Graph Linear Inequalities in Two Variables

Transcription

1 .8 Graph Linear Inequalities in Two Variables Before You solved linear inequalities in one variable. Now You will graph linear inequalities in two variables. Wh? So ou can model data encoding, as in Eample 4. Ke Vocabular linear inequalit in two variables solution of a linear inequalit graph of a linear inequalit half-plane A linear inequalit in two variables can be written in one of these forms: A B < C A B C A B > C A B C An ordered pair (, ) is a solution of a linear inequalit in two variables if the inequalit is true when the values of and are substituted into the inequalit. E XAMPLE Standardized Test Practice Which ordered pair is a solution of 3 4 > 8? A (6, 3) B (0, ) C (, ) D (3, 5) Solution Ordered Pair Substitute Conclusion (6, 3) 3(6) 4(3) 5 6 >/ 8 (6, 3) is not a solution. (0, ) 3(0) 4() 5 8 >/ 8 (0, ) is not a solution. (, ) 3() 4() 5 0 >/ 8 (, ) is not a solution. (3, 5) 3(3) 4(5) 5 > 8 (3, 5) is a solution. c The correct answer is D. A B C D GUIDED PRACTICE for Eample Tell whether the given ordered pair is a solution of (0, 4). (, ) 3. (3, 8) 4. (, 7) INTERPRET GRAPHS A dashed boundar line means that points on the line are not solutions. A solid boundar line means that points on the line are solutions. GRAPHING INEQUALITIES The graph of a linear inequalit in two variables is the set of all points in a coordinate plane that represent solutions of the inequalit. All solutions of 3 > lie on one side of the boundar line > The boundar line divides the plane into two half-planes. The shaded half-plane is the graph of 3 >. 3 Chapter Linear Equations and Functions

2 KEY CONCEPT For Your Notebook Graphing a Linear Inequalit To graph a linear inequalit in two variables, follow these steps: STEP STEP Graph the boundar line for the inequalit. Use a dashed line for < or > and a solid line for or. Test a point not on the boundar line to determine whether it is a solution of the inequalit. If it is a solution, shade the half-plane containing the point. If it is not a solution, shade the other half-plane. E XAMPLE Graph linear inequalities with one variable Graph (a) 3 and (b) < in a coordinate plane. a. Graph the boundar line 5 3. b. Graph the boundar line 5. Use a solid line because the Use a dashed line because the inequalit smbol is. inequalit smbol is <. Test the point (0, 0). Because Test the point (0, 0). Because (0, 0) is not a solution of the (0, 0) is a solution of the inequalit, shade the half-plane inequalit, shade the half-plane that does not contain (0, 0). that contains (0, 0). (0, 0) 3 3 < (0, 0) 3 E XAMPLE 3 Graph linear inequalities with two variables Graph (a) > and (b) 5 4 in a coordinate plane. AVOID ERRORS It is often convenient to use (0, 0) as a test point. However, if (0, 0) lies on the boundar line, ou must choose a different test point. a. Graph the boundar line 5. b. Graph the boundar line Use a dashed line because the Use a solid line inequalit smbol is >. because the inequalit smbol is. Test the point (, ). Because Test the point (0, 0). Because (, ) is a solution of the (0, 0) is not a solution of the inequalit, shade the half-plane inequalit, shade the half-plane that contains (, ). that does not contain (0, 0). 5 4 (, ) (0, 0) 3 > 3 at classzone.com.8 Graph Linear Inequalities in Two Variables 33

