Solving Linear Inequalities Chapter Overview and Pacing

Transcription

1 Solving Linear Inequalities Chapter verview and Pacing PACING (das) Regular Block Basic/ Basic/ Average Advanced Average Advanced Solving Inequalities b Addition and Subtraction (pp. 8 ) Solve linear inequalities b using addition. Solve linear inequalities b using subtraction. LESSN BJECTIVES Year-long and two-ear pacing: pages T0 T. Solving Inequalities b Multiplication and Division (pp. ) 0. Preview: Use algebra tiles to solve inequalities. (with 6- (with 6- Solve linear inequalities b using multiplication. Preview) Preview) Solve linear inequalities b using division. Solving Multi-Step Inequalities (pp. 7) 0. Solve linear inequalities involving more that one operation. Solve linear inequalities involving the Distributive Propert. Solving Compound Inequalities (pp. 9 ) Solve compound inequalities containing the word and and graph their solution sets. Solve compound inequalities containing the word or and graph their solution sets. Solving pen Sentences Involving Absolute Value (pp. ) Solve absolute value equations. Solve absolute value inequalities. Graphing Inequalities in Two Variables (pp. 8). Graph inequalities on the coordinate plane. (with 6-6 (with 6-6 Solve real-world problems involving linear inequalities. Follow-Up) Follow-Up) Follow-Up: Use a graphing calculator to investigate graphs of inequalities. Stud Guide and Practice Test (pp. 9 6) 0. Standardized Test Practice (pp. 6 6) (with 6-6 Follow-Up) Chapter Assessment TTAL 7 6 An electronic version of this chapter is available on StudentWorks TM. This backpack solution CD-RM allows students instant access to the Student Edition, lesson worksheet pages, and web resources. 6A Chapter 6 Solving Linear Inequalities

2 Timesaving Tools Chapter Resource Manager All-In-ne Planner and Resource Center See pages T and T. Stud Guide and Intervention CHAPTER 6 RESURCE MASTERS Practice (Skills and Average) Reading to Learn Mathematics Enrichment Assessment Prerequisite Skills Workbook Applications* Parent and Student Stud Guide Workbook -Minute Check Transparencies SC Interactive Chalkboard AlgePASS: Tutorial Plus (lessons) Materials SC (Preview: algebra tiles, equation mat, self-adhesive notes) , graphing calculator , GCS stopwatch GCS (Follow-Up: graphing calculator) 79 9, *Ke to Abbreviations: GCS Graphing Calculator and Spreadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual ELL Stud Guide and Intervention, Skills Practice, Practice, and Parent and Student Stud Guide Workbooks are also available in Spanish. Chapter 6 Solving Linear Inequalities 6B

3 Mathematical Connections and Background Continuit of Instruction Prior Knowledge Students solved open sentence inequalities in Chapter. Chapter had them graphing rational numbers and eploring absolute value. Solving single-step and multi-step equations was developed in Chapter. Students learned to graph linear equations in Chapter. This Chapter Students develop the properties for solving inequalities. The appl the Addition and Subtraction Properties of Inequalities to solve inequalities. The also use the Multiplication and Division Properties of Inequalities, where the sign is sometimes reversed, to solve inequalities. The solve single-step and multistep inequalities. Students solve compound inequalities, as well as equations and inequalities that contain absolute values. The chapter ends with graphing inequalities in two variables. Future Connections In future math studies, students solve and graph inequalities of other tpes of functions, such as quadratics. The also appl solving and graphing inequalities in Algebra to linear programming where the maimum profit for a situation is determined. Inequalities are used in man biological areas, such as determining appropriate parameters for populations of species in various regions. Solving Inequalities b Addition and Subtraction To solve an equation, isolate the variable so that it has a coefficient of on one side of the equal sign. An inequalit is solved the same wa. The Addition Propert of Inequalit is used in the same wa as the Addition Propert of Equalit. It states that an number can be added to each side of an inequalit and the result is a true inequalit. The same is true for the Subtraction Propert of Inequalit: a number can be subtracted from each side of an inequalit and the result is a true inequalit. There are infinitel man solutions to an inequalit. The solutions to inequalities can be written in set builder notation, for eample { ). This is read as the set of all numbers such that is greater than. The number found when solving an inequalit is a boundar that is sometimes included in the solution and sometimes not. It is included in the solution if the inequalit sign is or, but it is not included if the smbol is or. If the boundar number is included, a solid dot is placed at that point on the number line. If the number is not included, use an open circle. Then draw an arrow to the right if the rest of the solution set is greater than the boundar, or to the left if the rest of the solution set is less than the boundar. Solving Inequalities b Multiplication and Division Inequalities that include multiplication or division of the variable can also be solved. The same principles as found in the Multiplication and Division Properties of Equalit are used, with one main difference. If an inequalit is multiplied or divided b the same negative number on each side, the inequalit smbol is reversed. The smbol must be reversed to result in a true inequalit. The inequalit sign is not reversed if each side is multiplied or divided b the same positive number. You multipl or divide b a negative number onl if the coefficient of the variable is negative. Solving Multi-Step Inequalities Solve multi-step inequalities using the same process as for solving multi-step equations. Work backward using inverse operations to undo the operations. After each side is simplified using the Distributive Propert and/or combining like terms, work in the opposite order of the order of operations. The Addition and Subtraction Properties of Inequalit are applied first, followed b the Multiplication and Division Properties of Inequalities. 6C Chapter 6 Solving Linear Inequalities

4 If the solution is an untrue statement, such as 8, there is no solution. If the solution results in a statement that is alwas true, such as, then the solution is the set of all real numbers. A solution can alwas be checked b substituting it back into the inequalit. Solving Compound Inequalities In a compound inequalit, one variable is related to two different amounts with two inequalit signs. The signs ma be the same or the ma be different. If and is written between the inequalities, or the variable epression is between the two inequalit signs, the graph is the intersection of the two inequalities. This is because the solution must be true for both inequalities. If or is written between the two inequalities, the graph is the union of the two inequalities. This is because the solution can be true for either inequalit. Solving pen Sentences Involving Absolute Value An absolute value open sentence can be an equation or an inequalit. The value inside the absolute value smbols could be positive or negative. The absolute value represents the distance a number is from zero on a number line. Absolute value equations can be solved b graphing them or b writing them as a compound sentence and solving algebraicall. To solve algebraicall, write the epression inside the smbol as equal to the given value and then equal to the opposite of the given value. Solve each equation. Write both solutions inside one set of brackets. An absolute value inequalit is written as a compound inequalit. If the absolute value is on the left and the inequalit smbol is or, the compound sentence is written with and. If the absolute value is on the left and the inequalit smbol is or, the compound sentence is written with or. To solve the first case, write the epression from inside the absolute value smbol, the or, and the value to the right of the sign. Then write the epression, the opposite inequalit sign, and the opposite value. Solve both inequalities and write the solution set as an intersection. To solve the second case ou follow the same process, onl write the solution set as a union using or before solving both inequalities. Graphing Inequalities in Two Variables The solution set of an inequalit, like that of an equation, is all ordered pairs that make the statement true. Similar to the solution set of an equation in two variables, the solution set of an inequalit in two variables is graphed on a coordinate plane. However, the solution set of an inequalit is not linear. It does have a linear boundar, but it covers a region called a half-plane. First graph the inequalit as if it contained an equal sign like an equation. This is the boundar line. If the inequalit is or, then the line is dashed. A solid line is graphed for and. These relate to the circle and dot on a number line. Select a point in either half-plane and test it in the inequalit. (0, 0) is a good point to use if it is not on the boundar line. If the resulting statement is true, shade the half-plane that contains the point. If the statement is false, shade the other half plane. Quick Review Math Handbook Hot Words includes a glossar of terms while Hot Topics consists of eplanations of ke mathematical concepts with eercises to test comprehension. This valuable resource can be used as a reference in the classroom or for home stud. Lesson Hot Topics Section Lesson Hot Topics Section GS6 6., , P , 6.6 GS = Getting Started, P = Preview Additional mathematical information and teaching notes are available at Chapter 6 Solving Linear Inequalities 6D

5 and Assessment Ke to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters Tpe Student Edition Teacher Resources Technolog/Internet ASSESSMENT INTERVENTIN ngoing Prerequisite Skills, pp. 7,,, 7,, Practice Quiz, p. Practice Quiz, p. Mied Review Error Analsis Standardized Test Practice pen-ended Assessment Chapter Assessment pp.,, 7,,, 7 Find the Error, pp. 9, 8 Common Misconceptions, p. 6 pp., 8, 9,, 7,,, 7, 6, 6 6 Writing in Math, pp.,, 7,,, 7 pen Ended, pp., 8,,, 8, Standardized Test, p. 6 Stud Guide, pp. 9 6 Practice Test, p. 6 -Minute Check Transparencies Prerequisite Skills Workbook, pp , 8 8 Quizzes, CRM pp. 9 9 Mid-Chapter Test, CRM p. 9 Stud Guide and Intervention, CRM pp., 9 0, 6, 6 6, 67 68, 7 7 Cumulative Review, CRM p. 96 Find the Error, TWE pp. 9, 8 Unlocking Misconceptions, TWE pp., Tips for New Teachers, TWE pp., TWE pp. 6 6 Standardized Test Practice, CRM pp Modeling: TWE pp., Speaking: TWE pp. 7, 7 Writing: TWE pp., pen-ended Assessment, CRM p. 9 Multiple-Choice Tests (Forms, A, B), CRM pp Free-Response Tests (Forms C, D, ), CRM pp Vocabular Test/Review, CRM p. 9 AlgePASS: Tutorial Plus, Lessons and Standardized Test Practice CD-RM standardized_test EamView Pro (see below) MindJogger Videoquizzes vocabular_review Algebra Lesson For more information on Yearl ProgressPro, see p. 88. Yearl ProgressPro Skill Lesson(s) 6- Solving Inequalities b Addition and Subtraction Graphing Inequalities 6- Solving Inequalities b Multiplication and Division 6- Solving Multi-Step Inequalities 6- Solving Compound Inequalities Graphing Compound Inequalities 6- Solving pen Sentences Involving Absolute Value 6-6 Graphing Inequalities in Two Variables EamView Pro Use the networkable EamView Pro to: Create multiple versions of tests. Create modified tests for Inclusion students. Edit eisting questions and add our own questions. Use built-in state curriculum correlations to create tests aligned with state standards. Change English tests to Spanish and vice versa. For more information on Intervention and Assessment, see pp. T8 T. 6E Chapter 6 Solving Linear Inequalities

6 Reading and Writing in Mathematics Glencoe Algebra provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition Foldables Stud rganizer, p. 7 Concept Check questions require students to verbalize and write about what the have learned in the lesson. (pp., 8,,, 8, ) Reading Mathematics, p. 8 Writing in Math questions in ever lesson, pp.,, 7,,, 7 Reading Stud Tip, pp. 9, 9, 0 WebQuest, p. 7 Teacher Wraparound Edition Foldables Stud rganizer, pp. 7, 9 Stud Notebook suggestions, pp.,, 8,, 8,, 8, Modeling activities, pp., Speaking activities, pp. 7, 7 Writing activities, pp., Differentiated Instruction, (Verbal/Linguistic), p. 0 ELL Resources, pp. 6, 0,, 0, 6, 8,, 0, 6, 9 Additional Resources Vocabular Builder worksheets require students to define and give eamples for ke vocabular terms as the progress through the chapter. (Chapter 6 Resource Masters, pp. vii-viii) Reading to Learn Mathematics master for each lesson (Chapter 6 Resource Masters,pp. 7,, 9, 6, 7, 77) Vocabular PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabular lists that ou can customize. Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. Reading and Writing in the Mathematics Classroom WebQuest and Project Resources For more information on Reading and Writing in Mathematics, see pp. T6 T7. Taking good notes will help students become activel involved in the learning process. For each lesson, have students read and then write notes about the topic. You ma wish to show them the sample notes for Lesson 6- at the right to use as a guide. Afterward, allow class time for students to discuss their notes. Encourage students to talk about the procedures used in the lesson for solving the problems and how the addressed those procedures in their notes. Lesson 6- Set up an absolute value inequalit. Eample: +8 0 Case : The value inside the absolute value smbols is less than Case : The value inside the absolute value smbols is greater than Chapter 6 Solving Linear Inequalities 6F

