Content Standards Two-Variable Inequalities

Transcription

1 -8 Content Standards Two-Variable Inequalities A.CED. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate aes with labels and scales. Also F.IF.7.b bjective To graph two-variable inequalities You have a gift card for a store that sells pre-owned CDs and paperback books. You want to spend as much of the gift card as possible. How man of each item can ou bu? Eplain. The words as much c as possible are important here. MATHEMATICAL PRACTICES D IC AC ES AM YN TIVITI D Dnamic Activit Linear Inequalities In some situations ou need to compare quantities. You can use inequalities for situations that involve these relationships: less than, less than or equal to, greater than, and greater than or equal to. Essential Understanding Graphing an inequalit in two variables is similar to graphing a line. The graph of a linear inequalit contains all points on one side of the line and ma or ma not include the points on the line. LLesson Vocabular V llinear inequalit boundar half-plane test point A linear inequalit is an inequalit in two variables whose graph is a region of the coordinate plane bounded b a line. This line is the boundar of the graph. The boundar separates the coordinate plane into two half-planes, one of which consists of solutions of the inequalit. Chapter _hsmase_cc_8.indd boundar half-plane Functions, Equations, and Graphs 3/8/ :: PM

2 To determine which half-plane to shade, pick a test point that is not on the boundar. Check whether that point satisfies the inequalit. If it does, shade the half-plane that includes the test point. If not, shade the other half-plane. The origin, (, ), is usuall an eas test point as long as it is not on the boundar. Problem Graphing Linear Inequalities What is the graph of each inequalit? A S 3 Step Step Graph the boundar line 5 3. Choose a test point, (, ). Substitute 5 Use a dashed boundar line because the and 5 into. 3. inequalit is greater than, and the points. 3() on the line do not satisf the inequalit.. Since. is true, shade the half plane that includes (, ). (, ) B K 3 Can ou use the graph of S 3 to help graph K 3? If ou shaded above the line for. 3, then shade below the line for # 3. The boundar line is again 5 3, but it is solid because the inequalit is less than or equal to. Shade the region opposite the region shaded above (for.) because the inequalit is less than or equal to. You can also check the point (, ). # 3() # Since # is false, (, ) is not part of the solution. (, ) Got It?. What is the graph of each inequalit? a. $ b., You can also inspect inequalities solved for, such as. m b to determine which half-plane describes the solution. Since describes vertical position, the solution of. m b will be above the boundar line. The solution of, m b will be below the boundar line. Lesson -8 Two-Variable Inequalities 5

3 Problem Using a Linear Inequalit Entertainment The map shows the number of tickets needed for small or large rides at the fair. You do not want to spend more than $5 on tickets. How man small or large rides can ou ride? You can bu 6 tickets with $5. What are the unknowns? The unknowns are the number of small rides and the number of large rides ou can get on. Relate the number of tickets for small rides plus the number of tickets for large rides Define Let 5 the number of small rides. Let 5 the number of large rides. Write 3 5 is less than or equal to 6 # 6 Step Find the intercepts of the boundar line. Use the intercepts to graph the boundar line. When 5, 3 5() 5 6. When 5, 3() Graph the line that connects the intercepts (, ) and (, ). Since the inequalit is #, use a solid boundar line. Step Number of Large Rides The region above the boundar line represents combinations of rides that require more than 6 tickets. You purchased a finite number of tickets, 6, so ou will not be able to go on an infinite number of rides. Shade the region below the boundar line. The number of small rides and the number of large rides are whole numbers. In math, such a situation is called discrete. All points with whole number coordinates in the shaded region represent possible combinations of small and large rides. 6 (, ) 8 Rides at the Fair (, ) 8 6 Number of Small Rides Got It?. a. Suppose that ou decide to spend no more than $3 for tickets. What are the possible combinations of small and large rides that ou can ride now? Use a graph to find our answer. b. Reasoning Wh did the graph of the solution in Problem onl include Quadrant I? 6 Chapter _hsmase_cc_8.indd 6 Functions, Equations, and Graphs 3/8/ :5:8 PM

