that can haul the products, as shown in the table. Truck Standard Delue Large Medium Find and graph a sstem of inequalities describing the numbers of trucks of each size that are needed to deliver the order. 75. Nutrition A dietitian is asked to design a special dietar supplement using two different foods. 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. The minimum dail requirements of the diet are 00 units of calcium, 50 units of iron, and 00 units of vitamin B. (a) Write a sstem of inequalities describing the different amounts of food X and food Y that can be used. (b) Sketch a graph of the region corresponding to the sstem in part (a). (c) Find two solutions of the sstem and interpret their meanings in the contet of the problem. 7. Health A person s maimum heart rate is 0, where is the person s age in ears for When a person eercises, it is recommended that the person strive for a heart rate that is at least 50% of the maimum and at most 75% of the maimum. (Source: American Heart Association) (a) Write a sstem of inequalities that describes the eercise target heart rate region. (b) Sketch a graph of the region in part (a). (c) Find two solutions to the sstem and interpret their meanings in the contet of the problem. 77. Data Analsis: Prescription Drugs The table shows the retail sales (in billions of dollars) of prescription drugs in the United States from 999 to 00. (Source: National Association of Chain Drug Stores) 78. Phsical Fitness Facilit An indoor running track is to be constructed with a space for bod-building equipment inside the track (see figure). The track must be at least 5 meters long, and the bod-building space must have an area of at least 500 square meters. Year Model It Retail sales, (a) Use the regression feature of a graphing utilit to find a linear model for the data. Let t represent the ear, with t 9 corresponding to 999. (b) The total retail sales of prescription drugs in the United States during this five-ear period can be approimated b finding the area of the trapezoid bounded b the linear model ou found in part (a) and the lines 0, t 8.5, and t.5. Use a graphing utilit to graph this region. (c) Use the formula for the area of a trapezoid to approimate the total retail sales of prescription drugs. Bod-building equipment (a) Find a sstem of inequalities describing the requirements of the facilit. (b) Graph the sstem from part (a).

11 0_0705.qd /5/05 9:5 AM Page 55 Section 7.5 Sstems of Inequalities 55 Snthesis True or False? In Eercises 79 and 80, determine whether the statement is true or false. Justif our answer. 79. The area of the figure defined b the sstem is 99 square units. 80. The graph below shows the solution of the sstem 5 9 >. 8. Writing Eplain the difference between the graphs of the inequalit on the real number line and on the rectangular coordinate sstem. 8. 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? 8. Graphical Reasoning Two concentric circles have radii and, where >. The area between the circles must be at least 0 square units. (a) Find a sstem of inequalities describing the constraints on the circles. (b) Use a graphing utilit to graph the sstem of inequalities in part (a). Graph the line in the same viewing window. (c) Identif the graph of the line in relation to the boundar of the inequalit. Eplain its meaning in the contet of the problem. 8. The graph of the solution of the inequalit < is shown in the figure. Describe how the solution set would change for each of the following. (a) (b) > In Eercises 85 88, match the sstem of inequalities with the graph of its solution. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) (b) (d) Skills Review In Eercises 89 9, find the equation of the line passing through the two points. 89.,,, , 5,, , 5.,., Data Analsis: Cell Phone Bills The average monthl cell phone bills (in dollars) in the United States from 998 to 00, where t is the ear, are shown as data points t,. (Source: Cellular Telecommunications & Internet Association) 998, 9., 00, 7.7, 999,., 00, 8.0, 8, 0,,,, 0,.,.8,.9, , , 9.9 (a) Use the regression feature of a graphing utilit to find a linear model, a quadratic model, and an eponential model for the data. Let t 8 correspond to 998. (b) Use a graphing utilit to plot the data and the models in the same viewing window. (c) Which model is the best fit for the data? (d) Use the model from part (c) to predict the average monthl cell phone bill in 008.