7 Notes Have students read over the list of objectives and make a list of an words with which the are not familiar. Point out to students that this is onl one of man reasons wh each objective is important. thers are provided in the introduction to each lesson. Solving Linear Inequalities Lessons 6- through 6- Solve linear inequalities. Lesson 6- Solve compound inequalities and graph their solution sets. Lesson 6- Solve absolute value equations and inequalities. Lesson 6-6 Graph inequalities in the coordinate plane. Inequalities are used to represent various real-world situations in which a quantit must fall within a range of possible values. For eample, figure skaters and gmnasts frequentl want to know what the need to score to win a competition. That score can be represented b an inequalit. You will learn how a competitor can determine what score is needed to win in Lesson 6-. Ke Vocabular set-builder notation (p. 9) compound inequalit (p. 9) intersection (p. 9) union (p. 0) half-plane (p. ) NCTM Local Lesson Standards bjectives 6-, 6, 8, 9, 0 6-, 6, 8, 9, 0 Preview 6-, 6, 8, 9, 0 6-, 6, 8, 9, 0 6-,, 6, 8, 9, 0 6-,, 6, 8, 9, 0 6-6, 6, 8, 9, 0 6-6, 6, 8, 9, 0 Follow-Up Ke to NCTM Standards: =Number & perations, =Algebra, =Geometr, =Measurement, =Data Analsis & Probabilit, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 0=Representation 6 Chapter 6 Solving Linear Inequalities Vocabular Builder ELL The Ke Vocabular list introduces students to some of the main vocabular terms included in this chapter. For a more thorough vocabular list with pronunciations of new words, give students the Vocabular Builder worksheets found on pages vii and viii of the Chapter 6 Resource Masters. Encourage them to complete the definition of each term as the progress through the chapter. You ma suggest that the add these sheets to their stud notebooks for future reference when studing for the Chapter 6 test. 6 Chapter 6 Solving Linear Inequalities

8 Prerequisite Skills To be successful in this chapter, ou ll need to master these skills and be able to appl them in problem-solving situations. Review these skills before beginning Chapter 6. For Lessons 6- and 6- Solve each equation. (For review, see Lessons -, -, and -.) 6. 9 Solve Equations. t 8. b f 9. d 6. r r 6. m 7 m a a 9. k 7 0..b s (s ). n n 7. For Lesson 6- Evaluate Absolute Values Find each value. (For review, see Lesson -.) For Lesson 6-6 Graph Equations with Two Variables Graph each equation. (For review, see Lesson -.) 8. See pp. 6A 6D ( ) 8. Solving Linear Inequalities Make this Foldable to help ou organize our notes. Begin with two sheets of notebook paper. This section provides a review of the basic concepts needed before beginning Chapter 6. Page references are included for additional student help. Additional review is provided in the Prerequisite Skills Workbook, pp and 8 8. Prerequisite Skills in the Getting Read for the Net Lesson section at the end of each eercise set review a skill needed in the net lesson. For Prerequisite Lesson Skill 6- Multiplication and division equations (p. ) 6- Multi-step equations (p. ) 6- Graphing integers on a number line (p. 7) 6- Absolute values (p. ) 6-6 Graphing linear equations (p. ) Fold and Cut Fold one sheet in half along the width. Cut along the fold from each edge to the margin. Fold in half along the width. Cut along the fold between the margins. Fold a New Paper and Cut Fold Insert the first sheet through the second sheet and align the folds. Label Label each page with a lesson number and title. Solving Linear Inequalities Reading and Writing As ou read and stud the chapter, fill the journal with notes, diagrams, and eamples of linear inequalities. Chapter 6 Solving Linear Inequalities 7 For more information about Foldables, see Teaching Mathematics with Foldables. TM rganization of Data and Journal Writing After students make their Foldable journals, have them label each page to correspond to a lesson in the chapter. Students can use their Foldables to take notes, record concepts, and define terms. The can also use them to record the direction and progress of learning, to describe positive and negative eperiences during learning, to write about personal associations and eperiences, and to list eamples of was in which new knowledge has or will be used in their dail life. Chapter 6 Solving Linear Inequalities 7

9 Lesson Notes Solving Inequalities b Addition and Subtraction Focus -Minute Check Transparenc 6- Use as a quiz or a review of Chapter. Mathematical Background notes are available for this lesson on p. 6C. Building on Prior Knowledge In Chapter, students learned to solve equations using addition and subtraction. In this lesson, the should recognize that solving inequalities b addition and subtraction is a ver similar process. are inequalities used to describe school sports? Ask students: Is the number of schools that offer volleball greater than or less than the number of schools that offer track and field? less than Suppose 00 schools added track and field, and 00 added volleball. Would there be more schools offering track and field, or volleball? track and field Sports The kinds of sports offered in schools often reflects the sports that are popular in the region. Which sports are offered at our school? Which have the most participants? Vocabular set-builder notation STANDARD A. The student will solve multistep linear equations and inequalities in one variable, solve literal equations (formulas) for a given variable, and appl these skills to solve practical problems. Graphing calculators will be used to confirm algebraic solutions. Stud Tip Look Back To review inequalities, see Lesson -. 8 Chapter 6 Solving Linear Inequalities Solve linear inequalities b using addition. Solve linear inequalities b using subtraction. are inequalities used to describe school sports? In the school ear, more high schools offered girls track and field than girls volleball.,87,6 If 0 schools added girls track and field and 0 schools added girls volleball the net school ear, there would still be more schools offering girls track and field than schools offering girls volleball.,87 0?,6 0,607,6 SLVE INEQUALITIES BY ADDITIN Addition Propert of Inequalities. USA TDAY Snapshots Girls gear up for high school sports High school girls are plaing sports in record numbers, almost.7 million in the school ear. Most popular girls sports b number of schools offering each program: 6,6 Basketball,87 Track and field,6 Volleball,009,77 Softball Cross countr Source: National Federation of State High School Associations B Ellen J. Horrow and Alejandro Gonzalez, USA TDAY The sports application illustrates the Addition Propert of Inequalities Words If an number is added to each side of a true inequalit, the resulting inequalit is also true. Smbols For all numbers a, b, and c, the Eample 7 following are true If a b, then a c b c. 8. If a b, then a c b c. Eample This propert is also true when and are replaced with and. Solve b Adding Solve t. Then check our solution. t riginal inequalit t Add to each side. t 8 This means all numbers less than or equal to 8. CHECK Substitute 8, a number less than 8, and a number greater than 8. Let t 8. Let t 0. Let t 60. 8? 0? 60? The solution is the set {all numbers less than or equal to 8}. Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters Stud Guide and Intervention, pp. Skills Practice, p. Practice, p. 6 Reading to Learn Mathematics, p. 7 Enrichment, p. 8 Parent and Student Stud Guide Workbook, p. 6 School-to-Career Masters, p. Transparencies -Minute Check Transparenc 6- Answer Ke Transparencies Technolog Interactive Chalkboard

10 Stud Tip Reading Math {t t 8} is read the set of all numbers t such that t is less than or equal to 8. The solution of the inequalit in Eample was epressed as a set. A more concise wa of writing a solution set is to use set-builder notation. The solution in set-builder notation is {t t 8}. The solution to Eample can also be represented on a number line The heav arrow pointing to the left shows that the inequalit includes all numbers less than 8. Eample Graph the Solution The dot at 8 shows that 8 is included in the inequalit. Solve 7. Then graph it on a number line. 7 riginal inequalit 7 Add to each side. Simplif. Since is the same as, the solution set is { }. Teach SLVE INEQUALITIES BY ADDITIN In-Class Eamples Power Point Solve s 6. Then check our solution. s 77 or {all numbers greater than 77} Solve 9. Then graph it on a number line. { } The circle at shows that is not included in the inequalit. SLVE INEQUALITIES BY SUBTRACTIN solve inequalities. The heav arrow pointing to the right shows that the inequalit includes all numbers greater than. Subtraction can also be used to Subtraction Propert of Inequalities Words If an number is subtracted from each side of a true inequalit, the resulting inequalit is also true. Smbols For all numbers a, b, and c, the Eample 7 8 following are true If a b, then a c b c.. If a b, then a c b c. SLVE INEQUALITIES BY SUBTRACTIN In-Class Eample Power Point Solve q. Then graph the solution. {q q 9} This propert is also true when and are replaced with and. Eample Solve b Subtracting Solve 9 r 6. Then graph the solution. 9 r 6 riginal inequalit 9 r Subtract 9 from each side. r Simplif. The solution set is {r r } nline Lesson Plans Lesson 6- Solving Inequalities b Addition and Subtraction 9 USA TDAY Education s nline site offers resources and interactive features connected to each da s newspaper. Eperience TDAY, USA TDAY s dail lesson plan, is available on the site and delivered dail to subscribers. This plan provides instruction for integrating USA TDAY graphics and ke editorial features into our mathematics classroom. Log on to Interactive Chalkboard PowerPoint Presentations This CD-RM is a customizable Microsoft PowerPoint presentation that includes: Step-b-step, dnamic solutions of each In-Class Eample from the Teacher Wraparound Edition Additional, Your Turn eercises for each eample The -Minute Check Transparencies Hot links to Glencoe nline Stud Tools Lesson 6- Solving Inequalities b Addition and Subtraction 9

11 In-Class Eamples 6 Power Point Solve n n. Then graph the solution. {n n } 6 0 Teaching Tip When students solve inequalities with variables on both sides, suggest that the subtract the term with the lesser coefficient from each side so the remaining coefficient of the variable will be positive. Write an inequalit for the sentence below. Then solve the inequalit. Seven times a number is greater than si times that number minus two. 7a 6a ; {a a } ENTERTAINMENT Alicia wants to bu season passes to two theme parks. If one season pass costs $.99, and Alicia has $00 to spend on passes, the second season pass must cost no more than what amount? The second season pass must cost no more than $.0. Concept Check Solving Inequalities Ask students what the one phrase is that the will never see in a verbal inequalit problem. Sample answer: equal, or is equal to lmpics Yulia Barsukova of the Russian Federation won the gold medal in rhthmic gmnastics at the 000 Summer lmpics in Sdne, and Yulia Raskina of Belarus won the silver medal. Source: 0 Chapter 6 Solving Linear Inequalities Terms with variables can also be subtracted from each side to solve inequalities. Eample Variables on Both Sides Solve p 7 6p. Then graph the solution. p 7 6p riginal inequalit p 7 p 6p p Subtract p from each side. 7 p Simplif. Since 7 p is the same as p 7, the solution set is {p p 7} Verbal problems containing phrases like greater than or less than can often be solved b using inequalities. The following chart shows some other phrases that indicate inequalities. Eample Write and Solve an Inequalit Write an inequalit for the sentence below. Then solve the inequalit. Four times a number is no more than three times that number plus eight. Four times is no three times a number more than that number plus eight. n n 8 n n 8 n n n 8 n n 8 riginal inequalit Subtract n from each side. Simplif. The solution set is {n n 8}. Eample 6 Inequalities less than greater than at most at least fewer than more than no more than no less than less than or greater than equal to or equal to Write an Inequalit to Solve a Problem LYMPICS Yulia Raskina scored a total of 9.8 points in the four events of rhthmic gmnastics. Yulia Barsukova scored 9.88 in the rope competition, in the hoop competition, and 9.96 in the ball competition. How man points did Barsukova need to score in the ribbon competition to surpass Raskina and win the gold medal? Words Barsukova s total must be greater than Raskina s total. Variable Let r Barsukova s score in the ribbon competition. Barsukova s total is greater than Raskina s total. Inequalit r 9.8 Differentiated Instruction ELL Verbal/Linguistic If students are having difficult choosing the correct smbol for the problem s wording, have them use the chart above to write the common inequalit phrases on inde cards and the appropriate inequalit smbol on the back of each card. As students solve verbal problems such as Eamples and 6, the can pick the card that has the same wording as the problem. The back of the card will reveal the appropriate inequalit smbol to use. 0 Chapter 6 Solving Linear Inequalities

