Solving Systems of Linear Inequalities

Transcription

1 Locker LESSON 5. Solving Sstems of Linear Inequalities Teas Math Standards The student is epected to: A.3.F Solve sstems of two or more linear inequalities in two variables. Also A.3.E, A.3.G Mathematical Processes A.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including smbols, diagrams, graphs, and language as appropriate. Language Objective.C.,.D.1,.E.3,.I. Eplain to a partner what a sstem of linear inequalities in two variables is and how this sstem s solutions differ from those of a sstem of linear equations. ENGAGE Essential Question: How can ou represent the solution of a sstem of two or more linear inequalities? Graph the solutions of each individual inequalit; then find the region where all solution areas overlap. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how to use a sstem of linear inequalities to represent how much a person can spend on presents. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing Compan Name Class Date 5. Solving Sstems of Linear Inequalities Essential Question: How can ou represent the solutions of a sstem of two or more linear inequalities? Eplore A.3.F Solve sstems of two or more linear inequalities in two variables. Also A.3.E, A.3.G Turning a Sstem of Equations into a Sstem of Inequalities What does the graph of a sstem of inequalities = + look like? Start with a sstem of equations. = Graph the lines on the coordinate plane. How man regions do the two lines divide the graph into? Consider what it means to replace the equals sign with an inequalit. Replace the top equation b the inequalit +. How is this different? It means that for an -value, the solution includes not onl the corresponding point on the line = +, but also ever point in the coordinate plane with the same -value that has a greater -value than the corresponding point on the line. Indicate this on the graph b lightl shading the two regions above the line represented b = +. Answer shown on graph in (A). Now replace the second equation with the inequalit < How is this different? It means that for an -value, the solution includes ever point in the coordinate plane that has a lesser -value than does the point on the line with the same -value. Indicate this b lightl shading the two regions below the line. - Resource Locker However, because this inequalit is non-inclusive, the solution will not contain the points on the line itself. Indicate this b converting the solid line to a dashed line. Answer shown on graph in (A). How man regions of the graph were shaded in both steps? Pick a point that is in the darkest region of the graph and check that it agrees with both inequalities. - + True or False? < True of False? True True Module 5 9 Lesson Name Class Date 5. Solving Sstems of Linear Inequalities Essential Question: How can ou represent the solutions of a sstem of two or more linear inequalities? Houghton Mifflin Harcourt Publishing Compan A.3.F Solve sstems of two or more linear inequalities in two variables. Also A.3.E, A.3.G Eplore Turning a Sstem of Equations into a Sstem of Inequalities What does the graph of a sstem of inequalities look like? Start with a sstem of equations. = + Graph the lines on the coordinate plane. = How man regions do the two lines divide the graph into? Consider what it means to replace the equals sign with an inequalit. Replace the top equation b the inequalit +. How is this different? It means that for an -value, the solution includes not onl the corresponding point on the line = +, but also ever point in the coordinate plane with the same -value that has a greater -value than the corresponding point on the line. Indicate this on the graph b lightl shading the two regions above the line represented b = +. Answer shown on graph in (A). Now replace the second equation with the inequalit < Four Resource How is this different? It means that for an -value, the solution includes ever point in the coordinate plane that has a lesser -value than does the point on the line with the same -value. Indicate this b lightl shading the two regions below the line. However, because this inequalit is non-inclusive, the solution will not contain the points on the line itself. Indicate this b converting the solid line to a dashed line. How man regions of the graph were shaded in both steps? One Pick a point that is in the darkest region of the graph and check that it agrees with both inequalities. + True or False? < True of False? - - True True Four Answer shown on graph in (A). One Module 5 9 Lesson HARDCOVER PAGES 05 1 Turn to these pages to find this lesson in the hardcover student edition. 9 Lesson 5.

