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1 of 27 8/19/ :50 AM Answers Teacher Copy Lesson 2-2 Graphing Systems of Inequalities Plan Pacing: 1 class period Chunking the Lesson Example A #1 Example C #2 Example B p. 21 Check Your Understanding Lesson Practice Teach Bell-Ringer Activity Students should recall that an absolute value of a number is its distance from zero on a number line. Have students evaluate the following: [6] 6 [6] Then have students solve the following equation. 3. x = 6 [x = 6 or x = 6] Example A Marking the Text, Interactive Word Wall Point 2014 out the College Math Tip Board. to reinforce All rights why two reserved. solutions exist. Work through the solutions to the equation algebraically. Remind students that

2 of 27 8/19/ :50 AM solutions to an equation make the equation a true statement. This mathematical understanding is necessary for students to be able to check their results. Developing Math Language An absolute value equation is an equation involving an absolute value of an expression containing a variable. Just like when solving algebraic equations without absolute value bars, the goal is to isolate the variable. In this case, isolate the absolute value bars because they contain the variable. It should be emphasized that when solving absolute value equations, students must think of two cases, as there are two numbers that have a specific distance from zero on a number line. 1 Identify a Subtask, Quickwrite When solving absolute value equations, students may not see the purpose in creating two equations. Reviewing the definition of the absolute value function as a piecewise-defined function with two rules may enable students to see the reason why two equations are necessary. Have students look back at Try These A, parts c and d. Have volunteers construct a graph of the two piecewise-defined functions used to write each equation and then discuss how the solution set is represented by the graph. Example B Marking the Text, Simplify the Problem, Critique Reasoning, Group Presentation Start with emphasizing the word vary in the Example, discussing what it means when something varies. You may wish to present a simpler example such as: The average cost of a pound of coffee is $8. However, the cost sometimes varies by $1. This means that the coffee could cost as little as $7 per pound or as much as $9 per pound. Now have students work in small groups to examine and solve Example B by implementing an absolute value equation. Additionally, ask them to take the problem a step further and graph its solution on a number line. Have groups present their findings to the class. ELL Support For those students for whom English is a second language, explain that the word varies in mathematics means changes. There are different ways of thinking about how values can vary. Values can vary upward or downward, less than or greater than, in a positive direction or a negative direction, and so on. However, the importance comes in realizing that there are two different directions, regardless of how you think of it. Also address the word extremes as it pertains to mathematics. An extreme value is a maximum value if it is the largest possible amount (greatest value), and an extreme value is a minimum value if it is the smallest possible amount (least value). Developing Math Language An absolute value inequality is basically the same as an absolute value equation, except that the equal sign is now an inequality symbol: <,

3 of 27 8/19/ :50 AM >,,, or. It still involves an absolute value expression that contains a variable, just like before. Use graphs on a number line of the solutions of simple equations and inequalities and absolute value equations and inequalities to show how these are all related. Example C Simplify the Problem, Debriefing Before addressing Example C, discuss the following: Inequalities with A > b, where b is a positive number, are known as disjunctions and are written as A < b or A > b. For example, x > 5 means the value of the variable x is more than 5 units away from the origin (zero) on a number line. The solution is x < 5 or x > 5. See graph A. This also holds true for A b. Inequalities with A < b, where b is a positive number, are known as conjunctions and are written as b < A < b, or as b < A and A < b. For example: x < 5; this means the value of the variable x is less than 5 units away from the origin (zero) on a number line. The solution is 5 < x < 5. See graph B. This also holds true for A b. Students can apply these generalizations to Example C. Point out that they should proceed to solve these just as they would an algebraic equation, except in two parts, as shown above. After they have some time to work through parts a and b, discuss the solutions with the whole class. Teacher to Teacher Another method for solving inequalities relies on the geometric definition of absolute value x a as the distance from x to a. Here s how you can solve the inequality in the example: Thus, the solution set is all values of x whose distance from is greater than. The solution can be represented on a number line and written as x < 4 or x > 1. 2 Quickwrite, Self Revision/Peer Revision, Debriefing Use the investigation regarding the restriction c > 0 as an opportunity to discuss the need to identify impossible situations involving