12 Concept Check. Sample answers:,, 0 Guided Practice GUIDED PRACTICE KEY Eercises Eamples 0, 6 0. See pp. 6A 6D. Application indicates increased difficult Practice and Appl For See Eercises Eamples Etra Practice See page 8. Solve the inequalit r 9.8 riginal inequalit r 9.8 Simplif r Subtract from each side. r 9.89 Simplif. Barsukova needed to score more than 9.89 points to win the gold medal.. PEN ENDED List three inequalities that are equivalent to.. Compare and contrast the graphs of a and a. See margin.. Eplain what {b b } means. The set of all numbers b such that b is greater than or equal to.. Which graph represents the solution of m 7? a a. b c. d Solve each inequalit. Then check our solution, and graph it on a number line.. a {a a } 6. 9 b {b b } 7. t 7 {t t } r r 0. 7p 6p {.6} {r r 6.7} {p p } Define a variable, write an inequalit, and solve each problem. Then check our solution.. Sample answer: Let n the number.. A number decreased b 8 is at most. n 8 ; {n n }. A number plus 7 is greater than. n 7 ; {n n }. HEALTH Chapa s doctor recommended that she limit her fat intake to no more than 60 grams per da. This morning, she ate two breakfast bars with grams of fat each. For lunch she ate pizza with grams of fat. If she follows her doctor s advice, how man grams of fat can she have during the rest of the da? no more than g Match each inequalit with its corresponding graph.. d a f b. 6. a c c d e e b f Lesson 6- Solving Inequalities b Addition and Subtraction Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 6. write a solution set in set-builder notation with an eplanation of how to read it. include an eplanation of the difference between an inequalit graph with a dot and one with a circle. include an other item(s) that the find helpful in mastering the skills in this lesson. About the Eercises rganization b bjective Solve Inequalities b Addition: Solve Inequalities b Subtraction: dd/even Assignments Eercises are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: odd, odd, 6 76 Average: odd, 6 76 Advanced: even, -68 (optional: 69 76) Unlocking Misconceptions Rewriting Inequalities An equation such as can be rewritten as because of the Smmetric Propert of Equalit. Because of this propert, students ma incorrectl assume that the can rewrite an inequalit such as as. Remind students that the inequalit sign alwas points to the smaller value. In, it points to, so to write the epression with on the left, use. Answer. In both graphs, the line is darkened to the left. In the graph of a, there is a circle at to indicate that is not included in the graph. In the graph of a, there is a dot at to indicate that is included in the graph. Lesson 6- Solving Inequalities b Addition and Subtraction

13 Stud Stud Guide and and Intervention Intervention, p. (shown) and p. 6- NAME DATE PERID Solving Inequalities b Addition and Subtraction Solve Inequalities b Addition Addition can be used to solve inequalities. If an number is added to each side of a true inequalit, the resulting inequalit is also true. Addition Propert of Inequalities For all numbers a, b, and c, if a b, then a c b c, and if a b, then a c b c. The propert is also true when and are replaced with and. Eample Solve 8 6. Eample Solve a a. Then Then graph it on a number line. graph it on a number line. 8 6 riginal inequalit a a riginal inequalit Add 8 to each side. a a a a Add a to each side. Simplif. a Simplif. a a is the same as a. The solution in set-builder notation is { }. The solution in set-builder notation is {a a }. Number line graph: Number line graph: 0 Eercises 0 6 Solve each inequalit. Then check our solution, and graph it on a number line.. t 6 {t t 8}. n 6 {n n 8}. 6 g {g g 9} n 8 {n n }. { 0} 6. 6 s 8 {s s } Lesson See pp. 6A 6D. 8. { 8}. {w w }. {v v }. {a a }. {h h 0.6}. { 0.6} 6. a a 8 7. p p 9 Solve each inequalit. Then check our solution, and graph it on a number line. 0. t 8 {t t }. d 7 {d d }. n 7 {n n }. s {s s }. g {g g }. 8 r {r r } 6. q 7 {q q } 7. m {m m } f f {f f } 0. b b {b b }. w w. v (). a (). 0. h (0.) a 8 7. p 9 8. If d 7, then complete each inequalit. a. d? b. d? 0 8 c. d? 7 9. If z 0, then complete each inequalit. a. z? b. z? 7 c. z? Solve each inequalit. Then check our solution n 0.n 9. 8k 9k { 8} {n n } {k k } z. b b { } z z {b b } Define a variable, write an inequalit, and solve each problem. Then check our solution.. Sample answer: Let n the number.. A number decreased b is less than. n ; {n n 8} 0. The difference of two numbers is more than, and one of the numbers is. n ; {n n }. Fort is no greater than the difference of a number and. 0 n ; {n n } Skills Practice Practice, p. and (Average) Practice, p. 6 (shown) 6- NAME DATE PERID Solving Inequalities b Addition and Subtraction Match each inequalit with its corresponding graph.. 8 b a d b a c c d n (8); {n n 8}. n n ; {n n }. n ( 7) 8; {n n } Define a variable, write an inequalit, and solve each problem. Then check our solution. 0. Sample answer: Let n the number. 0. The sum of a number and is at least 7. n 7; {n n }. A number decreased b is less than. n ; {n n 8}. Thirt is no greater than the sum of a number and 8.. Twice a number is more than the sum of that number and.. The sum of two numbers is at most 8, and one of the numbers is 7.. Four times a number is less than or equal to the sum of three times the number and. n n (); {n n } 6. BILGY Adult Nile crocodiles weigh up to 00 pounds. If a oung Nile crocodile weighs 7 pounds, how man pounds might it be epected to gain in its lifetime? no more than 0 lb Solve each inequalit. Then check our solution, and graph it on a number line.. r () {r r 7} 6. 8 { 8} n. {n n.} 8.. { 0.} 0 9. z z z 0. c c c 0 Define a variable, write an inequalit, and solve each problem. Then check our solution.. Sample answer: Let n the number.. The sum of a number and 7 is no less than 6. n 7 6; {n n 9}. Twice a number minus is less than three times the number. n n; {n n }. Twelve is at most a number decreased b 7. n 7; {n n 9} 0. Eight plus four times a number is greater than five times the number. 8 n n; {n n 8}. ATMSPHERIC SCIENCE The troposphere etends from the earth s surface to a height of 6 miles, depending on the location and the season. If a plane is fling at an altitude of.8 miles, and the troposphere is 8.6 miles deep in that area, how much higher can the plane go without leaving the troposphere? no more than.8 mi 6. EARTH SCIENCE Mature soil is composed of three laers, the uppermost being topsoil. Jamal is planting a bush that needs a hole 8 centimeters deep for the roots. The instructions suggest an additional 8 centimeters depth for a cushion. If Jamal wants to add even more cushion, and the topsoil in his ard is 0 centimeters deep, how much more cushion can he add and still remain in the topsoil laer? no more than cm Reading to Learn Mathematics, p NAME DATE PERID Reading to Learn Mathematics Solving Inequalities b Addition and Subtraction Pre-Activit How are inequalities used to describe school sports? Read the introduction to Lesson 6- at the top of page 8 in our tetbook. Use the information in the graph to write an inequalit statement about participation in two sports. Sample answer: For softball and track and field,,009,87 Reading the Lesson Rewrite our inequalit statement to show that 0 schools added both of the sports. Is the statement still true? Sample answer:,09,67; es Write the letter of the graph that matches each inequalit.. b a. 0. d b. 0. a c. 0. c d. 0. Use the chart to write a sentence that could be described b the inequalit n n 7. Then solve the inequalit. Inequalities than greater less than at most at least fewer than more than no more than no less than less than or equal to greater than or equal to Sample answer: Three times a number is at least two times the number plus 7; n 7 ELL Biolog ne common species of bees is the honebee. A honebee colon ma have 60,000 to 80,000 bees. Source: Penn State, Cooperative Etension Service NAME DATE PERID 6- Enrichment, p. 8 Triangle Inequalities Chapter 6 Solving Linear Inequalities Recall that a line segment can be named b the letters of its endpoints. Line segment AB (written as AB) has points A and B for endpoints. The length of AB is written without the bar as AB. AB BC ma mb The statement on the left above shows that AB is shorter than BC. The statement on the right above shows that the measure of angle A is less than that of angle B. 7. ASTRNMY There are at least 00 billion stars in the Milk Wa. If 00 of these stars can be seen in a rural area without the aid of a telescope, how man stars in the gala cannot be seen in this wa? at least 99,999,998,900 stars 8. BILGY There are 00 species of bees and more than 600,000 species of insects. How man species of insects are not bees? more than 96,00 species 9. BANKING Cit Bank requires a minimum balance of $00 to maintain free checking services. If Mr. Haashi knows he must write checks for $00 and $97, how much mone should he have in his account before writing the checks? at least $77 0. GEMETRY The length of the base of the triangle at the right is less than the height of the triangle. What are the possible values of? more than 8 in.. SHPPING Terrell has $6 to spend at the mall. He bought a T-shirt for $8 and a belt for $. If Terrell still wants to bu a pair of jeans, how much can he spend on the jeans? no more than $. SCCER The Centerville High School soccer team plas 8 games in the season. The team has a goal of winning at least 60% of its games. After the first three weeks of the season, the team has won games. How man more games must the team win to meet their goal? at least 7 more games in. in. Helping You Remember 6. Teaching someone else can help ou remember something. Eplain how ou would teach another student who missed class to solve the inequalit. Subtract from each side. Simplif. These three inequalities are true for an triangle ABC, no matter how long the sides. a. AB BC AC b. If AB AC, then mc mb. c. If mc mb, then AB AC. B A C Use the three triangle inequalities for these problems. iangle D Chapter 6 Solving Linear Inequalities

14 SL/EC Practice Standardized Test Practice Maintain Your Skills Mied Review 66. {(, ), (, 6), (, 0)} 67. {(, 8), (, ), (, )} 68. {(, 8), (, 0), (, )} Getting Read for the Net Lesson. CRITICAL THINKING Determine whether each statement is alwas, sometimes, or never true. a. If a b and c d, then a c b d. alwas b. If a b and c d, then a c b d. never c. If a b and c d, then a c b d. sometimes HEALTH For Eercises and, use the following information. Hector s doctor told him that his cholesterol level should be below 00. Hector s cholesterol is.. Let p represent the number of points Hector should lower his cholesterol. Write an inequalit with p on one side. p 00. Solve the inequalit. {p p } 6. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 6A 6D. How are inequalities used to describe school sports? Include the following in our answer: an inequalit describing the number of schools needed to add girls track and field so that the number is greater than the number of schools currentl participating in girls basketball. 7. Which inequalit is not equivalent to? C A 7 B 6 C D 8. Which statement is modeled b n 6? A A The sum of a number and si is at least five. B The sum of a number and si is at most five. C The sum of a number and si is greater than five. D The sum of a number and si is no greater than five. 9. Would a scatter plot for the relationship of a person s height to the person s grade on the last math test show a positive, negative, or no correlation? (Lesson -7) no Write an equation in slope-intercept form of the line that passes through the given point and is parallel to the graph of each equation. (Lesson -6) 60. (, ); 6. (0, ); 6. (, ); 6 Find the net two terms in each sequence. (Lesson -8) 6. 7,, 9,, 6., 8, 7, 9, 6., 6,,,, 7, 8, 96 Solve each equation if the domain is {,, }. (Lesson -) PREREQUISITE SKILL Solve each equation. (For review of multiplication and division equations, see Lesson -.) t 69. 6g m g.. 7. a p.7. Lesson 6- Solving Inequalities b Addition and Subtraction Assess pen-ended Assessment Modeling Draw a large number line on an overhead transparenc. Using a washer or a coin, and a ribbon or a strip of paper, graph several different inequalities on the overhead. Use the coin to indicate a dot on the number line and the washer to indicate a circle. The ribbon is the ra representing points greater or less than the starting value. Have students identif the inequalities ou graph, identifing the significance of the circle or dot. Intervention Students should alwas check their solutions, but the often hurr to finish their assignments and omit this step. Remind students that checking solutions is especiall important with inequalities because the direction of the inequalit sign often gets changed when writing solutions in set-builder notation. New Getting Read for Lesson 6- PREREQUISITE SKILL In Lesson 6-, students learn how to solve inequalities using multiplication and division. The process is almost identical to the process for solving equations using multiplication and division. Use Eercises to determine our students familiarit with solving these tpes of equations. Lesson 6- Solving Inequalities b Addition and Subtraction

