12.2 Graphing Systems of Linear Inequalities

Transcription

1 Locker LESSON 1. Graphing Sstems of Linear Inequalities Common Core Math Standards The student is epected to: A-REI.1 Graph the solutions to a linear inequalit in two variables as a half-plane (ecluding the boundar in the case of a strict inequalit), and graph the solution set to a sstem of linear inequalities in two variables as the intersection of the corresponding half-planes. Also A-CED.3 Mathematical Practices MP. Modeling Language Objective Eplain to a partner how to determine whether a point is a solution to a sstem of inequalities. ENGAGE Essential Question: How do ou solve a sstem of linear inequalities? Graph each inequalit in the sstem on the same coordinate plane. The solution set will be all points in the area where the solutions of the inequalities overlap. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss how to graph an inequalit on a coordinate grid, and consider what the graph of all points that make two inequalities true at the same time might look like. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing Compan Name Class Date 1. Graphing Sstems of Linear Inequalities Essential Question: How do ou solve a sstem of linear inequalities? Eplore Determining Solutions of Sstems of Linear Inequalities Resource Locker A sstem of linear inequalities consists of two or more linear inequalities that have the same variables. The solutions of a sstem of linear inequalities are all the ordered pairs that make all the inequalities in the sstem true. Solve the sstem of equations b graphing. + 3 > First look at + 3 > 3. The equation of the boundar line is + 3 = 3. A B C intercept: 3 What are the -and -intercepts? intercept: 1 The inequalit smbol is > so use a dashed line. D Shade E F G H above Graph + 3 > 3. the boundar line for solutions that are greater than the inequalit. Look at The equation of the boundar line is - + = 6. intercept: 6 What are the -and -intercepts? intercept: 6 The inequalit smbol is so use a line. - solid Module 1 57 Lesson Name Class Date 1. Graphing Sstems of Linear Inequalities Essential Question: How do ou solve a sstem of linear inequalities? A-REI.1 For the full tet of these standards, see the table starting on page CA. Also A-CED.3 Eplore Determining Solutions of Sstems of Linear Inequalities Resource A sstem of linear inequalities consists of two or more linear inequalities that have the same variables. The solutions of a sstem of linear inequalities are all the ordered pairs that make all the inequalities in the sstem true. Solve the sstem of equations b graphing. + 3 > = 3 First look at + 3 > 3. The equation of the boundar line is. What are the -and -intercepts? The inequalit smbol is > so use a line. Shade the boundar line for solutions that are greater than the inequalit. Graph + 3 > 3. above intercept: 3 intercept: 1 dashed HARDCOVER PAGES 3 Turn to these pages to find this lesson in the hardcover student edition. Houghton Mifflin Harcourt Publishing Compan Look at The equation of the boundar line is. What are the -and -intercepts? The inequalit smbol is so use a line. - + = 6 intercept: 6 intercept: 6 solid Module 1 57 Lesson 57 Lesson 1.

2 I Shade J K L below the boundar line for solutions that are less than the inequalit. Graph on the same graph as + 3 > Identif the solutions. The are represented b the Check our answer b using a point in each region. Complete the table. - - overlapping shaded regions. EXPLORE Determining Solutions of Sstems of Linear Inequalities INTEGRATE TECHNOLOGY Students can use graphing calculators to help them graph sstems of linear inequalities. The calculator will show the boundar line of each graph, and students must specif whether the region above or below each line should be shaded. Ordered Pair Satisfies + 3 > 3? Satisfies - + 6? (0, 0) Yes (, 3) Yes Yes (-, ) (-, 6) Yes In the overlapping shaded regions? Yes QUESTIONING STRATEGIES What does the graph of the solutions of a sstem of two linear inequalities look like? It is a region on the coordinate plane. Reflect 1. Discussion Wh is (0, 0) a good point to use for checking the answer to this sstem of linear inequalities? The point (0, 0) does not lie on a boundar line, and it is eas to evaluate each inequalit in the sstem for = 0 and = 0. Houghton Mifflin Harcourt Publishing Compan Module 1 5 Lesson PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunit to address Mathematical Practice MP., which calls for students to use modeling. Students learn to graph sstems of linear inequalities, including both sstems with intersecting boundar lines and sstems with parallel boundar lines. Students also learn to interpret the graphs to determine which points are solutions and which points are not solutions for a sstem of linear inequalities. Graphing Sstems of Linear Inequalities 5

