Essential Question How can you graph a system of linear inequalities?

Transcription

1 5.7 Sstems of Linear Inequalities Essential Question How can ou graph a sstem of linear inequalities? Graphing Linear Inequalities Work with a partner. Match each linear inequalit with its graph. Eplain our reasoning. + Inequalit 0 Inequalit A. B. Graphing a Sstem of Linear Inequalities MAKING SENSE OF PROBLEMS To be proficient in math, ou need to eplain to ourself the meaning of a problem. Work with a partner. Consider the linear inequalities given in Eploration. + Inequalit 0 Inequalit a. Use two different colors to graph the inequalities in the same coordinate plane. What is the result? b. Describe each of the shaded regions of the graph. What does the unshaded region represent? Communicate Your Answer 3. How can ou graph a sstem of linear inequalities?. When graphing a sstem of linear inequalities, which region represents the solution of the sstem? 5. Do ou think all sstems of linear inequalities have a solution? Eplain our reasoning. 6. Write a sstem of linear inequalities 6 represented b the graph. Section 5.7 Sstems of Linear Inequalities 73

2 5.7 Lesson What You Will Learn Core Vocabular sstem of linear inequalities, p. 7 solution of a sstem of linear inequalities, p. 7 graph of a sstem of linear inequalities, p. 75 Previous linear inequalit in two variables Check solutions of sstems of linear inequalities. Graph sstems of linear inequalities. Write sstems of linear inequalities. Use sstems of linear inequalities to solve real-life problems. Sstems of Linear Inequalities A sstem of linear inequalities is a set of two or more linear inequalities in the same variables. An eample is shown below. < + Inequalit Inequalit A solution of a sstem of linear inequalities in two variables is an ordered pair that is a solution of each inequalit in the sstem. Checking Solutions Tell whether each ordered pair is a solution of the sstem of linear inequalities. < Inequalit + Inequalit a. (3, 5) b. (, 0) a. Substitute 3 for and 5 for in each inequalit. Inequalit Inequalit < + 5 <? (3) 5? < 6 5 Because the ordered pair (3, 5) is a solution of each inequalit, it is a solution of the sstem. b. Substitute for and 0 for in each inequalit. Inequalit Inequalit < + 0 <? ( ) 0? + 0 < 0 Because (, 0) is not a solution of each inequalit, it is not a solution of the sstem. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Tell whether the ordered pair is a solution of the sstem of linear inequalities.. (, 5); < 5 >. (, ); 3 + > 7 Chapter 5 Solving Sstems of Linear Equations

3 Graphing Sstems of Linear Inequalities The graph of a sstem of linear inequalities is the graph of all the solutions of the sstem. Core Concept Graphing a Sstem of Linear Inequalities Step Graph each inequalit in the same coordinate plane. Step Find the intersection of the half-planes that are solutions of the inequalities. This intersection is the graph of the sstem. < + 6 Check Verif that (, ) is a solution of each inequalit. Inequalit 3 3 Inequalit > + >? + > Graph the sstem of linear inequalities. 3 Inequalit > + Inequalit Step Graph each inequalit. Step Find the intersection of the half-planes. One solution is (, ). Graphing a Sstem of Linear Inequalities The solution is the purple-shaded region. (, ) Graph the sstem of linear inequalities. Graphing a Sstem of Linear Inequalities: No Solution + < Inequalit + > 3 Inequalit 3 Step Graph each inequalit. Step Find the intersection of the half-planes. Notice that the lines are parallel, and the half-planes do not intersect. So, the sstem has no solution. Monitoring Progress Graph the sstem of linear inequalities. Help in English and Spanish at BigIdeasMath.com > < > Section 5.7 Sstems of Linear Inequalities 75