15 Algebra Activit A Preview of Lesson 6- Getting Started bjective Use algebra tiles to solve inequalities. Materials algebra tiles equation mat self-adhesive notes Teach You ma wish to do the eample as a demonstration. Make sure the inequalit sign on the self-adhesive note is pointed in the correct direction to match the inequalit. nce the have isolated the tiles, remind students to separate the tiles into equal groups to correspond to the number of tiles. If the tiles end up on the right side of the inequalit, students ma rotate the mat 80 degrees to read the inequalit with the variable on the left side. Assess Have students work in small groups for Eercises 7. bserve to determine if the are able to verbalize the activities in Eercises. Students should conclude after Eercises 6 7 that when the multipl or divide both sides of an inequalit b a negative number, the direction of the inequalit sign changes. Solving Inequalities You can use algebra tiles to solve inequalities. Solve 6. Model the inequalit. Remove zero pairs. Chapter 6 Solving Linear Inequalities A Preview of Lesson 6- Virginia SL STANDARD A. The student will justif steps used in simplifing epressions and solving equations and inequalities. Justifications will include the use of concrete objects; pictorial representations; and the properties of real numbers, equalit, and inequalit. Use a self-adhesive note to cover the equals sign on the equation mat. Then write a smbol on the note. Model the inequalit. 6 6 Add 6 negative tiles to each side to isolate the tiles. Remove the zero pairs. Group the tiles. Model and Analze Use algebra tiles to solve each inequalit.. { }. 8 { }. 6 { }. { }. In Eercises, is the coefficient of in each inequalit positive or negative? negative 6. Compare the inequalit smbols and locations of the variable in Eercises with those in their solutions. What do ou find? 6 7. See pp. 6A 6D. 7. Model the solution for 6. What do ou find? How is this different from solving 6? Remove zero pairs. Since ou do not want to solve for a negative tile, eliminate the negative tiles b adding positive tiles to each side. Remove the zero pairs. Separate the tiles into groups. 6 or Stud Notebook You ma wish to have students summarize this activit and what the learned from it. Resource Manager Teaching Algebra with Manipulatives pp. 0 (master for algebra tiles) p. 6 (master for equation mat) p. (student recording sheet) Glencoe Mathematics Classroom Manipulative Kit algebra tiles equation mat Chapter 6 Solving Linear Inequalities

16 Solving Inequalities b Multiplication and Division Lesson Notes Virginia SL STANDARD A. The student will solve multistep linear equations and inequalities in one variable, solve literal equations (formulas) for a given variable, and appl these skills to solve practical problems. Graphing calculators will be used to confirm algebraic solutions. Solve linear inequalities b using multiplication. Solve linear inequalities b using division. SLVE INEQUALITIES BY MULTIPLICATIN If each side of an inequalit is multiplied b a positive number, the inequalit remains true ()? () Multipl each side b. ()? 9() Multipl each side b This is not true when multipling b negative numbers. 6 8 ()? () Multipl each side b. 6()? 8() Multipl each side b If each side of an inequalit is multiplied b a negative number, the direction of the inequalit smbol changes. These eamples illustrate the Multiplication Propert of Inequalities. Words Smbols are inequalities important in landscaping? Isabel Franco is a landscape architect. To beautif a garden, she plans to build a decorative wall of either bricks or blocks. Each brick is inches high, and each block is inches high. Notice that. in. in. in. Awall bricks high would be lower than a wall blocks high.? 8 8 in. Multipling b a Positive Number If each side of a true inequalit is multiplied b the same positive number, the resulting inequalit is also true. If a and b are an numbers and c is a positive number, the following are true. If a b, then ac bc, and if a b, then ac bc. Focus -Minute Check Transparenc 6- Use as a quiz or a review of Lesson 6-. Mathematical Background notes are available for this lesson on p. 6C. Building on Prior Knowledge The process of solving inequalities is identical to the process of solving equations ecept when multipling or dividing b a negative value. Students should understand that the do not have to learn a whole new process, but just a special rule when using negatives. are inequalities important in landscaping? Ask students: What is similar about the two walls? What is different? Both walls are rows high. Each row of bricks is inches high and each row of blocks is inches high. B what number are both sides of the inequalit multiplied to ield 8? After ou multipl both sides of the inequalit b the same number to ield 8, is the inequalit still true? Eplain. Yes; is less than 8. Lesson 6- Solving Inequalities b Multiplication and Division Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters Stud Guide and Intervention, pp. 9 0 Skills Practice, p. Practice, p. Reading to Learn Mathematics, p. Enrichment, p. Assessment, p. 9 Parent and Student Stud Guide Workbook, p. 7 School-to-Career Masters, p. Transparencies -Minute Check Transparenc 6- Answer Ke Transparencies Technolog Interactive Chalkboard Lesson - Lesson Title

17 Teach SLVE INEQUALITIES BY MULTIPLICATIN In-Class Eamples Power Point Teaching Tip Remind students b that is the same as b. To 7 7 isolate b, multipl b the reciprocal of, which is 7. 7 g Solve. Then check our solution. {g g 6} Solve d 6. {d d 8} Write an inequalit for the sentence below. Then solve the inequalit. Four-fifths of a number is at most twent. r 0; {r r } Words Smbols Multipling b a Negative Number If each side of a true inequalit is multiplied b the same negative number, the direction of the inequalit smbol must be reversed so that the resulting inequalit is also true. If a and b are an numbers and c is a negative number, the following are true. If a b, then ac bc, and if a b, then ac bc. This propert also holds for inequalities involving and. You can use this propert to solve inequalities. Eample Multipl b a Positive Number Solve b 7. Then check our solution. 7 b riginal inequalit b (7) (7) Multipl each side b 7. Since we multiplied b a positive 7 number, the inequalit smbol stas the same. b 7 CHECK To check this solution, substitute 7, a number less than 7, and a number greater than 7 into the inequalit. Let b 7. Let b 0. Let b 0. 7? 0? 0? The solution set is {b b 7}. Stud Tip Common Misconception A negative sign in an inequalit does not necessaril mean that the direction of the inequalit should change. For eample, when solving, do not change 6 the direction of the inequalit. Eample Solve p. p Multipl b a Negative Number riginal inequalit p () Multipl each side b and change to. Eample p The solution set is {p p }. Write and Solve an Inequalit Write an inequalit for the sentence below. Then solve the inequalit. ne fourth of a number is less than 7. ne fourth of a number is less than 7. n 7 n 7 () n ()(7) n 8 riginal inequalit Multipl each side b and do not change the inequalit s direction. The solution set is {n n 8}. 6 Chapter 6 Solving Linear Inequalities 6 Chapter 6 Solving Linear Inequalities

18 SLVE INEQUALITIES BY DIVISIN Dividing each side of an inequalit b the same number is similar to multipling each side of an equalit b the same number. Consider the inequalit 6. Divide each side b. Divide each side b ? 6 ()? () Since each side is divided b a positive number, the direction of the inequalit smbol remains the same. These eamples illustrate the Division Propert of Inequalities. Words Smbols Words Smbols Eample Solve h 9. Since each side is divided b a negative number, the direction of the inequalit smbol is reversed. Dividing b a Positive Number If each side of a true inequalit is divided b the same positive number, the resulting inequalit is also true. If a and b are an numbers and c is a positive number, the following are true. If a b, then a c b c, and if a b, then a c b c. Divide b a Positive Number Dividing b a Negative Number If each side of a true inequalit is divided b the same negative number, the direction of the inequalit smbol must be reversed so that the resulting inequalit is also true. If a and b are an numbers and c is a negative number, the following are true. If a b, then a c b c, and if a b, then a c b c. This propert also holds for inequalities involving and. SLVE INEQUALITIES BY DIVISIN Assessment Challenge students to eplain how the might alread know how to solve inequalities b division. Students should suggest that since the know how to solve inequalities b multiplication, and since division is the same as multipling b a reciprocal, then the alread know how to solve inequalities b division. New In-Class Eample Teaching Tip Point out to students that the rules for the Division Propert of Inequalities state that each side of an inequalit can be divided b a positive or negative number. In neither case is zero included because division b zero is an undefined operation. Solve s 60. {s s } Power Point h 9 riginal inequalit h 9 Divide each side b and do not change the direction of the inequalit sign. h 6. CHECK Let h 6.. Let h 7. Let h 6. h 9 h 9 h 9 (6.)? 9 (7)? 9 (6)? The solution set is {h h 6.}. Since dividing is the same as multipling b the reciprocal, there are two methods to solve an inequalit that involve multiplication. Lesson 6- Solving Inequalities b Multiplication and Division 7 Differentiated Instruction Kinesthetic Have students write an inequalit involving a negative coefficient of the variable on their paper, using a self-adhesive note for the inequalit smbol, such as. Tell them that the are going to change all the signs in the inequalit, so everthing is its opposite. The epression becomes. Students now can divide without having to worr about the inequalit sign. Lesson 6- Solving Inequalities b Multiplication and Division 7

19 In-Class Eamples 6 Power Point Teaching Tip Point out to students that it ma be easier to solve an inequalit using division when the inequalit involves whole numbers, and easier to solve using multiplication b reciprocals when the inequalit involves fractions. Solve 8q 6 using two methods. {q q 7} Teaching Tip Another wa to check the solution is to rework the problem using a different method. Which inequalit does not have the solution { 6}? B A B 6 7 C 6 D 8 SL/EC Practice Standardized Test Practice Eample Divide b a Negative Number Solve t 7 using two methods. Method t 7 t 7 t Method Divide. t 7 riginal inequalit Divide each side b and change to. Simplif. Multipl b the multiplicative inverse. riginal inequalit (t) 7 Multipl each side b and change to. t Simplif. The solution set is {t t }. You can use the Multiplication Propert and the Division Propert for Inequalities to solve standardized test questions. Eample 6 The Word not Multiple-Choice Test Item Which inequalit does not have the solution { }? A 7 B 0 C 7 7 D Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 6. cop the rules for the Multiplication Propert of Inequalities and Division Propert of Inequalities. include an other item(s) that the find helpful in mastering the skills in this lesson. Answer. You could solve the inequalit b multipling each side b or 7 b dividing each side b 7. In either case, ou must reverse the direction of the inequalit smbol. Test-Taking Tip Alwas look for the word not in the questions. This indicates that ou are looking for the one incorrect answer, rather than looking for the one correct answer. The word not is usuall in italics or uppercase letters to draw our attention to it. Concept Check. Sample answer: Three fourths of a number is greater than 9. 8 Chapter 6 Solving Linear Inequalities Standardized Test Practice Read the Test Item You want to find the inequalit that does not have the solution set { }. Solve the Test Item Consider each possible choice. A 7 B C 7 7 D (7) () () The answer is C.. Eplain wh ou can use either the Multiplication Propert of Inequalities or the Division Propert of Inequalities to solve 7r 8. See margin.. PEN ENDED Write a problem that can be represented b the inequalit c 9. Eample 6 Some questions can be answered without solving each equation or inequalit given. Have students eamine each inequalit in Eample 6 to determine what inequalit sign should be included in the solution set without working it out. For A and D, the sign becomes. For B, it stas. C is the remaining choice. 8 Chapter 6 Solving Linear Inequalities

20 . Ilonia; when ou divide each side of an inequalit b a negative number, ou must reverse the direction of the inequalit smbol. Guided Practice GUIDED PRACTICE KEY Eercises Eamples,, 0, 6 9,,, 6 Standardized Test Practice indicates increased difficult Practice and Appl For See Eercises Eamples 8, 9 9 8,,, 6 Etra Practice See page 8.. FIND THE ERRR Ilonia and Zachar are solving 9b 8. Who is correct? Eplain our reasoning.. Which statement is represented b 7n? a a. Seven times a number is at least. b. Seven times a number is greater than. c. Seven times a number is at most. d. Seven times a number is less than.. Which inequalit represents five times a number is less than? c a. n b. n c. n d. n Solve each inequalit. Then check our solution. t 6. g b 9 9. f 9 9 {g g } {t t 08} {b b.} {f f 0.6} Define a variable, write an inequalit, and solve each problem. Then check our solution. 0. Sample answer: Let n the number. 0. The opposite of four times a number is more than. n ; {n n }. Half of a number is at least 6. n 6; {n n }. Which inequalit does not have the solution set { }? B A Ilonia 9b 8 9b b Zachar 9b 8 9b b 0 B 6 C D Match each inequalit with its corresponding statement.. n 0 d a. Five times a number is less than or equal to ten.. n 0 a b. ne fifth of a number is no less than ten.. n 0 e c. Five times a number is less than ten. 6. n 0 f d. ne fifth of a number is greater than ten. 7. n 0 b e. Five times a number is greater than ten. 8. n 0 c f. Negative five times a number is less than ten. FIND THE ERRR Tell students to look first at the solutions from Ilonia and Zachar. The onl difference is the inequalit smbol. Since the solution involves division b a negative number, the inequalit smbol of the solution must be reversed from the original inequalit. About the Eercises rganization b bjective Solve Inequalities b Multiplication: 0 Solve Inequalities b Division: 0 dd/even Assignments Eercises 0 are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: 9 odd,, 9,,, 7, 9,, 78 Average: odd,,, 78 Advanced: even, 7 (optional: 7 78) All: Practice Quiz ( 0) Solve each inequalit. Then check our solution. 9. See margin. 9. 6g 0. 7t 8. d 8. 6z 6. m 7. b r a v 6 9. q 0. p 0..w s 6.. c 7. m Lesson 6- Solving Inequalities b Multiplication and Division 9 Answers 9. {g g } 0. {t t }. {d d 6}. {z z }. {m m }. {b b 0}. {r r 9} 6. {a a 99} 7. { } 8. {v v 9} 9. {q q } 0. {p p }. {w w.7}. {s s 8}. c c 0. m m Lesson 6- Solving Inequalities b Multiplication and Division 9

