Systems of Linear Inequalities

Transcription

1 . Sstems of Linear Inequalities sstem of linear inequalities? How can ou sketch the graph of a ACTIVITY: Graphing Linear Inequalities Work with a partner. Match the linear inequalit with its graph. + Inequalit 0 Inequalit a. b. ACTIVITY: Graphing a Sstem of Linear Inequalities COMMON CORE Sstems of Inequalities In this lesson, ou will write and graph sstems of linear inequalities in two variables. solve real-life problems. Learning Standards A.CED. A.REI. Work with a partner. Consider the sstem of linear inequalities given in Activit. + Inequalit 0 Inequalit Use colored pencils to shade the solutions of the two linear inequalities. When ou graph both inequalities in the same coordinate plane, what do ou get? Describe each of the shaded regions in the graph at the right. What does the unshaded region represent? 8 Chapter Solving Sstems of Linear Equations

2 ACTIVITY: Writing a Sstem of Linear Inequalities Work with a partner. Write a sstem of linear inequalities whose solution is the traffic sign at the right. () + () + () () ACTIVITY: Representing a State b a Linear Sstem Math Practice Use Clear Definitions What is a sstem of linear inequalities? How can this definition help ou identif the states? Two states can be represented as the graph of a sstem of linear inequalities. Identif the two states. Eplain our reasoning.. IN YOUR OWN WORDS How can ou sketch the graph of a sstem of linear inequalities? 6. When graphing a sstem of linear inequalities, which region represents the solution of the sstem? Do ou think all sstems have a solution? Eplain. Use what ou learned about sstems of linear inequalities to complete Eercises 7 9 on page 89. Section. Sstems of Linear Inequalities 8

3 . Lesson Lesson Tutorials Ke Vocabular sstem of linear inequalities, p. 86 solution of a sstem of linear inequalities, p. 86 graph of a sstem of linear inequalities, p. 86 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. EXAMPLE Checking Solutions Tell whether each ordered pair is a solution of the sstem. < Inequalit + Inequalit a. (, ) b. (, 0) Inequalit Inequalit Inequalit Inequalit < + < + <? ()? + 0 <? ( ) 0? + < 6 0 < 0 (, ) is a solution of both inequalities. So, it is a solution of the sstem. (, 0) is not a solution of both inequalities. So, it is not a solution of the sstem. Eercises 6 Tell whether the ordered pair is a solution of the sstem of linear inequalities.. <. + > ; (, ) < + ; (0, ) The graph of a sstem of linear inequalities is the graph of all of the solutions of the sstem. Graphing a Sstem of Linear Inequalities Step Graph each inequalit in the same coordinate plane. Step Find the intersection of the half-planes. This intersection is the graph of the sstem Chapter Solving Sstems of Linear Equations

4 EXAMPLE Graphing a Sstem of Linear Inequalities Stud Tip For help with graphing linear inequalities, see Section.. Graph the sstem. Step : Graph each inequalit. Inequalit > + Inequalit Step : Find the intersection of the half-planes. One solution is (, ). (, ) Check Verif that (, ) is a solution of each inequalit. Inequalit Inequalit > + >? + > The solution is the purple shaded region. EXAMPLE Graphing a Sstem of Linear Inequalities: No Solution Graph the sstem. + < Inequalit + > Inequalit Step : Graph each inequalit. Step : Find the intersection of the half-planes. The lines are parallel and the half-planes do not intersect. So, the sstem has no solution. Eercises 7 Graph the sstem of linear inequalities.. +. >. + < > Section. Sstems of Linear Inequalities 87

5 EXAMPLE Writing a Sstem of Linear Inequalities Write a sstem of linear inequalities represented b the graph. The horizontal boundar line passes through (0, ). So, an equation of the line is. The slope of the other boundar line is and the -intercept is 0. So, an equation of the line is. The shaded region is above the solid boundar line, so the inequalit is. The shaded region is below the dashed boundar line, so the inequalit is <. The sstem is and <. EXAMPLE Real-Life Application 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 could ou spend at each location? Use the constraints to write a sstem of linear inequalities. 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. One ordered pair in the solution region is (., ). So, ou could spend. hours at the mall and hours at the beach. Eercises 6 Write a sstem of linear inequalities represented b the graph WHAT IF? In Eample, ou want to spend at least hours at the mall. How does this change the sstem? Is (., ) still a solution? Eplain. 88 Chapter Solving Sstems of Linear Equations

6 . Eercises Help with Homework. VOCABULARY How can ou verif that an ordered pair is a solution of a sstem of linear inequalities?. WRITING How are solving sstems of linear inequalities and sstems of linear equations similar? How are the different?. REASONING Is the point shown a solution of the sstem of linear inequalities? Eplain. (, ) 9+(-6)= +(-)= +(-9)= 9+(-)= Tell whether the ordered pair is a solution of the sstem of linear inequalities.. <. > > + ; (, ) ; (, ) + ; (0, 0) Graph the sstem of linear inequalities. 7. < 8. > > + 0 < 0. <.. + > > < + > 9. + <. +. < + > MUFFINS You can spend at most $ on fruit. Blueberries cost $ per pound and strawberries cost $ per pound. You need at least pounds to make muffins. a. Define the variables. b. Write a sstem of linear inequalities that represents this situation. c. Graph the sstem of linear inequalities. ies. d. Is it possible to bu pounds of blueberries and pound of strawberries in this situation? Justif our answer. Section. Sstems of Linear Inequalities 89

7 ERROR ANALYSIS Describe and correct the error in graphing the sstem of linear inequalities < > + 6 Match the graph with the corresponding sstem of linear inequalities A. < + B. + C. + < + > + >. REASONING Describe the intersection of the half-planes of the sstem shown.. JOBS You earn $ per hour working as a manager at a grocer store. You also coach a soccer team for $0 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 this situation. b. Identif and interpret one solution of the sstem. 90 Chapter Solving Sstems of Linear Equations

8 Write a sstem of linear inequalities represented b the graph Graph the sstem of linear inequalities. 7. > < > > < STRUCTURE Write a sstem of linear inequalities that is equivalent to < where > 0. Graph the sstem.. REPEATED REASONING One inequalit in a sstem is + > 6. Write another inequalit so the sstem has (a) no solution and (b) infinitel man solutions.. AMUSEMENT PARK You have at most 8 hours to spend at an amusement park. You want to spend less than hours plaing games and at least hours on rides. How much time can ou spend on each activit?. ROAD TRIP On a road trip, ou drive about 70 miles per hour and our friend drives about 60 miles per hour. The plan is to drive less than hours and at least 600 miles each da. Your friend will drive more hours than ou. Identif and interpret one solution of this situation.. The following points are the vertices of a triangle. (, ), (6, ), (, ) a. Write a sstem of linear inequalities that represents the triangle. b. Find the area of the triangle. Evaluate the epression when a =, b =, and c =. (Skills Review Handbook). a bc 6. ab + c 7. c ac 8. MULTIPLE CHOICE What is the solution of ( ) = ( + )? (Section.) A = B = C = D = 7 Section. Sstems of Linear Inequalities 9