4 You can graph two-variable absolute value inequalities in the same wa that ou graph linear inequalities. Problem 3 Graphing an Absolute Value Inequalit What is the graph of R»? Absolute value inequalit Boundar Solve the inequalit for. Graph the related equation. Shade the solution., u u, u u Subtract from each side..u u Multipl both sides b. The graph of 5u u is the graph of 5 u u, reflected in the -ais and translated left units and up unit. Since the inequalit is solved for and.u u, shade the region above the boundar. Got It? 3. What is the graph of $ u u? You can use the transformations discussed in previous lessons to help draw the boundar graphs more quickl. You can also use them to write an inequalit based on a graph. Problem Writing an Inequalit Based on a Graph How can ou tell that the graph is not a stretch or compression of the graph of 5»? The slopes of the branches are and. What inequalit does this graph represent? The boundar is the graph of the absolute value function 5 u u, translated. The verte of 5 u u is translated to (3, ), so the boundar is the graph of 5 u 3 u. The solution is shaded above the boundar, so the inequalit is either. or $. Since the boundar is a dashed line, the correct inequalit is. u 3 u. Got It?. a. What inequalit does this graph represent? b. Reasoning You can tell from looking at the inequalit. 5 3 to shade above the boundar line to represent the solution. Can ou use the same technique to show the solution of an inequalit like.? Eplain. 6 Lesson -8 Two-Variable Inequalities 7

5 Lesson Check Do ou know HW? What is the graph of each inequalit?. 9 #. 7 $ 8 What is the graph of each absolute value inequalit? 3. # u u. $ u 3 u Do ou UNDERSTAND? MATHEMATICAL PRACTICES 5. Do the points on the boundar line of the graph of an inequalit help determine the shaded area of the graph? Eplain. 6. Compare and Contrast How is graphing a linear inequalit in two variables different from graphing a linear equation in two variables? 7. Reasoning Is the ordered pair Q 3, R a solution of 3. 3? Eplain. Practice and Problem-Solving Eercises MATHEMATICAL PRACTICES A Practice Graph each inequalit. See Problem , 3. #. # 5. 3 $ 3. $ 6. 3 # 9 5., 6. 5 $ 7. Cooking Th e time needed to roast a chicken depends on its weight. Allow at least min/lb for a chicken weighing as much as 6 lb. Allow at least 5 min/lb for a chicken weighing more than 6 lb. a. Write two inequalities to represent the time needed to roast a chicken. b. Graph the inequalities. See Problem. Graph each absolute value inequalit. 8. $ u u 9. # u 3 u. # u u.. u u. 7. u u 3. # u u. 3 $u u 5., u 3 u 6. 3 # u 3 u Write an inequalit for each graph. The equation for the boundar line is given u 6 u See Problem 3. See Problem Chapter Functions, Equations, and Graphs

6 B Appl Graph each inequalit on a coordinate plane $ 3. 5, ( ) # 6 3. u u u u # # $ 38. Think About a Plan The graph at the right relates the number of hours ou spend on the phone to the number of hours ou spend studing per week. Describe the domain for this situation. Write an inequalit for the graph. What is the least amount of time ou can spend on the phone per week? What is the most? What is the least amount of time ou can spend studing per week? What is the most? What is the greatest amount of time ou can spend either on the phone or studing per week? Write an inequalit for each graph Time Studing (hours) 3 Time per Week Time on phone (hours) Which graph best represents the solution of the inequalit $ u u? 6. The graph at the right relates the amount of gas in the tank of our car to the distance ou can drive. a. Describe the domain for this situation. b. Wh does the graph stop? c. Wh is onl the first quadrant shown? d. Reasoning Would ever point in the solution region be a solution? e. Write an inequalit for the graph. f. What does the coefficient of represent? Distance (mi) 3 Miles to Travel 8 6 Gas in tank (gallons) Lesson -8 Two-Variable Inequalities 9

7 C Challenge 7. Writing When ou graph an inequalit, ou can often use the point (, ) to test which side of the boundar line to shade. Describe a situation in which ou could not use (, ) as a test point. Graphing Calculator Graph each inequalit on a graphing calculator. Then sketch the graph. 8. # u u u u 9.. u u u 3 u 5., u 3 u u 3 u 5., 7 u u u u Standardized Test Prep SAT/ACT 5. Suppose varies directl with. If is 3 when is, what is when is 9? Which equation represents a line with slope and -intercept 3? What is the verte of 5 u u 5? (5, ) (5, ) (, 5) (, 5) Etended Response 55. The amount of a commission is directl proportional to the amount of a sale. A realtor received a commission of $8, on the sale of an $8, house. How much would the commission be on a $65, house? Mied Review Graph each function b translating its parent function. See Lesson u 5u u u f () 5 u 6 u 59. f () 5 u u 6. 5 u u 6. 5 u u 5 Determine whether varies directl with. If so, find the constant of variation. See Lesson Make a scatter plot and describe the correlation. 7. 5(, 6), (, ), (, ), (, ), (5, )6 7. 5(, 5), (5, 5), (, ), (, 3), (5, )6 Get Read! To prepare for Lesson 3-, do Eercises 7 7. Graph each equation. Use one coordinate plane for all three graphs See Lesson -5. See Lesson -3. Chapter Functions, Equations, and Graphs