2 Reflect 1. Discussion Wh did a dashed line replace the solid line in onl the second inequalit? The second inequalit is non-inclusive: points on the line are NOT solutions to the second inequalit (but the were to the original equation) while the first inequalit is inclusive and points on the line are solutions to it. EXPLORE Turning a Sstem of Equations into a Sstem of Inequalities Eplain 1 Graphing a Sstem of Linear Inequalities in Two Variables Graphing a sstem of linear inequalities is similar to graphing a single inequalit, but ever point in the solution region must make all the inequalities in the sstem true. The boundar line of each inequalit in the sstem is the graph of the related equation for the inequalit, using a solid line if the inequalit is inclusive ( or ) or a dashed line if the inequalit is eclusive (> or <). Eample 1 Graph the sstem of inequalities on the grid. Give the coordinates of two points in the solution set. - 1_ + < Graph the boundar line = Since the inequalit is inclusive, the line should be solid. Shade the half-plane that represents the solutions to the inequalit (above and to the right of the line). Graph the boundar line =. Since the inequalit is eclusive, the line should be dashed. Shade the half-plane that represents the solutions to the inequalit < (below and to the right of the line). The region that is in both half-planes will be shaded the darkest. This represents the solutions of the sstem. The regions that were shaded onl once are useful to help find the solution but the do not represent valid solutions of the sstem; the are regions that represent solutions to onl one of the two inequalities Points: (, ) and (, ) are in the solution set. - Module 5 90 Lesson PROFESSIONAL DEVELOPMENT Learning Progressions Students have alread learned to graph linear equations, linear inequalities, and sstems of linear equations. This lesson epands upon their abilit to graph linear sstems. Eventuall, students will use graphs of multiple linear inequalities as constraints in linear programming optimization problems. - Houghton Mifflin Harcourt Publishing Compan INTEGRATE TECHNOLOGY Students can graph a sstem of equations on their graphing calculators and then decide which area should be shaded for a corresponding sstem of inequalities. AVOID COMMON ERRORS Students ma revert to using all solid lines when graphing boundar lines in sstems of inequalities. Remind students that the line representing the boundar of an inequalit with a < or > sign is dotted, while the line representing the boundar of an inequalit with a or sign is solid. EXPLAIN 1 Graphing a Sstem of Linear Inequalities in Two Variables INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning Shade below the line for an inequalit with a < or sign. Shade above the line for an inequalit with a > or sign. Solving Sstems of Linear Inequalities 90

3 QUESTIONING STRATEGIES How can ou find points in the solution set? Pick an points that lie within the dark-shaded region where the lightl-shaded regions overlap. B > 3 + < Graph a [solid/dashed] boundar line for = 3. The half-plane that represents the solutions to the inequalit > 3 lies [above/below/left of/right of] the line. Graph a [solid/dashed] boundar line for = - +. The half-plane that represents the solutions to the inequalit + < lies [above/below] the line. Shade each half-plane. The region that is in both half-planes will be double-shaded. Sample Points: (, -), (6, -6) ` Your Turn Graph the sstem of inequalities on the grid. Give the coordinates of two points in the solution set. + < < < 1 Houghton Mifflin Harcourt Publishing Compan Possible points: (0, 0), (-, ) Possible points: (0, 0), (, -) Module 5 91 Lesson COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs. Have one student in each pair write a sstem of linear inequalities, and have the second student graph the solution. Then, have students change roles and repeat the process. 91 Lesson 5.

4 Eplain Eample Solving a Real-World Problem Use the four-step problem solving method (Analze, Formulate a Plan, Solve, Justif and Evaluate) to solve the problem. Sand makes $ profit on ever cup of juice that she sells and $1 on ever fruit bar that she sells. She wants to sell at least 5 fruit bars per da and at least 5 cups of juice per da. She wants to earn at least $5 per da. Show and describe all the possible combinations of juice and fruit bars that Sand needs to sell to meet her goals, and pick two possible combinations that meet her goals. Analze Information Identif the important information Profit on juice is $ per cup. Profit on fruit bars is $1 each. Profit should be $5 or more. She wants to sell at least 5 fruit bars. She wants to sell at least 5 cups of juice. EXPLAIN Solving a Real-World Problem AVOID COMMON ERRORS Students ma etend real world solutions into more than one quadrant of the graph. Remind students that most real-world situations require that 0 and 0, so the solution set eists onl in the first quadrant. Formulate a Plan You want to find an equation relating profit to number of fruit bars and number of cups of juice sold. Then convert Sand s sales and profit goals into inequalities and graph them to see where the solutions are. Solve Let c represent the number of juice cups sold, and let b represent the number of fruit bars sold. Profit (p) is given b the equation p = c + b. The inequalities that represent Sand s dail sales and profit goals are given b c 5 Sell at least 5 cups of juice. b 5 Sell at least 5 fruit bars. c + b 5 Earn at least $5 profit. c Cups of Juice Houghton Mifflin Harcourt Publishing Compan b Fruit Bars Module 5 9 Lesson DIFFERENTIATE INSTRUCTION Visual Cues Show students that the solution set for a sstem of inequalities consists of the points in one of the four regions into which the boundar lines divide the coordinate plane. Illustrate that the boundar ras for a given region are either included in the solution set or not, depending on whether the inequalities include or eclude equalit. Solving Sstems of Linear Inequalities 9