4 Writing Sstems of Linear Inequalities Writing a Sstem of Linear Inequalities Write a sstem of linear inequalities represented b the graph. Inequalit The horizontal boundar line passes through (0, ). So, an equation of the line is =. The shaded region is above the solid boundar line, so the inequalit is. Inequalit The slope of the other boundar line is, and the -intercept is 0. So, an equation of the line is =. The shaded region is below the dashed boundar line, so the inequalit is <. The sstem of linear inequalities represented b the graph is Inequalit <. Inequalit Writing a Sstem of Linear Inequalities Write a sstem of linear inequalities represented b the graph. Inequalit The vertical boundar line passes through (3, 0). So, an equation of the line is = 3. The shaded region is to the left of the solid boundar line, so the inequalit is 3. Inequalit The slope of the other boundar line is, and the -intercept is. 3 So, an equation of the line is =. The shaded region is above 3 the dashed boundar line, so the inequalit is > 3. The sstem of linear inequalities represented b the graph is 3 Inequalit 3 >. Inequalit Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write a sstem of linear inequalities represented b the graph Chapter 5 Solving Sstems of Linear Equations

5 Solving Real-Life Problems Modeling with Mathematics You have at most 8 hours to spend at the mall and at the beach. You want to spend at least hours at the mall and more than hours at the beach. Write and graph a sstem that represents the situation. How much time can ou spend at each location?. Understand the Problem You know the total amount of time ou can spend at the mall and at the beach. You also know how much time ou want to spend at each location. You are asked to write and graph a sstem that represents the situation and determine how much time ou can spend at each location.. Make a Plan Use the given information to write a sstem of linear inequalities. Then graph the sstem and identif an ordered pair in the solution region. 3. Solve the Problem Let be the number of hours at the mall and let be the number of hours at the beach. + 8 > at most 8 hours at the mall and at the beach at least hours at the mall more than hours at the beach Graph the sstem. Time at the Mall and at the Beach Check ? Hours at the beach Hours at the mall > 5 > One ordered pair in the solution region is (.5, 5). So, ou can spend.5 hours at the mall and 5 hours at the beach.. Look Back Check our solution b substituting it into the inequalities in the sstem, as shown. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 8. Name another solution of Eample WHAT IF? You want to spend at least 3 hours at the mall. How does this change the sstem? Is (.5, 5) still a solution? Eplain. Section 5.7 Sstems of Linear Inequalities 77

6 5.7 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY How can ou verif that an ordered pair is a solution of a sstem of linear inequalities?. WHICH ONE DOESN T BELONG? Use the graph shown. Which of the ordered pairs does not belong with the other three? Eplain our reasoning. 5 (, ) (0, ) (, 6) (, ) 6 Monitoring Progress and Modeling with Mathematics In Eercises 3 6, tell whether the ordered pair is a solution of the sstem of linear inequalities. 3. (, 3). (, ) 5. (, 5) 6. (, ) In Eercises 7 0, tell whether the ordered pair is a solution of the sstem of linear inequalities. (See Eample.) 7. ( 5, ); < > (0, 0); (, ); > > 5 0. (, ); + 5 In Eercises 0, graph the sstem of linear inequalities. (See Eamples and 3.). >. < 5 > 9. < > + > In Eercises 6, write a sstem of linear inequalities represented b the graph. (See Eamples and 5.) <. < > > < > < + 78 Chapter 5 Solving Sstems of Linear Equations