4 of 27 8/19/ :50 AM inequalities. Check Your Understanding Debrief students answers to these items to ensure that they understand concepts related to absolute value equations. Have groups of students present their solutions to Item 4. Assess Students answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. Adapt Check students answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving absolute value equations and inequalities and graphing the solutions of absolute value equations and inequalities. If students are still having difficulty, review the process of rewriting an absolute value equation or inequality as two equations or inequalities. Activity Standards Focus In Activity 2, students represent constraints using equations and/or inequalities. They graph these constraints on a coordinate plane. Then they use their graphs to determine solutions to a system of equations or system of inequalities. Throughout this activity, emphasize the process of writing equations and inequalities from verbal descriptions and generating solutions once the constraints are graphed on the coordinate plane. Plan Pacing: 1 class period Chunking the Lesson #1 #2 4 #5 6 #7 #8 10 Check Your Understanding Lesson Practice

5 of 27 8/19/ :50 AM Teach Bell-Ringer Activity Have students write a function for each situation. 1. cost of plumbing repairs: $35/hr initial fee for repair: $50 [C(h) = h] 2. descent of hot air balloon: 5 ft/min initial height of balloon: 250 ft [H(m) = 250 5m] 3. number of students: 18 per bus other students: 135 [S(b) = b] Discuss with students the method they used to write the functions and the definitions of the variables they chose. 1 Debriefing This item is designed as an entry-level question. It will be used throughout the activity, so a debriefing is important to make sure that all students have the correct answer as they progress through the activity. Common Core State Standards for Activity 2 HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. HSA-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Differentiating Instruction Items 2 10 are designed to review how to work with linear equations and activate students prior knowledge. It may be possible to work through these items quickly without using the mini-lessons on this page and the next. If students need more support, take the time to have discussions about rate, y-intercept, slope, slope-intercept form, and point-slope form as needed. 2 4 Create Representations, Look for a Pattern, Quickwrite

6 of 27 8/19/ :50 AM Ensure that students know the difference between discrete and continuous data. In this case, only a whole number of tickets can be purchased, so the graph is individual points rather than a line. Watch for students that have 760, 660, and 560 as their last three entries in the table. These students may not realize that the numbers of tickets in the last three rows of the table are no longer increasing by 1 each time. When students look for the pattern in Item 3, make sure they relate the fact that the amount of money decreasing represents a negative rate of change. 5 Create Representations, Group Presentation, Debriefing Ask students to share their answers. Review slope as a constant rate of change for linear functions. Have students look at successive differences in the table, and use the graph to review how to find the slope of a linear function. Mini-Lesson: Slope-Intercept Form of the Equation of a Line If students need additional help with finding slope or finding the equation of a line in slope-intercept form, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. 6 Think-Pair-Share, Quickwrite Ensure students make the connection between the constant term and the y-intercept of a line in slope-intercept form. 7 Create Representations Students will replicate what they did in Items 2 6 without the scaffolding. Use this time to help individual students who need more support in the review process Look for a Pattern, Activating Prior Knowledge, Interactive Word Wall, Quickwrite, Debriefing Review the concept of domain and clarify the idea of the contextual domain. Discuss why negative values have no meaning in the situation. Connect to Technology For additional technology resources, visit SpringBoard Digital.

7 of 27 8/19/ :50 AM Mini-Lesson: Point-Slope Form of the Equation of a Line If students need additional help with writing an equation using point-slope form, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. Check Your Understanding Debrief students answers to these items to ensure that they understand concepts related to interpreting the rate of change and intercepts of an equation. Assess Students answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. Adapt Check students answers to the Lesson Practice to ensure that they understand basic concepts related to writing and graphing linear equations, as well as how to interpret rate of change and intercepts of equations. If students are continuing to having difficulty writing equations to model a given situation, have them practice writing word expressions and translating the word expressions into algebraic expressions. Plan Pacing: 1 class period Chunking the Lesson #1 #2 #3 #4 5 #6 #7 Check Your Understanding #11 #12 13 #14 15 Check Your Understanding