3 EXPLAIN 1 Solving Sstems of Linear Inequalities b Graphing INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Discuss with students the differences between finding a solution to a sstem of linear equations and finding solutions to a sstem of inequalities. Students should understand that when the lines given b two linear equations intersect, the solution is a single point, but when the boundar lines for the graphs of two inequalities intersect, the sstem of inequalities has an infinite number of solutions. QUESTIONING STRATEGIES What must be true for a point to be a solution to a sstem of linear inequalities? The point must make both inequalities true. If the point does not make both inequalities true, it is not a solution. Houghton Mifflin Harcourt Publishing Compan Eplain 1 Solving Sstems of Linear Inequalities b Graphing You can use a graph of a sstem of linear inequalities to determine and identif solutions to the sstem of linear inequalities. Eample 1 Graph the sstem of linear inequalities. Give two ordered pairs that are solutions and two that > - 3 Solve the first inequalit for Graph the sstem. + > - 3 (0, 0) and (, ) (-6, -) and (-, ) > _ 3 - Reflect. Is ( 6, 6) a solution of the sstem?, a solution must satisf both inequalities. Your Turn Graph the sstem of linear inequalities. Give two ordered pairs that are solutions and two that Possible answers are given < -3 Possible answers are given.. > - + < Solve the first inequalit for. Graph the sstem > _ (0, 0) (-, 0) (0, -) (, ) and and (, -) and (-, -) (0, 0) and (-, -) (0, 0) and (-, -) (-, ) and (, -6) Module 1 59 Lesson COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in groups of two. Tell each student to write a sstem of two linear inequalities. Each student should graph the sstem of linear inequalities written b the partner, and the partner should verif that the graph is correct. Students should either clearl shade their graphs to indicate the region where solutions lie or label each region contains solution points or does not contain solution points. 59 Lesson 1.

4 Eplain Graphing Sstems of Inequalities with Parallel Boundar Lines If the lines in a sstem of linear equations are parallel, there are no solutions. However, if the boundar lines in a sstem of linear inequalities are parallel, the sstem ma or ma not have solutions. Eample < - 3 > + - Graph each sstem of linear inequalities. Describe the solutions > This sstem has no solution. The solutions are all points between the parallel lines and on the solid line. - - EXPLAIN Graphing Sstems of Inequalities with Parallel Boundar Lines AVOID COMMONS ERRORS When solving sstems of inequalities involving parallel boundar lines, it ma be eas to assume that the solutions are the points that lie between the lines. Remind students that it is possible for such sstems to have no solutions or for the solutions to be the same as the solutions to one of the two inequalities. Students should alwas check their solutions b making sure the make both original statements true. Your Turn Graph each sstem of linear inequalities. Describe the solutions < The solutions are the same as the solutions of This sstem has no solution. Houghton Mifflin Harcourt Publishing Compan QUESTIONING STRATEGIES Without graphing, how can ou determine whether the functions in a sstem of inequalities will produce parallel boundar lines? If the slopes of the boundar lines are the same, then the lines will be parallel. How do the solutions for a sstem of inequalities with parallel lines compare to the solutions for a sstem of equations with parallel lines? For a sstem of inequalities with parallel boundar lines, either the solution is a region of the coordinate plane or there is no solution. For a sstem of equations with parallel lines, there is never a solution. Module Lesson DIFFERENTIATE INSTRUCTION Kinesthetic Eperience Some students ma find the different was of shading the coordinate plane confusing. The ma benefit from first graphing one inequalit, then folding the coordinate grid along the boundar line so that onl the part of the grid with solutions is showing. Students can then graph the second inequalit and again fold along the boundar line. The section that remains visible should be the part of the coordinate grid that contains points that Graphing Sstems of Linear Inequalities 550