21 Stud Stud Guide and and Intervention Intervention, p. 9 (shown) and p NAME DATE PERID Solving Inequalities b Multiplication and Division Solve Inequalities b Multiplication If each side of an inequalit is multiplied b the same positive number, the resulting inequalit is also true. However, if each side of an inequalit is multiplied b the same negative number, the direction of the inequalit must be reversed for the resulting inequalit to be true. Multiplication Propert of Inequalities For all numbers a, b, and c, with c 0,. if c is positive and a b, then ac bc; if c is positive and a b, then ac bc;. if c is negative and a b, then ac bc; if c is negative and a b, then ac bc. The propert is also true when and are replaced with and. Eample Solve. Eample Solve k. 8. Solve 8. Then graph the solution. { }; See margin for graph. 6. Solve m 9. Then graph the solution. {m m }; See margin for graph. 7. If a 7, then complete each inequalit. a. a?. b. a? c.? a 6 8. If t, then complete each inequalit. a. t? 0. b. 8t? c.? t 8 riginal equation 8 (8) (8) Multipl each side b 8; change to Simplif. The solution is { 96}. Eercises k riginal equation k Multipl each side b. k 0 Simplif. The solution is {k k 0}. Solve each inequalit. Then check our solution. n p h. 6 { } {n n 00} {h h } {p p 6} m h 0. n 0 6. b {n n 0} b b m m {h h.0} g 9p n a {g g 0} p p {n n } {a a } Define a variable, write an inequalit, and solve each problem. Then check our solution.. Sample answer: Let n the number.. Half of a number is at least. n ; {n n 8}. The opposite of one-third a number is greater than 9. n 9; {n n 7}. ne fifth of a number is at most 0. n 0; {n n 0} Skills Practice Practice, p. and (Average) Practice, p. (shown) 6- NAME DATE PERID Solving Inequalities b Multiplication and Division Match each inequalit with its corresponding statement.. n d a. Negative four times a number is less than five.. n f b. Four fifths of a number is no more than five.. n e c. Four times a number is fewer than five.. n b d. Negative four times a number is no less than five.. n c e. Four times a number is at most five. 6. n a f. Four fifths of a number is more than five. Solve each inequalit. Then check our solution. a s h p {a a 70} {h h } {s s 96} {p p } 0 9. n. t. m 6. k 0 {n n 8} {t t } {m m 0} {k k }. b c {b b 0.} {c c 0} { 0} {j j 9.} v u. f 8 {v v 8} {u u 0} 8 f f Define a variable, write an inequalit, and solve each problem. Then check our solution.. Sample answer: Let n the number.. Negative three times a number is at least 7. n 7; {n n 9}. Two thirds of a number is no more than 0. n 0; {n n }. Negative three fifths of a number is less than 6. n 6; {n n 0} 6. FLDING A river is rising at a rate of inches per hour. If the river rises more than feet, it will eceed flood stage. How long can the river rise at this rate without eceeding flood stage? no more than 8 h 7. SALES Pet Supplies makes a profit of $.0 per bag on its line of natural dog food. If the store wants to make a profit of no less than $, how man bags of dog food does it need to sell? at least 90 bags Reading to Learn Mathematics, p. 6- NAME DATE PERID Reading to Learn Mathematics Solving Inequalities b Multiplication and Division Pre-Activit Wh are inequalities important in landscaping? Read the introduction to Lesson 6- at the top of page in our tetbook. Would a wall 6 bricks high be lower than a wall 6 blocks high? Wh? es; 6() 6() Reading the Lesson Would a wall n bricks high be lower than a wall n blocks high? Eplain. es; When one quantit is less than another quantit, multipling both quantities b the same positive number does not change the truth of the inequalit.. Write an inequalit that describes each situation. a. A number n divided b 8 is greater than. n 8 b. Twelve times a number k is at least 7. k 7 j ELL 7 Lesson n 90; {n n 60} Zoos Dr. Harr Wegeforth founded the San Diego Zoo in 96 with just 0 animals. Toda, the zoo has over 800 animals. Source: a. Sample answer:, but 9. b. Sample answer: and, but. Define a variable, write an inequalit, and solve each problem. Then check our solution. 9. Sample answer: Let n the number. 9. Seven times a number is greater than 8. 7n 8; {n n } 0. Negative seven times a number is at least. 7n ; {n n }. Twent-four is at most a third of a number. n; {n n 7}. Two thirds of a number is less than. n ; {n n.}. Twent-five percent of a number is greater than or equal to 90.. Fort percent of a number is less than or equal to. 0.0n ; {n n.}. GEMETRY The area of a rectangle is less than 8 square feet. The length of the rectangle is 0 feet. What is the width of the rectangle? less than ft 6. FUND-RAISING The Middletown Marching Mustangs want to make at least $000 on their annual mulch sale. The band makes $.0 on each bag of mulch that is sold. How man bags of mulch should the band sell? at least 800 bags 0 Chapter 6 Solving Linear Inequalities 7. LNG-DISTANCE CSTS Juan s long-distance phone compan charges him 9 for each minute or an part of a minute. He wants to call his friend, but he does not want to spend more than $.0 on the call. How long can he talk to his friend? no more than 7 min 8. EVENT PLANNING The Countr Corner Reception Hall does not charge a rental fee as long as at least $000 is spent on food. Shaniqua is planning a class reunion. If she has chosen a buffet that costs $8.9 per person, how man people must attend the reunion to avoid a rental fee for the hall? at least 9 people 9. LANDSCAPING Matthew is planning a circular flower garden with a low fence around the border. If he can use up to 8 feet of fence, what radius can he use for the garden? (Hint: C r) up to about 6 ft 0. DRIVING Average speed is calculated b dividing distance b time. If the speed limit on the interstate is 6 miles per hour, how far can a person travel legall in hours? no more than 97 mi. ZS The earl membership to the San Diego Zoo for a famil with adults and children is $. The regular admission to the zoo is $8 for each adult and $8 for each child. How man times should such a famil plan to visit the zoo in a ear to make a membership less epensive than paing regular admission? at least times. CRITICAL THINKING Give a countereample to show that each statement is not alwas true. a. If a b, then a b. b. If a b and c d, then ac bd.. CITY PLANNING The cit of Santa Clarita requires that a parking lot can have no more than 0% of the parking spaces limited to compact cars. If a certain parking lot has spaces for compact cars, how man spaces must the lot have to conform to the code? at least 7 spaces c. A number divided b 0 is less than or equal to 0. (0) 0 d. Three fifths of a number n is at most. n e. Nine is greater than or equal to one half of a quantit m. 9 m. Use words to tell what each inequalit sas. a. 6n is less than 6 times a number n. t b. A number t divided b is greater than or equal to. c. times a number is at most. Helping You Remember. In our own words, write a rule for multipling and dividing inequalities b positive and negative numbers. Sample answer: When ou multipl or divide each side of a true inequalit b a positive number, the result is true. When ou multipl or divide a true inequalit b a negative number, ou must reverse the direction of the inequalit sign. 6- Enrichment, p. The Maa NAME DATE PERID 'I'he Maa were a Native American people who lived from about 00 B.C. to about 00 A.D. in the region that toda encompasses much of Central America and southern Meico. Their man accomplishments include eceptional architecture, potter, painting, and sculpture, as well as significant advances in the fields of astronom and mathematics. The Maa developed a sstem of numeration that was based on the number twent. The basic smbols of this sstem are shown in the table at the right. The places in a Maan numeral are written verticall the bottom place represents ones, the place above represents twenties, the place above that represents 0 0, or four hundreds, and so on. For instance, this is how to write the number 997 in Maan numerals Answers Chapter 6 Solving Linear Inequalities

22 Maintain Your Skills P SL/EC Practice Standardized Test Practice Mied Review Getting Read for the Net Lesson. CIVICS For a candidate to run for a count office, he or she must submit a petition with at least 6000 signatures of registered voters. Usuall onl 8% of the signatures are valid. How man signatures should a candidate seek on a petition? at least 709 signatures. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. Wh are inequalities important in landscaping? Include the following in our answer: an inequalit representing a brick wall that can be no higher than feet, and an eplanation of how to solve the inequalit. 6. The solution set for which inequalit is not represented b the following graph? B A B C 9 D.. 7. Solve 7 8 t. C t t 6 A t t 6 t t 6 B C D t t Solve each inequalit. Then check our solution, and graph it on a number line. (Lesson 6-) See margin for graphs. 8. s 7 {s s 9} 9. g {g g 7}60. 7 n {n n } See 6. Draw a scatter plot that shows a positive correlation. (Lesson -7) pp. 6A 6D. Write an equation of the line that passes through each pair of points. (Lesson -) 6. (, ), (, ) 0 6. (, ), (, ) 6. (, ), (, ) If h(), find each value. (Lesson -6) 9 6. h() h() h(w) w 68. h(r 6) r 6 Solve each proportion. (Lesson -6) t w PREREQUISITE SKILL Solve each equation. (To review multi-step equations, see Lessons - and -.) t g a 6 9a (7a 8) 78. (p ) 7(p ) Lessons 6- and 6- ractice Quiz. See pp. 6A 6D for graphs. Solve each inequalit. Then check our solution, and graph it on a number line. (Lesson 6-). h 6. r. p 9. a. 7g 6g {h h } {r r } {p p } {a a } {g g } Solve each inequalit. Then check our solution. (Lesson 6-) 6 0. See pp. 6A 6D. 6. z 0 7. v q 9. 6 r 0. w 6 Assess pen-ended Assessment Writing Have students write a one-paragraph summar of what the think is the most important thing to remember about solving inequalities b multiplication or division. Students will likel suggest that the most important thing to remember is to change the direction of the inequalit smbol when multipling or dividing b negative numbers. Ask student volunteers to read their paragraphs to the class. Getting Read for Lesson 6- PREREQUISITE SKILL Lesson 6- presents multi-step inequalities, which builds on what students have learned about solving multi-step equations. Use Eercises 7 78 to determine our students familiarit with solving multi-step equations. Assessment ptions Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons 6- and 6-. Lesson numbers are given to the right of the eercises or instruction lines so students can review concepts not et mastered. Quiz (Lessons 6- and 6-) is available on p. 9 of the Chapter 6 Resource Masters. Lesson 6- Solving Inequalities b Multiplication and Division Answers. Inequalities can be used to compare the heights of walls. Answers should include the following. If represents the number of bricks and the wall must be no higher than ft or 8 in., then 8. To solve this inequalit, divide each side b and do not change the direction of the inequalit. The wall must be 6 bricks high or fewer Lesson 6- Solving Inequalities b Multiplication and Division