3 GUIDED PRACTICE for Eamples and 3 Graph the inequalit in a coordinate plane. 5. > < < > E XAMPLE 4 Solve a multi-step problem MOVIE RECORDING A film class is recording a DVD of student-made short films. Each student group is allotted up to 300 megabtes (MB) of video space. The films are encoded on the DVD at two different rates: a standard rate of 0.4 MB/sec for normal scenes and a high-qualit rate of. MB/sec for comple scenes. Write an inequalit describing the possible amounts of time available for standard and high-qualit video. Graph the inequalit. Identif three possible solutions of the inequalit. Solution STEP Write an inequalit. First write a verbal model. Standard rate (MB/sec) p Standard time (sec) High-qualit rate (MB/sec) p High-qualit time (sec) Total space (MB) 0.4 p. p 300 An inequalit is STEP STEP 3 Graph the inequalit. First graph the boundar line Use a solid line because the inequalit smbol is. Test the point (0, 0). Because (0, 0) is a solution of the inequalit, shade the half-plane that contains (0, 0). Because and cannot be negative, shade onl points in the first quadrant. (50, 00) 00 (300, 0) (600, 5) Standard (sec) Identif solutions. Three solutions are given below and on the graph. (50, 00) 50 seconds of standard and 00 seconds of high qualit (300, 0) 300 seconds of standard and 0 seconds of high qualit (600, 5) 600 seconds of standard and 5 seconds of high qualit For the first solution, 0.4(50).(00) 5 300, so all of the available space is used. For the other two solutions, not all of the space is used. High qualit (sec) Chapter Linear Equations and Functions

4 ABSOLUTE VALUE INEQUALITIES Graphing an absolute value inequalit is similar to graphing a linear inequalit, but the boundar is an absolute value graph. E XAMPLE 5 Graph an absolute value inequalit Graph > 3 4 in a coordinate plane. Solution STEP STEP Graph the equation of the boundar, Use a dashed line because the inequalit smbol is >. Test the point (0, 0). Because (0, 0) is a solution of the inequalit, shade the portion of the coordinate plane outside the absolute value graph. (0, 0) > z 3 z 4 GUIDED PRACTICE for Eamples 4 and 5. WHAT IF? Repeat the steps of Eample 4 if each student group is allotted up to 40 MB of video space. Graph the inequalit in a coordinate plane < EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5, 5, and 45 5 STANDARDIZED TEST PRACTICE Es.,, 8, 39, 40, 4, 46, and 48. VOCABULARY Cop and complete: The graph of a linear inequalit in two variables is a(n)?.. WRITING Compare the graph of a linear inequalit in two variables with the graph of a linear equation in two variables. EXAMPLE on p. 3 for Es. 3 6 EXAMPLES and 3 on p. 33 for Es. 7 0 CHECKING SOLUTIONS Tell whether the given ordered pairs are solutions of the inequalit. 3. > 7; (0, 0), (8, 5) 4. 5; (3, ), (, ) 5. 4; (0, 4), (, 8) 6. < 3; (0, 0), (, ) GRAPHING INEQUALITIES Graph the inequalit in a coordinate plane. 7. < > < > } } 3 5. < > Graph Linear Inequalities in Two Variables 35

5 ERROR ANALYSIS Describe and correct the error in graphing the inequalit. 9. < MULTIPLE CHOICE Which ordered pair is not a solution of 3 5 < 30? A (0, 0) B (, 7) C (, 7) D (5, 5) EXAMPLE 5 on p. 35 for Es. 8 ABSOLUTE VALUE INEQUALITIES Graph the inequalit in a coordinate plane.. > 3. < 5 4. > } 6. < MULTIPLE CHOICE The graph of which inequalit is shown? A 3 B 3 C > 3 D 3 CHECKING SOLUTIONS Tell whether the given ordered pairs are solutions of the inequalit. 9. } 3 } ; (6, 8), (3, 3) <.6; (0.5, ), (3.8, 0) > ; (0.5, ), (3,.5) 3. } 4 > ; 4 } 3, 0, } 3, 4 GRAPHING INEQUALITIES Graph the inequalit in a coordinate plane < > > } 3 } > 37. } 5 3 } OPEN-ENDED MATH Write a linear inequalit in two variables that has (, 3) and (, 6) as solutions, but does not have (4, 0) as a solution. 40. WRITING Eplain wh it is not helpful when graphing a linear inequalit in two variables to choose a test point that lies on the boundar line. 4. SHORT RESPONSE Write an inequalit for the graph shown. Eplain how ou came up with the inequalit. Then describe a real-life situation that the first-quadrant portion of the graph could represent. 4. CHALLENGE Write an absolute value inequalit that has eactl one solution in common with 3 5. The common solution should not be the verte (3, 5) of the boundar. Eplain how ou found our inequalit. 5 WORKED-OUT SOLUTIONS 36 Chapter Linear on p. Equations WS and Functions 5 STANDARDIZED TEST PRACTICE