5 QUESTIONING STRATEGIES How do ou know whether to use a <,, >, or when writing a sstem of linear inequalities to model a real-world problem? Pa attention to certain words and phrases. At most and no more than indicate that ou should use. At least and no less than indicate that ou should use. Justif and Evaluate This solution seems reasonable because the solution region includes areas of large sales and profits increase with increasing sales. Sand can meet her sales goal if she sells at the points (, 9) or (6, 1), corresponding to $ from sales of fruit bars and $1 from sales of juice, or $6 from sales of fruit bars and $ from sales of juice. This tpe of solution is called an unbounded solution. The solution region is not contained in a finite area on the graph. Vance wants to fence in a rectangular area for his dog. He wants the length of the rectangle to be at least 30 feet and the perimeter to be no more than 150 feet. Graph all possible dimensions of the rectangle. Analze Identif the important information Rectangular area. The length is at least 30 feet. The perimeter is at most 150 feet. Formulate a Plan You are looking for a solution to the dimensions of the rectangle (width and length). To find the limits on width ou will need to relate width and length to perimeter with an equation. perimeter = length + width Solve Using l and w for length and width, write the inequalities that represent Vance s requirements for the fence: l 30 w 10 l + w Houghton Mifflin Harcourt Publishing Compan Width (feet) Length (feet) l Module 5 93 Lesson LANGUAGE SUPPORT Communicate Math Have students work in pairs, using cards with the equations = and = + 3 on them. Give each pair a card with,,, and on it. Have one student circle an two smbols and have the other student replace the = signs with the smbols, then describe the resulting graph. Then have the other student switch the placement of the inequalit smbols and describe that graph. Pairs can compare their results with others who chose different smbols to see all of the possible solutions. 93 Lesson 5.

6 Justif and Evaluate Check two points to see if the solution makes sense. (l, w) = (50, 0) or (0, 30) are both in the solution region. Reflect. Think about the difference between a real-world quantit like an amount of flour and one like the balance of a bank account. There is a boundar on cups of flour that is not stated eplicitl but does limit the choice of real numbers that can represent cups of flour. How is this different from an account balance? You cannot have negative cups of flour, but an account ma be overdrawn and thus have negative dollars. 5. In the first eample, think of a statement that would make the solution bounded or limit how much profit Sand could make. Possible answer: Sand can onl purchase 50 items (fruit bars or cups of juice) per da or less. Your Turn 6. Olivia is painting a logo for a billboard and wants to use a combination of blue paint and red paint to completel cover a 5000 square foot billboard. A gallon of paint can cover 500 square feet. Red paint costs $0 per gallon and blue paint costs $30 per gallon. She onl has $50 to spend. Make a graph showing all the possible combinations of paint that meet her goals. 1 b Blue Paint (gallons) 0r + 30b r + 500b r b b _-500r _ b _ 5_ 3r + b -r Red Paint (gallons) r Houghton Mifflin Harcourt Publishing Compan Module 5 9 Lesson Solving Sstems of Linear Inequalities 9

7 ELABORATE INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning The area that represents the solution set can be identified b choosing a point in each of the four areas formed b the intersecting lines and substituting it into both inequalities. SUMMARIZE THE LESSON How do ou graph the solution of a sstem of linear inequalities? Graph the corresponding sstem of linear equations, using dotted lines for inequalities with < or > and solid lines for inequalities with or. Shade the regions indicated b the inequalit signs. The solution set, if one eists, is the region where the shaded regions intersect. 7. Dustin decides to tr selling whole-wheat muffins and bagels at his farm stand. He has $60 to spend on ingredients and eight hours to prepare the muffins and bagels that he will sell on opening da. One dozen muffins costs $10 to make and takes 1 hour, while one dozen bagels costs $15 to make and takes hours. Make a graph showing all the possible combinations of food that meet his goals. Muffins (dozen) m b Bagels (dozen) m + -m_ b _ 10m 60_ + 15b _ 60 b + b 15-10m -m 15 b + b -m + 3 Elaborate. When graphing inequalities, how can ou check that ou shaded the correct areas? Pick a point in the shaded area and check it in the inequalit. Houghton Mifflin Harcourt Publishing Compan 9. Will the point where the two lines intersect alwas be a solution? Wh or wh not? No; if at least one of the inequalities is not inclusive than the point will not be a solution. It will onl be a solution of both inequalities are inclusive. 10. Essential Question Check-In What is the difference between the wa a less than inequalit is graphed and the wa a less than or equal to inequalit is graphed? Graph the boundar of each inequalit as a dashed line for eclusive inequalities (< or >) and a solid line for inclusive inequalities ( or ). Module 5 95 Lesson 95 Lesson 5.