23 Lesson Notes Focus -Minute Check Transparenc 6- Use as a quiz or a review of Lesson 6-. Mathematical Background notes are available for this lesson on p. 6C. Building on Prior Knowledge Solving multi-step inequalities is no different from solving multistep equations ecept when multipling or dividing b a negative value. Students should understand that the don t have to learn a whole new process, but just a special rule when using negatives. are linear inequalities used in science? Ask students: What would the inequalit F represent? The temperatures at which chlorine is not a gas. What epression was substituted for F to represent the temperature of the boiling point of chlorine in degrees 9 Celsius? C Solving Multi-Step Inequalities Virginia SL STANDARD A. The student will solve multistep linear equations and inequalities in one variable, solve literal equations (formulas) for a given variable, and appl these skills to solve practical problems. Graphing calculators will be used to confirm algebraic solutions. Solve linear inequalities involving more than one operation. Solve linear inequalities involving the Distributive Propert. The boiling point of a substance is the temperature at which the element changes from a liquid to a gas. The boiling point of chlorine is F. That means chlorine will be a gas for all temperatures greater than F. If F represents temperature in degrees Fahrenheit, the inequalit F represents the temperatures for which chlorine is a gas. If C represents degrees Celsius, then F 9 C. You can solve 9 C to find the temperatures in degrees Celsius for which chlorine is a gas. SLVE MULTI-STEP INEQUALITIES The inequalit 9 C involves more than one operation. It can be solved b undoing the operations in the same wa ou would solve an equation with more than one operation. Eample Solve a Real-World Problem SCIENCE Find the temperatures in degrees Celsius for which chlorine is a gas. 9 C riginal inequalit 9 C 9 C 6 Simplif. 9 9 C 9 (6) Subtract from each side. Multipl each side b 9. C Simplif. Chlorine will be a gas for all temperatures greater than C. When working with inequalities, do not forget to reverse the inequalit sign whenever ou multipl or divide each side b a negative number. Eample are linear inequalities used in science? Source: World Book Encclopedia Inequalit Involving a Negative Coefficient Solve 7b 9 6. Then check our solution. 7b 9 6 riginal inequalit 7b Subtract 9 from each side. 7b Simplif. 7 b 7 7 Divide each side b 7 and change to. b Simplif. Boiling Points argon chlorine bromine water iodine 0 F F 8 F F 6 F Chapter 6 Solving Linear Inequalities Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters Stud Guide and Intervention, pp. 6 Skills Practice, p. 7 Practice, p. 8 Reading to Learn Mathematics, p. 9 Enrichment, p. 60 Assessment, pp. 9, 9 Parent and Student Stud Guide Workbook, p. 8 Transparencies -Minute Check Transparenc 6- Answer Ke Transparencies Technolog AlgePASS: Tutorial Plus, Lesson Interactive Chalkboard Multimedia Applications

24 . ; for those values of for which the inequalit is true; 0 for those values of for which the inequalit is not true. CHECK To check this solution, substitute, a number less than, and a number greater than. Let b. Let b. Let b 6. 7b 9 6 7b 9 6 7b 9 6 7() 9? 6 7() 9? 6 7(6) 9? 6 9? 6 8 9? 6 9? The solution set is {b b }. Eample Write and Solve an Inequalit Write an inequalit for the sentence below. Then solve the inequalit. Three times a number minus eighteen is at least five times the number plus twent-one. Three times is at five times twent a number minus eighteen least the number plus one. n 8 n n 8 n riginal inequalit n 8 n n n Subtract n from each side. n 8 Simplif. n Add 8 to each side. n 9 Simplif. n 9 Divide each side b and change to. n 9. Simplif. The solution set is {n n 9.}. Agraphing calculator can be used to solve inequalities. Solving Inequalities You can find the solution of an inequalit in one variable b using a graphing calculator. n a TI-8 Plus, clear the Y list. Enter as Y. (The smbol is item on the TEST menu.) Press GRAPH. [0, 0] scl: b [0, 0] scl: Think and Discuss. Describe what is shown on the screen. part of the graph of. Use the TRACE function to scan the values along the graph. What do ou notice about the values of on the graph? if ; otherwise, 0. Solve the inequalit algebraicall. How does our solution compare to the pattern ou noticed in Eercise? Lesson 6- Solving Multi-Step Inequalities Teach SLVE MULTI-STEP INEQUALITIES In-Class Eamples Power Point SCIENCE The inequalit F represents the temperatures in degrees Fahrenheit for which water is a gas (steam). Similarl, the 9 inequalit C represents the temperatures in degrees Celsius for which water is a gas. Find the temperatures in degrees Celsius for which water is a gas. Water will be a gas for all temperatures greater than 00C. Solve d 79. Then check our solution. {d d 6} Teaching Tip Eamples and were solved in two steps. Eample requires three steps. ther problems will require even more steps. Remind students that when the solve multi-step inequalities, the should alwas undo operations in the reverse of the order of operations. This means undoing addition or subtraction first to isolate the variable, then multipling or dividing to make the coefficient. Write an inequalit for the sentence below. Then solve the inequalit. Four times a number plus twelve is less than a number minus. n n ; {n n } Solving Inequalities Graphed on a number line, would have a circle at, and would have a dot at. Since the calculator does not make this distinction on the graph, have students use the TABLE feature. For an -value of, the table shows that the corresponding -value is 0, meaning the graph does not include (which is the same as having a circle at ). Lesson 6- Solving Multi-Step Inequalities

25 Intervention Before ou introduce solving inequalities involving the Distributive Propert, review the Distributive Propert with students. Have volunteers eplain the Distributive Propert in their own words, and give eamples on the chalkboard or overhead projector. In-Class Eamples New SLVE INEQUALITIES INVLVING THE DISTRIBUTIVE PRPERTY Solve 8 (c ) 6c ( c). c c Solve 7(s ) s 8s (s ). Stud Notebook Power Point Concept Check Distributive Propert Ask students to identif the first step the must do when solving inequalities that have grouping smbols. Use the Distributive Propert to remove the grouping smbols. Practice/Appl Have students cop Eample or a similar problem to show how to solve inequalities using the Distributive Propert. include an other item(s) that the find helpful in mastering the skills in this lesson. Concept Check Guided Practice GUIDED PRACTICE KEY Eercises Eamples 8,, 9 0 Chapter 6 Solving Linear Inequalities SLVE INEQUALITIES INVLVING THE DISTRIBUTIVE PRPERTY When solving equations that contain grouping smbols, first use the Distributive Propert to remove the grouping smbols. Eample Distributive Propert Solve d (8d 9) (d 7). d (8d 9) (d 7) riginal inequalit d 6d 8 d 7 d 8 d d 8 d d d If solving an inequalit results in a statement that is alwas true, the solution is all real numbers. If solving an inequalit results in a statement that is never true, the solution is the empt set. The empt set has no members.. Compare and contrast the method used to solve h 6 7 and the method used to solve h 6 7. See margin.. PEN ENDED Write a multi-step inequalit with the solution graphed below Justif each indicated step. 8 d Simplif. (a 7) 9 a 9 a.? Distributive Propert a a b.? Add to each side. a a c.? Divide each side b. a Distributive Propert Combine like terms. Add d to each side. 8 d Add to each side. d Simplif. d Divide each side b. d Simplif. Since d is the same as d, the solution set is {d d }. Eample Empt Set Solve 8(t ) (t ) (t 7) 8. 8(t ) (t ) (t 7) 8 riginal inequalit 8t 6 t t 8 t 8 t 7 t 8 t t 7 t Distributive Propert Combine like terms. Subtract t from each side. 8 7 This statement is false. Since the inequalit results in a false statement, the solution set is the empt set. Unlocking Misconceptions 7 8 Sample answer: Students ma incorrectl assume that the solution of all inequalities in which the variable has been eliminated is the empt set. Remind students that the must simplif the inequalit to see whether it is a true statement. If the inequalit is true, the solution set is all real numbers. nl when the inequalit is untrue is the solution set the empt set. Chapter 6 Solving Linear Inequalities

26 . { 0.}. {r r 8} 6. {b b } Application indicates increased difficult Practice and Appl For See Eercises Eamples,, 8 9. Etra Practice See page 8. a. Subtract 7 from each side. b. Multipl each side b. 6. {f f 8} 7. {d d } 8. {w w 6} 9. q q 0. {a a 9}. {r r 9}. {k k 8} 7. {t t } 8. {h h 79} 0. {b b is a real number.}. n 0; 8 {n n 80} 6. n 8 ; {n n 6} Solve each inequalit. Then check our solution.. 9. r b 9b 7. (g ) (g ) 8. t (t ) ( t) {g g } {t t } 9. Define a variable, write an inequalit, and solve the problem below. Then check our solution. Seven minus two times a number is less than three times the number plus thirt-two. Sample answer: Let n the number; 7 n n ; {n n }. 0. SALES Asalesperson is paid $,000 a ear plus % of the amount of sales made. What is the amount of sales needed to have an annual income greater than $,000? more than $60,000 Justif each indicated step.. w 7 9. m m See margin. w a.? ()m () m a.? w 6 m m m m m m b.? w (6) b.? m w 0 ()(m) () c.? m. Solve (t 7) (t 9). Show each step and justif our work. {t t }. Solve (k ) (k ). Show each step and justif our work. {k k }. See margin and pp. 6A 6D for steps and justifications. Solve each inequalit. Then check our solution.. t 6 {t t } 6. 8f 9 7. d 8. w q q a 0a r 0r. k 7k 7. v 7 {v v 9}. a 8 0 {a a }. w w {w w } 6. b 8 b {b b } 7. 7 t (t ) ( t) 8. (h 6) 7(h 7) h 9. ( ) 0. (b ) (b 6)..v..v 6.7 {v v.}. 0.(d ) 0.8d. {d d 0}. Solve ( ) ( ) ( ). Then graph the solution. { }. Solve ( ) ( 6) ( ). Then graph the solution. { }. See pp. 6A 6D for graphs. Define a variable, write an inequalit, and solve each problem. Then check our solution. 8. Sample answer: Let n the number.. ne eighth of a number decreased b five is at least thirt. 6. Two thirds of a number plus eight is greater than twelve. 7. Negative four times a number plus nine is no more than the number minus twent-one. n 9 n ; {n n 6} 8. Three times the sum of a number and seven is greater than five times the number less thirteen. (n 7) n ; {n n 7} Differentiated Instruction Lesson 6- Solving Multi-Step Inequalities Interpersonal Some students benefit from working with a partner so that the can talk through the process being used. Group students in pairs to solve inequalities. nce both students agree on the solution, have them test several values to help verif that their solution is correct. About the Eercises rganization b bjective Solve Multi-Step Inequalities:,, 6,, 7, 9,, 8 Solve Inequalities Involving the Distributive Propert:,, 7 0,, 8,,, 7 dd/even Assignments Eercises 8 are structured so that students practice the same concepts whether the are assigned odd or even problems. Alert! Eercises 6 8 require a graphing calculator. Assignment Guide Basic: odd,, 7, 9, 6, 8 Average: 7 odd, 6, 8 Advanced: 8 even, 6 7 (optional: 7 8) Answers. To solve both the equation and the inequalit, ou first subtract 6 from each side and then divide each side b. In the equation, the equal sign does not change. In the inequalit, the inequalit smbol is reversed because ou divided b a negative number. a. Multipl each side b and change to. b. Add m to each side. c. Multipl each side b and change to.. (t 7) (t 9) riginal inequalit t 8 t 8 Distributive Propert t 8 t t 8 t Subtract t from each side. t 8 8 Simplif. t Add 8 to each side. t 6 Simplif. t 6 Divide each side b. t Simplif. {t t } Lesson 6- Solving Multi-Step Inequalities