5 ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Students should understand that a sstem of linear inequalities will never have all points in the coordinate plane as solutions. Because the solution set for a sstem of inequalities is the set of points that satisf both inequalities, it is alwas a subset of the solution to each inequalit. The graph of the solutions is the area of overlap between two half-planes. Elaborate 7. Is it possible for a sstem of two linear inequalities to have ever point in the plane as solutions? Wh or wh not? ; each boundar line divides the plane into two half-planes, one of each being the solution of each inequalit. It is not possible for the two solution half-planes to overlap and completel cover the plane.. Discussion How would ou write a sstem of linear inequalities from a graph? To write a sstem of linear inequalities from a graph, write the linear inequalit for each of the graphs that make up the sstem. 9. Essential Question Check-In How does testing specific ordered pairs tell ou that the solution ou graphed is correct? Ordered pairs of points from the overlapping shaded region will satisf both inequalities. QUESTIONING STRATEGIES Is it possible for the solutions of a sstem of two linear inequalities to form a line? If so, give an eample. Yes; possible answer: the solutions of the sstem of inequalities + and + are all the points on the line = +. SUMMARIZE THE LESSON How do ou use a graph to find solutions for a sstem of linear inequalities? Identif the points on the graph that make both inequalities true. If the boundar lines are parallel, it is possible that the sstem of linear equalities ma have no solutions. Houghton Mifflin Harcourt Publishing Compan Evaluate: Homework and Practice 1. Match the inequalit with the correct boundar line. Answers ma be used more than once. b a. = b. = d > b c. = d. = - + e _ e. = 3 - d - > f. = a 3 Online Homework Hints and Help Etra Practice Module Lesson LANGUAGE SUPPORT Connect Vocabular Discuss with students the meaning of the word satisf as it relates to sstems of linear inequalities. Students are probabl familiar with man uses for the word outside of the math classroom. Compare these uses to the wa the word is used mathematicall. Students should understand that an ordered pair satisfies a sstem of linear inequalities when it meets the conditions of the sstem or, in other words, when it is a solution of the sstem. 551 Lesson 1.

6 Determine if the given point satisfies either equation and is a solution of the sstem of inequalities < 6 ; (0, 0) 3. 5_ 5-10 (0, 0) satisfies (0, 0) satisfies - 0 < 6. The point is a solution of the sstem. + 5 > -10 ; (.5, -1.5) - (.5, -1.5) satisfies -. (.5, -1.5) satisfies + 5 > -10. The point is a solution of the sstem. EVALUATE Determine if the given point is a solution of the sstem of inequalities. If not, find a point that is.. (-9, ) 5. (6, -) The point (-9, ) is a solution. The point (6, -) is a solution. 6. (0, -) The point (0, -) is not a solution. The point (-, 0) is a solution Graph the sstem of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions. Possible answers are given. > > (, -) and (, -) (, ) and (0, ) (0, 0) and (, -) (0, 0) and (, -6) Houghton Mifflin Harcourt Publishing Compan ASSIGNMENT GUIDE Concepts and Skills Eplore Determining Solutions of Sstems of Linear Inequalities Eample 1 Solving Sstems of Linear Inequalities b Graphing Eample Graphing Sstems of Inequalities with Parallel Boundar Lines Practice Eercises 1 6 Eercises 7 1, 3 Eercises 15, 5 INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Students ma be unsure how to determine which ordered pairs are solutions for a sstem of linear inequalities if the are not shown on the coordinate grid. Discuss was to determine whether a point not shown on the grid is a solution, including etending the graph and substituting the coordinates of the point into the inequalities. Module 1 55 Lesson Eercise Depth of Knowledge (D.O.K.) Mathematical Practices Recall of Information MP. Reasoning 7 Skills/Concepts MP. Modeling 3 3 Strategic Thinking MP. Modeling 3 Strategic Thinking MP.6 Precision 5 3 Strategic Thinking MP.3 Logic Graphing Sstems of Linear Inequalities 55

7 AVOID COMMONS ERRORS When boundar lines are parallel, students ma arrive at incorrect solutions b not paing attention to the inequalit signs in the inequalities. Remind students to check their solutions b verifing that the points make both original inequalities true. < < (0, 0) and (, ) (, -) and (10, -) (0, ) and (, 6) (0, 0) and (, 6) VISUAL CUES Remind students that when an inequalit is written in the form > m + b, the inequalit sign determines which part of the graph should be shaded. If the sign > or is used, the half-plane above the line should be shaded; if the sign < or is used, it is the half-plane below the line that should be shaded. - 3_ > - - (0, 0) and (10, -6) (, -) and (, 6) < (-, -6) and (-6, -6) - (0, 0) and (, 6) Houghton Mifflin Harcourt Publishing Compan _ < - (, -6) and (6, -6) (0, 0) and (, 6) < 3 (0, 0) and (6, -6) (-10, 0) and (, 6) Module Lesson 553 Lesson 1.