8 Evaluate: Homework and Practice Graph the sstem of inequalities on the grid. Give the coordinates of two points in the solution set. 1. 1_ < - < - + Online Homework Hints and Help Etra Practice EVALUATE Possible points: (0, 0), (-, ) Possible points: (0, -), (1, -) 3. - > > 0 ASSIGNMENT GUIDE Concepts and Skills Eplore Turning a Sstem of Equations into a Sstem of Inequalities Eample 1 Graphing a Sstem of Inequalities in Two Variables Eample Solving a Real-World Problem Practice Eercises 1 Eercises 5 16 Eercises Possible points: (, -), (, -) Possible points: (, -), (6, -6) > Houghton Mifflin Harcourt Publishing Compan INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning In graphing an inequalit involving a vertical line, the area to the left or to the right of the line is shaded. For inequalities in the forms < a and a, the region to the left is shaded. For inequalities in the forms > a and a, the region to the right is shaded. - Possible points: (0, 0), (1, 1) - Possible points: (0, 0), (1, 1) Module 5 96 Lesson Eercise Depth of Knowledge (D.O.K.) Mathematical Processes Recall of Information 1.E Create and use representations 17 1 Skills/Concepts 1.A Everda life Skills/Concepts 1.D Multiple representations 3 3 Strategic Thinking 1.B Problem solving model 5 3 Strategic Thinking 1.D Multiple representations Solving Sstems of Linear Inequalities 96

9 AVOID COMMON ERRORS Students ma not recall the rules for multipling and dividing inequalities b negative numbers. Remind students that when the multipl or divide both sides an inequalit b a negative number, the direction of the inequalit sign changes < > Possible points: (, 0), (, ) Possible points: (0, 0), (-1, -1) 9. 1_ - 1_ > > No Solution No Solution 11. > 3 1. < <1 Houghton Mifflin Harcourt Publishing Compan + < Possible points: (, 0), (, 1) Possible points: (, ), (0, -) Module 5 97 Lesson 97 Lesson 5.

10 Write the sstem of inequalities shown b each graph CONNECT VOCABULARY Compare and contrast a sstem of linear inequalities in two variables with a sstem of three linear equations in three variables and the linear-quadratic sstems discussed in the previous lessons. Have students complete a chart with the similarities and differences among these three kinds of sstems. > - - 1_ _ < - 1_ _ > - < A surf shop makes profits of $150 for each surfboard and $100 for each wakeboard. The owner sells at least 3 surfboards and at least 6 wakeboards per month. The shop owner wants to earn at least $000 per month. Graph all possible combinations of surfboard and wakeboard sales that would satisf the store owner s earnings goal. Use a check point to justif the reasonableness of the solution. s 3 w s + 100w 000 s 0 w -_ 150s _ Surfboards w - 3_ s + 0 Check point (10, 10) : 150 (10) (10) = 500, so the owner sells at least 3 surfboards and 6 wakeboards and earns more than $000. Wakeboards w Houghton Mifflin Harcourt Publishing Compan Module 5 9 Lesson Solving Sstems of Linear Inequalities 9