8 of 27 8/19/ :50 AM Lesson Practice Teach Bell-Ringer Activity Write the statement You must be at least 13 years old and at least 54 inches tall to ride this ride. on the board. Write the general ordered pair (age, height) on the board. Ask students which of the following people could ride the ride: Anna(11, 48); Ben(13, 58); Candice(14, 41); Danielle(10, 55); Ed(15, 60). Have students discuss which constraint prevents students from riding the ride. 1 Interactive Word Wall Introduce students to the word feasible as it applies to the solution sets of equations and inequalities. When using algebraic expressions to find solution sets within an applied setting, students must always interpret the solution set for reasonableness within the context of the applied setting. To further illustrate the word feasible, pose the following to students: You and a friend are playing a game of Tic-Tac-Toe. What are the feasible results of the game? [you win, you lose, you tie] 2 Think-Pair-Share This item returns to the domain constraints of the problem. Discuss how those constraints affect the feasible options for the conditions given in Item 1. The statement in Item 2b assumes that Roy will buy some number of meals greater than zero. 3 Create Representations, Debriefing For this item, assume that Roy has no additional expenses. Students may incorrectly assume that everyone eats three meals per day. Therefore, they may be confused by m as a variable for the number of meals that Roy may eat. From their perspective, Roy will eat three meals per day, or 15 meals, during his entire stay in New York City. 4 5 Think-Pair-Share, Work Backward, Discussion Groups In Item 4, some students may give t 9.6 as the answer. Although this is a correct solution to the inequality, it does not answer the question asked, which requires an integer answer. Likewise, in Item 5, some students may give m 21.5 as the answer. Although this is a solution to the inequality, the question asks for a specific number of meals, which also must be an integer. ELL Support Support students whose first language is not English by pairing them with more-fluent speakers for Item 6. Pairs can practice

9 of 27 8/19/ :50 AM their listening and speaking skills as they take turns describing the process of graphing a linear inequality. Encourage students to use precise mathematical language in their discussions. 6 Create Representations, Activating Prior Knowledge, Quickwrite Review how to graph a linear inequality. Later, in Item 8, students will graph all the constraint inequalities on one grid. For now, the focus is on one inequality. This makes it easier to determine whether or not students have any misunderstandings about the procedure for graphing linear inequalities. If students need additional help, assign Mini-Lesson: Graphing Linear Inequalities. Mini-Lesson: Graphing Linear Inequalities If students need additional help with graphing linear inequalities, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. Technology Tip If students are using TI-Nspire technology, provide the following directions for how to graph the inequality given in Item 7: Step 1: Choose Graphs&Geometry from the home screen. Step 2: Change the equal sign to a less than sign by using the [CLEAR] key followed by the [<] key located in the leftmost column of white keys. Step 3: Enter the function f1(x) as [X] (use the green letter key for x). Step 4: Adjust the viewing window as needed to view the graph. For additional technology resources, visit SpringBoard Digital. 7 Activating Prior Knowledge, Interactive Word Wall, Debriefing Part a provides an opportunity to review the concept of independent and dependent variables. Discuss that t is replaced with x because t (the number of tickets) is the independent variable and m (the number of meals) depends on the number of tickets. Students should discuss why replacing t with y and m with x does not work. In part c, students can adjust the viewing window by pressing [WINDOW] and entering an Xmin of 0, an Xmax of 16, an Xscl of 1, a Ymin of 0, a Ymax of 36, and a Yscl of 1. Note that due to the height of the graph on the student page, the graph displayed on the calculator will

10 0 of 27 8/19/ :50 AM not completely match, but students should be able to relate one to the other. Check Your Understanding Debrief students answers to these items to ensure that they understand concepts related to graphing linear inequalities and interpreting solutions to situations involving linear inequalities. 11 Activating Prior Knowledge, Create Representations Students must work with the language of inequalities in such phrases as at least, at most, not more than, less than, more than, and so on. Ask students to distinguish between not more than threeand less than three, and to distinguish between at least one and more than one. For each pair of phrases, ask students to use a number line to identify numbers that satisfy both phrases and then numbers that satisfy just one phrase but not the other. Have students represent each phrase by writing an inequality Create Representations As students graph the various inequalities, be sure that they use the boundary lines and shading correctly to interpret the inequalities. Have several students plot the four inequalities on the board so that any differences can be discussed and reconciled. For Item 13, some students may choose points that are not actually solutions of the system of inequalities. Use this as an opportunity to discuss why graphically represented information is only as accurate as the care taken when graphing, and as the precision that a graphing tool allows. Reinforce what it means to be a member of the feasible region and also the importance of verification by substituting into all four inequalities Quickwrite, Discussion Groups, Debriefing Have students list four other points that are solutions and four points that are not solutions. In each case, have them identify why the point is a solution or why it is not. Tie in the graphic representation to the real-world context, and for points that are not solutions, have students write a few sentences explaining why the point is not a solution. Check Your Understanding Debrief students answers to these items to ensure that they understand concepts related to graphing linear inequalities and interpreting solutions to situations involving linear inequalities. Assess Students answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning.