27 Stud Stud Guide and and Intervention Intervention, p. (shown) and p NAME DATE PERID Solving Multi-Step Inequalities Solve Multi-Step Inequalities To solve linear inequalities involving more than one operation, undo the operations in reverse of the order of operations, just as ou would solve an equation with more than one operation. Eample Solve 6. Eample Solve a a. 6 riginal inequalit a a riginal inequalit 6 Subtract from a a a a Subtract a from GEMETRY For Eercises 9 and 0, use the following information. B definition, the measure of an acute angle is less than 90 degrees. Suppose the measure of an acute angle is a. 9. Write an inequalit to represent the situation. a Solve the inequalit. {a a } 6 each side. Simplif. Add to each side. Simplif. 6 Divide each side b. The solution is { }. Eercises Simplif. a a a 9 a 9 a 9 The solution is a a 9 Solve each inequalit. Then check our solution. q 7.. 8n 0 6 n.. each side. Simplif. Add to each side. Simplif. Divide each side b and change to. Simplif. n n {q q }. 6n 8 8n. d d 6. r 6 8r 8 {n n } {d d 0} {r r } m 8 m { 6} { 6} {m m } n n 0. {n n } Define a variable, write an inequalit, and solve each problem. Then check our solution.. Sample answer: Let n the number.. Negative three times a number plus four is no more than the number minus eight. n n 8; {n n }. ne fourth of a number decreased b three is at least two. n ; {n n 0}. The sum of twelve and a number is no greater than the sum of twice the number and 8. n n (8); {n n 0} Skills Practice Practice, p. 7 and (Average) Practice, p. 8 (shown) 6- NAME DATE PERID Solving Multi-Step Inequalities Justif each indicated step (8) a.? b.? c.? a. Multipl each side b 8. b. Subtract from each side. c. Divide each side b.. (h ) (h ) h 6h 0 a. h 6h h 6h 6h 6h b. h h c. h 6 h 6 d. h a. Distributive Propert b. Subtract 6h from each side. c. Subtract from each side. d. Divide each side b and change to. Solve each inequalit. Then check our solution. t u 6 6u 0. a {t t } {u u 7} {a a } w 6. 8 {w w 9} f {f f } 6h 8. h {h h } 9. (z ) (z ) {z z 8}???? Lesson 6- Phsical Science Mercur is a metal that is a liquid at room temperature. In fact, its melting point is 8 C. Mercur is used in thermometers because it epands evenl as it is heated. Source: World Book Encclopedia SCHL For Eercises and, use the following information. Carmen s scores on three math tests were 9, 9, and 88. The fourth and final test of the grading period is tomorrow. She needs an average (mean) of at least 9 to receive an A for the grading period s 9. If s is her score on the fourth test, write an inequalit to represent the situation.. If Carmen wants an A in math, what must she score on the test? at least 9 PHYSICAL SCIENCE For Eercises and, use the information at the left and the information below. The melting point for an element is the temperature where the element changes from a solid to a liquid. If C represents degrees Celsius and F represents degrees Fahrenheit, then C (F ). 9. (F ) 8 9. Write an inequalit that can be used to find the temperatures in degrees Fahrenheit for which mercur is a solid.. For what temperatures will mercur be a solid? temperatures less than 6. F. HEALTH Keith weighs 00 pounds. He wants to weigh less than 7 pounds. If he can lose an average of pounds per week on a certain diet, how long should he sta on his diet to reach his goal weight? more than weeks 6. CRITICAL THINKING Write a multi-step inequalit that has no solution and one that has infinitel man solutions. Sample answers: ; 7. PERSNAL FINANCES Nicholas wants to order a pizza. He has a total of $.00 to pa the deliver person. The pizza costs $7.0 plus $. per topping. If he plans to tip % of the total cost of the pizza, how man toppings can he order? or fewer toppings 0. e (e ) (6e ) {e e }. n (n 6) 0 {n n 9} Define a variable, write an inequalit, and solve each problem. Then check our solution.. Sample answer: Let n the number.. A number is less than one fourth the sum of three times the number and four. n n ;{n n }. Two times the sum of a number and four is no more than three times the sum of the number and seven decreased b four. (n ) (n 7) ; {n n 9}. GEMETRY The area of a triangular garden can be no more than 0 square feet. The base of the triangle is 6 feet. What is the height of the triangle? no more than ft. MUSIC PRACTICE Nabuko practices the violin at least hours per week. She practices for three fourths of an hour each session. If Nabuko has alread practiced hours in one week, how man sessions remain to meet or eceed her weekl practice goal? at least sessions Reading to Learn Mathematics, p NAME DATE PERID Reading to Learn Mathematics Solving Multi-Step Inequalities ELL LABR For Eercises 8 0, use the following information. A union worker made $00 per week. His union sought a one-ear contract and went on strike. nce the new contract was approved, it provided for a % raise. 8. Assume that the worker was not paid during the strike. Given his raise in salar, how man weeks could he strike and still make at least as much for the net weeks as he would have made without a strike? no more than weeks 9. How would our answer to Eercise 8 change if the worker had been making $600 per week? no change 0. How would our answer to Eercise 8 change if the worker s union provided him with $0 per week during the strike? up to.8 weeks Pre-Activit How are linear inequalities used in science? Read the introduction to Lesson 6- at the top of page in our tetbook. Then write an inequalit that could be used to find the temperatures in degrees Celsius for which each substance is a gas. 9 9 Argon: C 0 Bromine: C 8. NUMBER THERY Find all sets of two consecutive positive odd integers whose sum is no greater than 8. 7, 9;, 7;, ;, Reading the Lesson. What does the phrase undoing the operations in reverse of the order of operations mean? Sample answer: First add or subtract to undo subtraction or addition, then multipl or divide to undo division or multiplication.. Describe how checking the solution of an inequalit is different from checking the solution of an equation. Sample answer: Instead of substituting one value for the variable, there are infinitel man values that can be used to check. It is a good idea to use a value that is less than, the value equal to, and a value greater than the number in the solution to check an inequalit.. Describe how the Distributive Propert can be used to remove the grouping smbols in the inequalit 7( 8). Multipl 7 b both and 8.. Is it possible to have no solution when ou solve an inequalit? Eplain our answer and give an eample. Sample answer: Yes; if solving results in an inequalit that is never true (and the signs have been reversed if necessar), then there is no solution. Eample: (t ) 8 (t ) 8 Helping You Remember. Make a checklist of steps ou can use when solving inequalities. () Use the Distributive Propert to remove an grouping smbols. () Combine an like terms. () Add or subtract the same variable terms or constants on both sides. () Multipl or divide to undo operations. () Reverse the direction of the inequalit smbol if both sides were multiplied or divided b a negative number. (6) Be sure the variable is b itself on one side of the final inequalit. 6 Chapter 6 Solving Linear Inequalities NAME DATE PERID 6- Enrichment, p. 60 Carlos Montezuma During his lifetime, Carlos Montezuma (86? 9) was one of the most influential Native Americans in the United States. He was recognized as a prominent phsician and was also a passionate advocate of the rights of Native American peoples. The eercises that follow will help ou learn some interesting facts about Dr. Montezuma s life. Solve each inequalit. The word or phrase net to the equivalent inequalit will complete the statement correctl.. k 0. r 9 Montezuma was born in the state He was a Native American of the?? of. Yavapais, who are a people. a. a. r Navajo k Arizona b. k Montana b. r Mohawk c. k Utah c. r Mohave-Apache. NUMBER THERY Find all sets of three consecutive positive even integers whose sum is less than 0. 0,, ; 8, 0, ; 6, 8, 0;, 6, 8;,, 6 6 Chapter 6 Solving Linear Inequalities

28 SL/EC Practice Standardized Test Practice Graphing Calculator Maintain Your Skills Mied Review Getting Read for the Net Lesson. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How are linear inequalities used in science? Include the following in our answer: an inequalit for the temperatures in degrees Celsius for which bromine is a gas, and a description of a situation in which a scientist might use an inequalit.. What is the first step in solving? D 9 A Add to each side. B Subtract from each side. C Divide each side b 9. D Multipl each side b 9.. Solve t 8t (6t 0). C A {t t 6} B {t t 6} C {t t } D {t t } Use a graphing calculator to solve each inequalit ( ) ( ) { } { 8} { } 9. BUSINESS The charge per mile for a compact rental car at Great Deal Rentals is $0.. Mrs. Ludlow must rent a car for a business trip. She has a budget of $0 for mileage charges. How man miles can she travel without going over her budget? (Lesson 6-) up to 6 mi Solve each inequalit. Then check our solution, and graph it on a number line. (Lesson 6-) See margin for graphs. 60. d {d d 9} 6. t {t t 8} 6. 7 { } Write the point-slope form of an equation for a line that passes through each point with the given slope. (Lesson -) 6. (, ), m 6. (, ), m 6. (, 6), m ( ) ( ) Determine the slope of the line that passes through each pair of points. (Lesson -) 66. (, ), (, 6) 67. (, ), (, ) (0, ), (, ) Determine whether each equation is a linear equation. If an equation is linear, rewrite it in the form A B C. (Lesson -) no 7. es; 0 es; 7 Solve each equation. Then check our solution. (Lesson -) 7. ( ) ( ) 7. t 7 t. PREREQUISITE SKILL Graph each set of numbers on a number line. (To review graphing integers on a number line, see Lesson -.) 7 8. See pp. 6A 6D. 7. {,, } 7. {, 0,, } 76. {,,, } 77. {integers less than } 78. {integers greater than } 79. {integers between and 6} 80. {integers between and } 8. {integers greater than or equal to } 8. {integers less than 6 but greater than } Assess pen-ended Assessment Speaking Have students eplain to the class the different methods the now know for solving inequalities, including those learned in previous lessons. Ask them to start with the simplest methods and progress to the more comple. Record student responses on the chalkboard or overhead projector. As each student describes a method, call on another student to give an eample of how to use the method. Record this eample along with the corresponding method. Getting Read for Lesson 6- PREREQUISITE SKILL Students will learn how to solve compound inequalities and graph them in Lesson 6-. Use Eercises 7 8 to determine our students familiarit with graphing sets of integers on a number line. Assessment ptions Quiz (Lesson 6-) is available on p. 9 of the Chapter 6 Resource Masters. Mid-Chapter Test (Lessons 6- through 6-) is available on p. 9 of the Chapter 6 Resource Masters. Lesson 6- Solving Multi-Step Inequalities 7 Answers. Inequalities can be used to describe the temperatures for which an element is a gas or a solid. Answers should include the following. The inequalit for temperatures in degrees Celsius for which bromine is 9 a gas is C 8. Sample answer: Scientists ma use inequalities to describe the temperatures for which an element is a solid Lesson 6- Solving Multi-Step Inequalities 7

29 Reading Mathematics Getting Started Discuss the meaning of the adjective compound. Ask students the meaning of the word and to give an eample of something that is compound. Review the polgons listed on this page. In order to determine whether compound statements are true, students must be familiar with the number of sides each polgon has. Teach Sentence Structure Ask students to recall the definition of a compound sentence from their language arts studies. Students should recall that a compound sentence has two independent clauses that are joined b a coordinating conjunction, punctuation, or both. Eplain that the compound statements in this activit are compound sentences in which the two independent clauses are joined b the coordinating conjunctions and or or. Assess Stud Notebook Ask students to summarize what the have learned about compound statements. Compound Statements Two simple statements connected b the words and or or form a compound statement. Before ou can determine whether a compound statement is true or false, ou must understand what the words and and or mean. Consider the statement below. Atriangle has three sides, and a heagon has five sides. For a compound statement connected b the word and to be true, both simple statements must be true. In this case, it is true that a triangle has three sides. However, it is false that a heagon has five sides; it has si. Thus, the compound statement is false. Acompound statement connected b the word or ma be eclusive or inclusive. For eample, the statement With our dinner, ou ma have soup or salad, is eclusive. In everda language, or means one or the other, but not both. However, in mathematics, or is inclusive. It means one or the other or both. Consider the statement below. Atriangle has three sides, or a heagon has five sides. For a compound statement connected b the word or to be true, at least one of the simple statements must be true. Since it is true that a triangle has three sides, the compound statement is true. Reading to Learn Determine whether each compound statement is true or false. Eplain our answer.. See margin for eplanations.. Aheagon has si sides, or an octagon has seven sides. true. An octagon has eight sides, and a pentagon has si sides. false. Apentagon has five sides, and a heagon has si sides. true. Atriangle has four sides, or an octagon does not have seven sides. true. Apentagon has three sides, or an octagon has ten sides. false 6. Asquare has four sides, or a heagon has si sides. true 7. or 8 6 false 8. 0 and false 9. 0 and 0 true or true. or true. 0 and false 8 Chapter 6 Solving Linear Inequalities Triangle Square Pentagon Heagon ctagon Answers ELL English Language Learners ma benefit from writing ke concepts from this activit in their Stud Notebooks in their native language and then in English.. true or false. true and false. true and true. false or true. false or false 6. true or true 7. false or false 8. false and true 9. true and true 0. true or true. false or true. false and true 8 Chapter 6 Solving Linear Inequalities