11 1. Alice is serving pepper jack cheese and cheddar cheese on a platter. She wants to have more than pounds of each. Pepper jack cheese costs $ per pound and cheddar cheese costs $ per pound. Alice wants to spend at most $0 on cheese. Graph all possibile combinations of the two cheeses Alice could bu. Use a check point to justif the reasonableness of the solution. Cheddar Cheese (lb) 6 c 0 6 Pepper Jack Cheese (lb) p p > c > p + c 0 c - p_ 0_ + c -p + 10 Check point (3, 3) : (3) + (3) = 1, so Alice has at least pounds of each cheese and spends less than $ In one week, Ed can mow at most 9 lawns and rake at most 7 lawns. He charges $0 for mowing and $10 for raking. He needs to earn more than $10 in one week. Graph all the possible combinations of mowing and raking that Ed can do to meet his goal. Use a check point to justif the reasonableness of the solution. Raking Jobs 16 1 r m Mowing Jobs 0m + _ 10r > 10 m 9 r > - 0m _ r 7 r > -m + 1 Check point (6, 6) : 0 (6) + 10 (6) = 10, so Ed mows fewer than 9 lawns, rakes fewer than 7 lawns, and earns more than $10. Houghton Mifflin Harcourt Publishing Compan 0. Linda works at a pharmac for $15 an hour. She also bab-sits for $10 an hour. Linda needs to earn more than $90 per week, but she does not want to work more than 0 hours per week. Graph the number of hours Linda could work at each job to meet her goals. Use a check point to justif the reasonableness of the solution. Hours Babsitting 1 1 b 6 p Hours at Pharmac 15p + 10b > 90 15p p + b 0 b > -_ 90_ b - p + 0 b > - 3p_ + 9 Check point (9, 6) : 15 (9) + 10 (6) = 195, so Linda works no more than 0 hours and earns more than $90. Module 5 99 Lesson 99 Lesson 5.

12 1. Ton wants to plant at least 0 acres of corn and at least 50 acres of sobeans. He has 00 acres on which to plant. Graph all the possible combinations of the number of acres of corn and of sobeans Ton could plant. Use a check point to justif the reasonableness of the solution. Sobeans (acres) s Corn (acres). Match each set of inequalities with the correct graph. < c c 0 s 50 s + c 00 s -c + 00 Check point (0, 0) : = 160, so Ton can plant at least 0 acres of corn, at least 50 acres of sobeans, and less than 00 total acres. C. B. > INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling Students graph the solutions of linear sstems b graphing each linear inequalit and shading the overlapping region. Shading the individual inequalities is helpful but not essential, since none of the region outside the overlap areas is a solution. The solution of two inequalities whose equations form vertical angles could be shown as the V-shaped region and its boundar onl. A. B C. > D. A. > D Houghton Mifflin Harcourt Publishing Compan Module Lesson Solving Sstems of Linear Inequalities 300

13 JOURNAL Have students write a sstem of linear inequalities in two variables and graph the solution. H.O.T. Focus on Higher Order Thinking 3. Eplain the Error Two students wrote a sstem of linear inequalities to describe the graph. Which student is incorrect? Eplain the error Student A < 1_ Student B 1_ + < -3-6 Student B is incorrect. The signs of the inequalities are switched.. Critical Thinking Can the solutions of a sstem of linear inequalities be the points on a line? Give an eample or eplain wh not. Yes; if the inequalitites in each sstem are based on the same line and that line is included in the sstem, then the solutions of the sstem are the points on the line. For eample, the solutions of the sstem + 5 are represented b all the ordered pairs on the line = Make a Conjecture What must be true of the boundar lines in a sstem of two linear inequalities if there is no solution of the sstem? Eplain. There are two possibilities. One is that the boundar lines are parallel and the solutions of each inequalit go in opposite directions. The other is that the boundar lines are the same line but that the inequalities are non-inclusive and their solutions go in opposite directions. Houghton Mifflin Harcourt Publishing Compan Module Lesson 301 Lesson 5.

14 Lesson Performance Task Ingrid has 6 nephews and nieces and is going to bu them all presents. She wants to bu the same present for each of the nephews and the same present for each of the nieces. Ingrid plans to spend at least $10 but no more than $0. She wants the prices of the presents to be within $ of each other. Find and graph the solution set. What do ou notice about the solution region on the graph? Nieces Presents Nephews Presents This gives the solution set The solution region on the graph is a parallelogram. QUESTIONING STRATEGIES What do ou notice about the graphs of the boundar lines? The are two pairs of parallel lines. Suppose all of the inequalities in the sstem were reversed? Describe the solution region. Eplain. There would be no solution; for each pair of parallel lines, the solution regions would face awa from each other and would not intersect. INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling Have students consider the simplest quadrilateral that the could graph using inequalities and what tpes of inequalities could be used. Students should see that a set of inequalities like 5 and 1, which represent four simple inequalities, would re sult in a solution that is a rectangle. Houghton Mifflin Harcourt Publishing Compan Module 5 30 Lesson EXTENSION ACTIVITY Sstems of inequalities can be used to create other geometric shapes. Have students graph the given sstem. Students should find that the have shaded a pentagon. Encourage students to come up with different patterns based on different sstems of inequalities. The might also consider how to make three-dimensional shapes such as tetrahedrons b using sstems of inequalities in three variables Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Solving Sstems of Linear Inequalities 30