11 1 of 27 8/19/ :50 AM See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. Adapt Check students answers to the Lesson Practice to ensure that they understand how to graph inequalities and identify the solutions of a system of inequalities. Watch for students who choose points that satisfy either inequality as opposed to all inequalities in a system. Reinforce students understanding of the meaning of the solution to a system of inequalities. Learning Targets p. 21 Represent constraints by equations or inequalities. Use a graph to determine solutions of a system of inequalities. Think-Pair-Share (Learning Strategy) Definition Thinking through a problem alone, pairing with a partner to share ideas, and concluding by sharing results with the class Purpose Enables the development of initial ideas that are then tested with a partner in preparation for revising ideas and sharing them with a larger group Interactive Word Wall (Learning Strategy) Definition Visually displaying vocabulary words to serve as a classroom reference of words and groups of words as they are introduced, used, and mastered over the course of a year Purpose Provides a visual reference for new concepts, aids understanding for reading and writing, and builds word

12 2 of 27 8/19/ :50 AM knowledge and awareness Create Representations (Learning Strategy) Definition Creating pictures, tables, graphs, lists, equations, models, and /or verbal expressions to interpret text or data Purpose Helps organize information using multiple ways to present data and to answer a question or show a problem solution Work Backward (Learning Strategy) Definition Tracing a possible answer back through the solution process to the starting point Purpose Provides another way to check possible answers for accuracy Discussion Groups (Learning Strategy) Definition Working within groups to discuss content, to create problem solutions, and to explain and justify a solution Purpose Aids understanding through the sharing of ideas, interpretation of concepts, and analysis of problem scenarios Close Reading (Learning Strategy) Definition Reading text word for word, sentence by sentence, and line by line to make a detailed analysis of meaning

13 3 of 27 8/19/ :50 AM Purpose Assists in developing a comprehensive understanding of the text Debriefing (Learning Strategy) Definition Discussing the understanding of a concept to lead to consensus on its meaning Purpose Helps clarify misconceptions and deepen understanding of content Activating Prior Knowledge (Learning Strategy) Definition Recalling what is known about a concept and using that information to make a connection to a new concept Purpose Helps students establish connections between what they already know and how that knowledge is related to new learning Suggested Learning Strategies Think-Pair-Share, Interactive Word Wall, Create Representations, Work Backward, Discussion Groups, Close Reading, Debriefing, Activating Prior Knowledge Work with your group on Items 1 through 5. As needed, refer to the Glossary to review translations of p. 23p. 22 key terms. Incorporate your understanding into group discussions to confirm your knowledge and use of key mathematical language. Academic Vocabulary

14 4 of 27 8/19/ :50 AM The term feasible means that something is possible in a given situation. Discussion Group Tips As you share your ideas, be sure to use mathematical terms and academic vocabulary precisely. Make notes as you listen to group members to help you remember the meaning of new words and how they are used to describe mathematical concepts. Ask and answer questions clearly to aid comprehension and to ensure understanding of all group members ideas. 1. Roy s spending money depends on both the number of tickets t and the number of meals m. Determine whether each option is feasible for Roy and provide a rationale in the table below. Tickets (t) Meals(m) Total Cost Is it feasible? Rationale Yes Yes 1360 = No 1480 > No You cannot buy half a ticket. 2. Construct viable arguments. For all the ordered pairs (t, m) that are feasible options, explain why each statement below must be true.