7 ERROR ANALYSIS In Eercises 7 and 8, describe and correct the error in graphing the sstem of linear inequalities MODELING WITH MATHEMATICS You are fishing Dnamic Solutions for surfperch available and rockfish, at BigIdeasMath.com which are species of bottomfish. Gaming laws allow ou to catch no more than 5 surfperch per da, no more than 0 rockfish per da, and no more than 0 total bottomfish per da. a. Write and graph a sstem of linear inequalities that represents the situation. b. Use the graph to determine whether ou can catch surfperch and 9 rockfish in da > + surfperch rockfish 3. REASONING Describe the intersection of the half-planes of the sstem shown. 9. MODELING WITH MATHEMATICS You can spend at most $ on fruit. Blueberries cost $ per pound, and strawberries cost $3 per pound. You need at least 3 pounds of fruit to make muffins. (See Eample 6.) a. Write and graph a sstem of linear inequalities that represents the situation. b. Identif and interpret a solution of the sstem. c. Use the graph to determine whether ou can bu pounds of blueberries and pound of strawberries. 30. MODELING WITH MATHEMATICS You earn $0 per hour working as a manager at a grocer store. You are required to work at the grocer store at least 8 hours per week. You also teach music lessons for $5 per hour. You need to earn at least $0 per week, but ou do not want to work more than 0 hours per week. a. Write and graph a sstem of linear inequalities that represents the situation. b. Identif and interpret a solution of the sstem. c. Use the graph to determine whether ou can work 8 hours at the grocer store and teach hour of music lessons. 33. MATHEMATICAL CONNECTIONS The following points are the vertices of a shaded rectangle. (, ), (6, ), (6, ), (, ) a. Write a sstem of linear inequalities represented b the shaded rectangle. b. Find the area of the rectangle. 3. MATHEMATICAL CONNECTIONS The following points are the vertices of a shaded triangle. (, 5), (6, ), (, ) a. Write a sstem of linear inequalities represented b the shaded triangle. b. Find the area of the triangle. 35. PROBLEM SOLVING You plan to spend less than half of our monthl $000 pacheck on housing and savings. You want to spend at least 0% of our pacheck on savings and at most 30% of it on housing. How much mone can ou spend on savings and housing? 36. PROBLEM SOLVING On a road trip with a friend, ou drive about 70 miles per hour, and our friend drives about 60 miles per hour. The plan is to drive less than 5 hours and at least 600 miles each da. Your friend will drive more hours than ou. How man hours can ou and our friend each drive in da? Section 5.7 Sstems of Linear Inequalities 79

8 37. WRITING How are solving sstems of linear inequalities and solving sstems of linear equations similar? How are the different? 38. HOW DO YOU SEE IT? The graphs of two linear equations are shown. A 3 = + D = + B Replace the equal signs with inequalit smbols to create a sstem of linear inequalities that has point C as a solution, but not points A, B, and D. Eplain our reasoning USING STRUCTURE Write a sstem of linear inequalities that is equivalent to <, where > 0. Graph the sstem. 0. MAKING AN ARGUMENT Your friend sas that a sstem of linear inequalities in which the boundar lines are parallel must have no solution. Is our friend correct? Eplain.. CRITICAL THINKING Is it possible for the solution set of a sstem of linear inequalities to be all real numbers? Eplain our reasoning. OPEN-ENDED In Eercises, write a sstem of linear inequalities with the given characteristic.. All solutions are in Quadrant I. C 3. All solutions have one positive coordinate and one negative coordinate.. There are no solutions. 5. OPEN-ENDED One inequalit in a sstem is + > 6. Write another inequalit so the sstem has (a) no solution and (b) infinitel man solutions. 6. THOUGHT PROVOKING You receive a gift certificate for a clothing store and plan to use it to bu T-shirts and sweatshirts. Describe a situation in which ou can bu 9 T-shirts and sweatshirt, but ou cannot bu 3 T-shirts and 8 sweatshirts. Write and graph a sstem of linear inequalities that represents the situation. 7. CRITICAL THINKING Write a sstem of linear inequalities that has eactl one solution. 8. MODELING WITH MATHEMATICS You make necklaces and ke chains to sell at a craft fair. The table shows the amounts of time and mone it takes to make a necklace and a ke chain, and the amounts of time and mone ou have available for making them. Time to make (hours) Cost to make (dollars) Necklace Ke chain Available a. Write and graph a sstem of four linear inequalities that represents the number of necklaces and the number of ke chains that ou can make. b. Find the vertices (corner points) of the graph of the sstem. c. You sell each necklace for $0 and each ke chain for $8. The revenue R is given b the equation R = Find the revenue corresponding to each ordered pair in part (b). Which verte results in the maimum revenue? Maintaining Mathematical Proficienc Write the product using eponents. (Skills Review Handbook) () () () 5. Write an equation of the line with the given slope and -intercept. (Section.) Reviewing what ou learned in previous grades and lessons 5. slope: 53. slope: 5. slope: 55. slope: 3 -intercept: 6 -intercept: 5 -intercept: -intercept: 0 80 Chapter 5 Solving Sstems of Linear Equations