6 PROBLEM SOLVING EXAMPLE 4 on p. 34 for Es CALLING CARDS You have a $0 phone card. Calls made using the card cost $.03 per minute to destinations within the United States and $.06 per minute to destinations in Brazil. Write an inequalit describing the numbers of minutes ou can use for calls to U.S. destinations and to Brazil. 44. RESTAURANT MANAGEMENT A pizza shop has 300 pounds (4800 ounces) of dough. A small pizza uses ounces of dough and a large pizza uses 8 ounces of dough. Write and graph an inequalit describing the possible numbers of small and large pizzas that can be made. Then give three possible solutions. 45. CRAFTS Cotton lace costs $.50 per ard and linen lace costs $.50 per ard. You plan to order at most $75 of lace for crafts. Write and graph an inequalit describing how much of each tpe of lace ou can order. If ou bu 4 ards of cotton lace, what are the amounts of linen lace ou can bu? 46. SHORT RESPONSE You sell T-shirts for $5 each and caps for $0 each. Write and graph an inequalit describing how man shirts and caps ou must sell to eceed $800 in sales. Eplain how ou can modif this inequalit to describe how man shirts and caps ou must sell to eceed $600 in profit if ou make a 40% profit on shirts and a 30% profit on caps. 47. MULTI-STEP PROBLEM On a two week vacation, ou and our brother can rent one canoe for $ per da or rent two mountain bikes for $3 each per da. Together, ou have $0 to spend. a. Write and graph an inequalit describing the possible numbers of das ou and our brother can canoe or biccle together. b. Give three possible solutions of the inequalit from part (a). c. You decide that on one da ou will canoe alone and our brother will biccle alone. Repeat parts (a) and (b) using this new condition. 48. EXTENDED RESPONSE While camping, ou and a friend filter river water into two clindrical containers with the radii and heights shown. You then use these containers to fill the water cooler shown. a. Find the volumes of the containers and the cooler in cubic inches. b. Using our results from part (a), write and graph an inequalit describing how man times the containers can be filled and emptied into the water cooler without the cooler overflowing. c. Convert the volumes from part (a) to gallons ( in. 3 ø gal). Then rewrite the inequalit from part (b) in terms of these converted volumes. d. Graph the inequalit from part (c). Compare the graph with our graph from part (b), and eplain wh the results make sense..8 Graph Linear Inequalities in Two Variables 37

7 49. CHALLENGE A widescreen television image has a width w and a height h that satisf the inequalit w } h > 4 } 3. a. Does the television screen shown at the right meet the requirements of a widescreen image? b. Let d be the length of a diagonal of a television image. Write an inequalit describing the possible values of d and h for a widescreen image. MIXED REVIEW Look for a pattern in the table. Then write an equation that represents the table. (p. 34) PREVIEW Prepare for Lesson 3. in Es Graph the equation. (p. 89) Write an equation of the line that satisfies the given conditions. (p. 98) 58. m 5 } 4, passes through (0, ) m 5 3, passes through (3, 7) 60. passes through (0, ) and (5, 8) 6. passes through (4, ) and (7, 6) QUIZ for Lessons.7.8 Graph the function. Compare the graph with the graph of 5. (p. 3) f() 5 } 5 Write an equation of the graph. (p. 3) Graph the inequalit in a coordinate plane. (p. 3) 7. > MINI-CARS You have a 0 credit gift pass to a mini-car racewa. It takes credits to drive the cars on the Rall track and 3 credits to drive the cars on the Grand Pri track. Write and graph an inequalit describing how man times ou can race on the two tracks using our gift pass. Then give three possible solutions. (p. 3) 38 EXTRA PRACTICE for Lesson.8, p. 0 ONLINE QUIZ at classzone.com