15 5 of 27 8/19/ :50 AM a. All coordinates in the ordered pairs are integer values. Sample explanation: Both meals and tickets are integral values, so only coordinate pairs that are integers will be solutions. b. If graphed in the coordinate plane, all ordered pairs would fall either in the first quadrant or on the positive m-axis. Sample explanation: Tickets must be greater than or equal to zero, and meals must be greater than zero. Math Terms A linear inequality is an inequality that can be written in one of these forms, where A and B are not both equal to 0: Ax + By < C, Ax + By > C, Ax + By C, or Ax + By C. 3. Write a linear inequality that represents all ordered pairs (t, m) that are feasible options for Roy. 100t + 40m 1360 for t 0 and m > 0 4. If Roy buys exactly two meals each day, determine the total number of tickets that he could purchase in five days. Show your work. 9 tickets. Sample answer:

16 6 of 27 8/19/ :50 AM 100t + 40(10) t 960 t 9.6 Because you cannot purchase part of a ticket, at most 9 tickets can be purchased. 5. If Roy buys exactly one ticket each day, find the maximum number of meals that he could eat in the five days. Show your work. 21 meals. Sample answer: 100(5) + 40m m 860 m 21.5 Because you cannot purchase part of a meal, at most 21 meals can be purchased. Math Tip Recall how to graph linear inequalities. First, graph the corresponding linear equation. Then choose a test point not on the line to determine which half-plane contains the set of solutions to the inequality. Finally, shade the half-plane that contains the solution set. 6. To see what the feasible options are, you can use a visual display of the values on a graph. a. Attend to precision. Graph your inequality from Item 3 on the grid below. b. What is the boundary line of the graph?

17 7 of 27 8/19/ :50 AM 100t + 40m = 1360 c. Which half-plane is shaded? How did you decide? The lower half-plane is shaded. Sample explanation: The test point (0, 0) satisfies the inequality. d. Write your response for each item as points in the form (t, m). Item 4 Item 5 (9, 10) (5, 21) e. Are both those points in the shaded region of your graph? Explain. Yes. Sample explanation: Both are near but below the boundary line. Technology Tip To enter an equation in a graphing calculator, start with [Y=]. Technology Tip To graph an inequality that includes or >, you would use the symbol or. You need to indicate whether the half-plane above or below the boundary line will be shaded. 7. Use appropriate tools strategically. Now follow these steps to graph the inequality on a graphing calculator.

18 8 of 27 8/19/ :50 AM a. Replace t with x, and replace m with y. Then solve the inequality for y. Enter this inequality into your graphing calculator. b. Use the left arrow key to move the cursor to the far left of the equation you entered. Press [ENTER] until the symbol to the left of Y1 changes to. What does this symbol indicate about the graph? Sample answer: This symbol indicates that the half-plane below the boundary line will be shaded. c. Now press [GRAPH]. Depending on your window settings, you may or may not be able to see the boundary line. Press [WINDOW] and adjust the viewing window so that it matches the graph from Item 6. Then press [GRAPH] again. Check students graphs. d. Describe the graph. Sample answer: The graph shows a line with negative slope, with shading below and to the left of the line. Check Your Understanding p Compare and contrast the two graphs of the linear inequality: the one you made using paper and pencil and the one on your graphing calculator. Describe an advantage of each graph compared to the other. Sample answer: Both graphs show the same inequality, but the paper graph uses the variables t

19 9 of 27 8/19/ :50 AM and m, and the calculator graph uses the variables x and y. One advantage of the paper graph is that you can add titles to the axes to show what the graph represents. One advantage of the calculator graph is that you do not need to determine the coordinates of points on the boundary line in order to graph it. Math Tip Use a solid boundary line for inequalities that include or. Use a dashed boundary line for inequalities that include > or <. 9. a. What part of your graphs represents solutions for which Roy would have no money left over? Explain. Solutions on the boundary line represent solutions for which Roy would have no money left over. For these solutions, the total cost of the tickets plus the total cost of the meals is equal to $1360, which is the amount of money Roy has. b. What part of your graphs represents solutions for which Roy would have money left over? Explain. Solutions below the boundary line represent solutions for which Roy would have money left over. For these solutions, the total cost of the tickets plus the total cost of the meals is less than $1360, which is the amount of money Roy has. 10. Explain how you would graph the inequality 2x + 3y < 12, either by using paper and pencil or by using a graphing calculator. Sample answer: Write the related linear equation. Then choose several values of x and substitute them into the equation to find the corresponding values of y. Graph the ordered pairs and draw a dashed line through them. Then choose a test point not on the boundary line and use it to determine which half-plane to shade.