8 Graph each sstem of linear inequalities. Describe the solutions < 3 - The solutions are the same as the solutions of < _ 5 + _ 5-6 5_ _ _ _ + 10 The solutions are the The solutions are all same as the points between the - parallel lines - and on the solid lines. solutions of 5. < The solutions are all points between the parallel lines, including points on the line This sstem has no solution. 9_ - 1 < 9_ - 9 This sstem has no solution Houghton Mifflin Harcourt Publishing Compan INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Some students will choose points that are not solutions to the sstem of linear equalities b selecting points that are in the unshaded portion of the graph. Students should understand that while this part of the graph contains points that are not solutions, regions of the graph that contain solutions to onl one of the two inequalities also contain non-solution points. INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. Warn students that the brace, {, is not alwas used when writing a sstem of inequalities. A standardized test might omit the smbol and simpl sa that the inequalities are a sstem. Module 1 55 Lesson Graphing Sstems of Linear Inequalities 55

9 JOURNAL Have students write the steps the use for finding points that are solutions to a sstem of inequalities. Steps should be written so that the appl to sstems that have intersecting boundar lines and to sstems that have parallel boundar lines, as well. < - 3_ _. 5 - > The solutions are all The solutions are the points between the same as the parallel lines and on solutions of the solid line. - > > H.O.T. Focus on Higher Order Thinking 3. Persevere in Problem Solving Write and graph a sstem of linear inequalities for which the solutions are all the points in the second quadrant, not including points on the aes. > 0 < Critical Thinking Can the solutions of a sstem of linear inequalities be the points on a line? Eplain. Yes; if the inequalities have the same boundar line and one is less than or equal to while the other is greater than or equal to then the solutions are points on the boundar line. The sstem - + and - + has onl the boundar line - + = as its solution. Houghton Mifflin Harcourt Publishing Compan 5. Eplain the Error A student was asked to graph the sstem < 3_ and describe the solution set. The student gave the following answer. Eplain what the student did wrong, then give the correct answer. The solutions are the same as the solutions of The student switched the inequalit signs when graphing them. The correct solution set is the same as the solutions of < 3_ -. - Module Lesson 555 Lesson 1.

10 Lesson Performance Task Successful stock market investors know a lot about inequalities. The know up to what point the are willing to accept losses, and at what point the are willing to lock in their profits and not subject their investments to additional risk. The often have these inequalities all mapped out at the time the purchase a stock, so the can tell instantl if the are sticking to their investment strateg. Graph the sstem of linear inequalities. Then describe the solution set and give two ordered pairs that are solutions and two that are not. Is there anthing particular to note about the shape of this sstem? < - 3_ 5 + 3_ + > - 3_ 5 - > 3_ - 6 QUESTIONING STRATEGIES Before graphing the sstem of inequalities, can ou tell whether an of the boundar lines will be parallel? If so, which ones? Eplain how ou know. Yes; the boundar line for 3 + is parallel to the boundar line for > _ 3-6 since the have the same slope, _ 3, and the line for < - _ 3 + is parallel to the line for 5 > - _ since the have the same slope, Will an of the boundar lines be perpendicular? Eplain how ou know. ; none of the inequalities have slopes that are opposite reciprocals of each other The solution set is a parallelogram. The sstem is two sets of parallel lines. Possible answers: (0, 0) and (-, -) (, 6) and (6, -6) Houghton Mifflin Harcourt Publishing Compan AVOID COMMON ERRORS Students ma incorrectl describe the graph as a square or a rectangle. Remind students that a figure with two pairs of opposite parallel sides is not necessaril a rectangle. For it to be a rectangle, all four interior angles must be right angles. Stress that observing a figure on a graph looks like it has right angles is not sufficient. The equations for the sides of the figure must be used to determine whether an sides are perpendicular. Module Lesson EXTENSION ACTIVITY Challenge students to sketch a closed figure on a coordinate grid, then write a sstem of linear inequalities whose solution set is all the points within that figure. Then have each student trade sstems of inequalities with a partner and have them graph each other s sstems. As a class, discuss what students discovered about the tpes of regions the can or cannot create with sstems of linear inequalities. Students ma find that the solution set of a sstem of linear inequalities must be a region with straight-line sides and interior angles less than 10 ; in other words, all the figures are conve polgons. 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. Graphing Sstems of Linear Inequalities 556