20 0 of 27 8/19/ :50 AM p. 25 Math Terms Constraints are the conditions or inequalities that limit a situation. 11. Roy realized that some other conditions or constraints apply. Write an inequality for each constraint described below. a. Roy eats lunch and dinner the first day. On the remaining four days, Roy eats at least one meal each day, but he never eats more than three meals each day. 6 m 14 b. There are only 10 performances playing that Roy actually wants to see while he is in New York City, but he may not be able to attend all of them. 0 t 10 c. Roy wants the number of meals that he eats to be no more than twice the number of performances that he attends. m 2t 12. Model with mathematics. You can use a graph to organize all the constraints on Roy s trip to New York City. a. List the inequalities you found in Items 3 and m 14, 0 t 10, m 2t, 100t + 40m 1360

21 1 of 27 8/19/ :50 AM b. Graph the inequalities from Items 3 and 11 on a single grid. The shaded region is the intersection of the four inequalities. Academic Vocabulary When you confirm a statement, you show that it is true or correct. 13. By looking at your graph, identify two ordered pairs that are feasible options to all of the inequalities. Confirm that these ordered pairs satisfy the inequalities listed in Item 12. a. First ordered pair (t, m): b. Second ordered pair (t, m): Answers will vary based on the points that students pick. 14. Label the point (6, 10) on the grid in Item 6. a. Interpret the meaning of this point. Roy will go to 6 performances and have 10 meals. b. Construct viable arguments. Is this ordered pair in the solution region common to all of the inequalities? Explain. Yes. Sample explanation: The ordered pair is in the shaded region of the graphs

22 2 of 27 8/19/ :50 AM of all four inequalities. 15. If Roy uses his prize money to purchase 6 tickets and eat 10 meals, how much money will he have left over for other expenses? Show your work. $360 Sample answer: 1360 ( ) = 360 Check Your Understanding 16. Given the set of constraints described earlier, how many tickets could Roy purchase if he buys 12 meals? Explain. 6, 7, or 8 tickets; Sample explanation: Substituting 12 for m and solving m 2t for t shows that t 6, so the number of tickets is at least 6. Substituting 12 for m and solving 100t + 40m 1360 for t shows that t 8.8, so the number of tickets is no more than a. If you were Roy, how many meals and how many tickets would you buy during the 5-day trip? Sample answer: 14 meals and 7 tickets b. Explain why you made the choices you did, and tell how you know that this combination of meals and tickets is feasible. Sample answer: If I were Roy, I would want to eat 3 meals per day whenever possible, so I would choose the maximum number of meals, which is 14. If Roy eats 14 meals, he can buy either 7 or 8 tickets and still meet all of the constraints. If I were Roy, I would pick 7 tickets so that I could have some money left over for souvenirs.

23 3 of 27 8/19/ :50 AM 18. Explain how you would graph this constraint on a coordinate plane: 2 x 5. Graph the vertical lines x = 2 and x = 5. Then shade the region between them. Lesson 2-2 Practice p Graph these inequalities on the same grid, and shade the solution region that is common to all of the inequalities: y 2, x 8, and. 20. Identify two ordered pairs that satisfy the constraints in Item 19 and two ordered pairs that do not satisfy the constraints. Sample answer: ordered pairs that satisfy the constraints: (4, 3) and (6, 2); ordered pairs that do not satisfy the constraints: (5, 1) and (9, 4) A snack company plans to package a mixture of almonds and peanuts. The table shows information about these types of nuts. The company wants the nuts in each package to have at least 60 grams of protein and to cost no more than $4. Use this information for Items Nut Protein (g/oz) Cost ($/oz) Almonds Peanuts

24 4 of 27 8/19/ :50 AM Math Tip When answering Item 21, remember that the number of ounces of each type of nut cannot be negative. 21. Model with mathematics. Write inequalities that model the constraints in this situation. Let x represent the number of ounces of almonds in each package and y represent the number of ounces of peanuts. 6x + 8y 60, 0.30x y 4, x 0, y Graph the constraints. Shade the solution region that is common to all of the inequalities. 23. a. Identify two ordered pairs that satisfy the constraints. Answers will vary but should be ordered pairs in the solution region. Sample answer: (10, 2) and (4, 12) b. Reason quantitatively. Which ordered pair represents the more expensive mixture? Which ordered pair represents the mixture with more protein? Explain your answer. Answers will vary depending on the answer to part a. Sample answer: (10, 2) represents a mixture that would cost $3.40 and have 76 g of protein. (4, 12) represents a mixture that would cost $3.60 and have 120 g of protein. So, (4, 12) represents both the more expensive mixture and the mixture with more protein.

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