30 Solving Compound Inequalities Virginia SL STANDARD A. The student will solve multistep linear equations and inequalities in one variable, solve literal equations (formulas) for a given variable, and appl these skills to solve practical problems. Graphing calculators will be used to confirm algebraic solutions. Lesson Notes Vocabular compound inequalit intersection union Stud Tip Reading Math The statement,0 c,00 can be read,0 is less than or equal to c, which is less than,00. Solve compound inequalities containing the word and and graph their solution sets. Solve compound inequalities containing the word or and graph their solution sets. are compound inequalities used in ta tables? Richard Kelle is completing his income ta return. He uses the table to determine the amount he owes in federal income ta. Let c represent the amount of Mr. Kelle s income. His income is at least $,0 and it is less than $,00. This can be written as c,0 and c,00. When considered together, these two inequalities form a compound inequalit. This compound inequalit can be written without using and in two was.,0 c,00 or,00 c,0 INEQUALITIES CNTAINING AND Acompound inequalit containing and is true onl if both inequalities are true. Thus, the graph of a compound inequalit containing and is the intersection of the graphs of the two inequalities. In other words, the solution must be a solution of both inequalities. The intersection can be found b graphing each inequalit and then determining where the graphs overlap. Eample If taable income is At least,000,00,00,0,00,0,00,0,00,0,00,0 Source: IRS Less than,00,00,0,00,0,00,0,00,0,00,0,600 Graph an Intersection Graph the solution set of and Ta Tables Single Married filing jointl Graph. Graph. Find the intersection. Married filing separatel Head of a household The solution set is { }. Note that the graph of includes the point. The graph of does not include. Focus -Minute Check Transparenc 6- Use as a quiz or a review of Lesson 6-. Mathematical Background notes are available for this lesson on p. 6D. Building on Prior Knowledge In Lesson 6-, students learned to graph inequalities on a number line. Those same skills will be used in this lesson to graph two inequalities and determine which part(s) of their graphs satisf the given compound inequalit. are compound inequalities used in ta tables? Ask students: What inequalit smbol represents the term at least, when we sa that Mr. Kelle s income is at least $,0? greater than or equal to () What is the least amount his income could be? $,0 What is the greatest amount his income could be? $,99.99 Lesson 6- Solving Compound Inequalities 9 Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters Stud Guide and Intervention, pp. 6 6 Skills Practice, p. 6 Practice, p. 6 Reading to Learn Mathematics, p. 6 Enrichment, p. 66 Parent and Student Stud Guide Workbook, p. 9 Transparencies -Minute Check Transparenc 6- Real-World Transparenc 6 Answer Ke Transparencies Technolog AlgePASS: Tutorial Plus, Lesson Interactive Chalkboard Lesson - Lesson Title 9

31 Teach INEQUALITIES CNTAINING AND In-Class Eamples In-Class Eample Teaching Tip The smbol for intersection is. The solution set in Eample could be written as { } { }. Graph the solution set of and. The solution set is { } Teaching Tip Students ma benefit from rewriting the compound inequalit as two separate inequalities before the attempt to solve it. Solve 7 z. Then graph the solution set. {z z 9} INEQUALITIES CNTAINING R Power Point Power Point TRAVEL A ski resort has several tpes of hotel rooms and several tpes of cabins. The hotel rooms cost at most $89 per night, and the cabins cost at least $09 per night. Write and graph a compound inequalit that describes the amount a guest would pa per night at the resort. {n n 89 or n 09}, where n is the amount a guest pas per night Stud Tip Reading Math When solving problems involving inequalities, within is meant to be inclusive. Use or. between is meant to be eclusive. Use or. Pilot Pilots check aviation weather forecasts to choose a route and altitude that will provide the smoothest flight. nline Research For information about a career as a pilot, visit: careers Eample Solve and Graph an Intersection Solve. Then graph the solution set. First epress using and. Then solve each inequalit. and 6 The solution set is the intersection of the two graphs The solution set is { 6} Graph or. Graph 6. Find the intersection. INEQUALITIES CNTAINING R Another tpe of compound inequalit contains the word or. A compound inequalit containing or is true if one or more of the inequalities is true. The graph of a compound inequalit containing or is the union of the graphs of the two inequalities. In other words, the solution of the compound inequalit is a solution of either inequalit, not necessaril both. The union can be found b graphing each inequalit. Eample Write and Graph a Compound Inequalit AVIATIN An airplane is eperiencing heav turbulence while fling at 0,000 feet. The control tower tells the pilot that he should increase his altitude to at least,000 feet or decrease his altitude to no more than 6,000 feet to avoid the turbulence. Write and graph a compound inequalit that describes the altitude at which the airplane should fl. Words Variables The pilot has been told to fl at an altitude of at least,000 feet or no more than 6,000 feet. Let a be the plane s altitude. The plane s is at,000 the is no 6,000 altitude least feet or altitude more than feet. Inequalit a,000 or a 6,000 Now, graph the solution set.,000 0,000,000,000 0,000,000,000 0,000,000 a,000 or a 6,000 Graph a,000. Graph a 6,000. Find the union Chapter 6 Solving Linear Inequalities Differentiated Instruction Visual/Spatial Prepare a large number line with two dashed horizontal lines above it as a transparenc or laminated sheet of paper. Give the students a compound inequalit written as two statements. Have students use a dr-erase marker to plot the solution for each inequalit on one of the dashed lines. If the inequalit involves and, have them wipe awa an parts that are not on both dashed lines. This leaves a clear picture of what part of the number line should be used for the solution. 0 Chapter 6 Solving Linear Inequalities

32 Eample Solve and Graph a Union Solve h 9 or 7h 8. Then graph the solution set. h 9 or 7h 8 h 9 7h 8 h 7h h 7 h 7 7 h h The solution set is the union of the two graphs. In-Class Eample Power Point Teaching Tip The smbol for union is. The solution set in Eample could be written as {h h } {h h }. Solve k 7 or 9k 0. Then graph the solution set. {k k 8} 6 0 Graph h Concept Check Guided Practice GUIDED PRACTICE KEY Eercises Eamples 7 8, 8. See margin for graphs. 8. {w w 8} 9. {n n or n 8} Application Graph h. Find the union. Notice that the graph of h contains ever point in the graph of h. So, the union is the graph of h. The solution set is {h h }.. Describe the difference between a compound inequalit containing and and a compound inequalit containing or. See margin.. Write 7 is less than t, which is less than as a compound inequalit. 7 t. PEN ENDED Give an eample of a compound inequalit containing and that has no solution. Sample answer: and Graph the solution set of each compound inequalit.. See margin.. a 6 and a. or 9 Write a compound inequalit for each graph or Solve each compound inequalit. Then graph the solution set w and w 9. n 7 or n 7 0. z or z {z z }. 8 { }. Define a variable, write a compound inequalit, and solve the following problem. Three times a number minus 7 is less than 7 and greater than. Sample answer: Let n the number; n 7 7; {n n 8}.. PHYSICAL SCIENCE According to Hooke s Law, the force F in pounds required to stretch a certain spring inches beond its natural length is given b F.. If forces between 0 and 0 pounds, inclusive, are applied to the spring, what will be the range of the increased lengths of the stretched spring? about Natural length Stretched inches Lesson 6- Solving Compound Inequalities Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 6. include an eample of an inequalit containing and, and the graph of the inequalit. include an eample of an inequalit containing or, and the graph of the inequalit. include an other item(s) that the find helpful in mastering the skills in this lesson. Answers. A compound inequalit containing and is true if and onl if both inequalities are true. A compound inequalit containing or is true if and onl if at least one of the inequalities is true Lesson 6- Solving Compound Inequalities

33 About the Eercises rganization b bjective Inequalities Containing and:,, 8, 6, 8, 9,, 8, 9,,, 6 8 Inequalities Containing or: 6, 7,, 7, 0,, 6, 7, 0,,, dd/even Assignments Eercises 7 are structured so that students practice the same concepts whether the are assigned odd or even problems. Alert! Eercise involves research on the Internet or other reference materials. Eercise 7 requires a graphing calculator. Assignment Guide Basic: odd, 7 7 odd, 9 odd, 80 Average: 9 odd, 80 Advanced: 8 even, 9 7 (optional: 7 80) All: Practice Quiz ( 0) Answers indicates increased difficult Practice and Appl For Eercises See Eamples 7 8, 6 8 Etra Practice See page or. 7 or 6. 0 or. or 8. See pp. 6A 6D for graphs. 8. {k 0 k 6} 9. {f f } 0. {d d or d 7}. {h h }. {q q 6} 6. { is a real number} 8. {p p }. n 8 ; {n n }. 8 n 0; {n n }. n or n 0; {n n 7 or n }. 0 n ; {n 0 n } Chapter 6 Solving Linear Inequalities Graph the solution set of each compound inequalit. 9. See margin.. and 9. s 7 and s 0 6. r 6 or r 6 7. m or m d 9. g Write a compound inequalit for each graph WEATHER The Fujita Scale (F-scale) is the official classification sstem for tornado damage. ne factor used to classif a tornado is wind speed. Use the information in the table to write an inequalit for the range of wind speeds of an F tornado. 8 w BILGY Each tpe of fish thrives in a specific range of temperatures. The optimum temperatures for sharks range from 8 C to C, inclusive. Write an inequalit to represent temperatures where sharks will not thrive. t 8 or t F-Scale Number Solve each compound inequalit. Then graph the solution set. 8. k and k 8 9. f 8 and f 9 0. d or d. h 0 or h. { 9}. 0 { 6}. t 7 and t 6. 8 q and q 6. or 7. n or n {n n } 8. p p 8 p 9. g 6 g g 8 0. c c 0 or c. 0.b 6 or b 6 8 b {c c or c } {b b or b } Define a variable, write an inequalit, and solve each problem.. Sample. Eight less than a number is no more than and no less than. answer: Let n the. The sum of times a number and is between 8 and 0. number.. The product of and a number is greater than or less than 0.. ne half a number is greater than 0 and less than or equal to. 6. HEALTH About 0% of the time ou sleep is spent in rapid ee movement (REM) sleep, which is associated with dreaming. If an adult sleeps 7 to 8 hours, how much time is spent in REM sleep? between. and.6 hours inclusive 7. SHPPING Astore is offering a $0 mail-in rebate on all color printers. Luisana is looking at different color printers that range in price from $7 to $60. How much can she epect to spend after the mail-in rebate? between $ and $0 inclusive F0 F F F F F Rating 0 7 mph 7 mph 7 mph 8 06 mph mph 6 8 mph Chapter 6 Solving Linear Inequalities

34 8. FUND-RAISING Rashid is selling chocolates for his school s fund-raiser. He can earn prizes depending on how much he sells. So far, he has sold $70 worth of chocolates. How much more does he need to sell to earn a prize in categor D? between $ and $0 inclusive Sales ($) Prize A B C D E 9. CRITICAL THINKING Write a compound inequalit that represents the values of which make the following epressions false. a. or 8 and 8 b. 6 and 6 or Stud Stud Guide and and Intervention Intervention, p. 6 (shown) and p NAME DATE PERID Solving Compound Inequalities Inequalities Containing and A compound inequalit containing and is true onl if both inequalities are true. The graph of a compound inequalit containing and is the intersection of the graphs of the two inequalities. Ever solution of the compound inequalit must be a solution of both inequalities. Eample Graph the solution Eample Solve using set of and. and. Then graph the solution set Graph. Graph. Find the intersection. The solution set is { }. Eercises and Graph the solution set of each compound inequalit. Graph. Graph. Find the intersection. The solution set is { }.. b and b. q. and HEARING For Eercises 0, use the following information. Humans hear sounds with sound waves within the 0 to 0,000 hertz range. Dogs hear sounds in the to 0,000 hertz range. 0. Write a compound inequalit for the hearing range of humans and one for the hearing range of dogs. 0 h 0,000; d 0,000. {h h 0,000};. What is the union of the two solution sets? the intersection? {h 0 h 0,000};. Write an inequalit or inequalities for the range of sounds that dogs can hear, but humans cannot. h 0 or 0,000 h 0, p. d and d 6. p 0 Solve each compound inequalit. Then graph the solution set. 7. w 8. p {w w } {p p 7} and { 6 } { } n and n 6. d 6d d {n n } {d d } 0 0 Lesson 6- SL/EC Practice Standardized Test Practice Graphing Calculator RESEARCH Use the Internet or other resource to find the altitudes in miles of the laers of Earth s atmosphere, troposphere, stratosphere, mesosphere, thermosphere, and eosphere. Write inequalities for the range of altitudes for each laer. Sample answer: troposphere: a 0, stratosphere: 0 a 0, mesosphere: 0 a 0, thermosphere: 0 a 00, eosphere: a 00. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 6A 6D. How are compound inequalities used in ta tables? Include the following in our answer: a description of the intervals used in the ta table shown at the beginning of the lesson, and a compound inequalit describing the income of a head of a household paing $70 in taes.. Ten pounds of fresh tomatoes make between 0 and cups of cooked tomatoes. How man cups does one pound of tomatoes make? A A between and cups B between and cups C between and cups D between and cups 6. Solve 7. B A 6 B 9 C D SLVE CMPUND INEQUALITIES In Lesson 6-, ou learned how to use a graphing calculator to find the values of that make a given inequalit true. You can also use this method to test compound inequalities. The words and and or can be found in the LGIC submenu of the TEST menu of a TI-8 Plus. Use this method to solve each of the following compound inequalities using our graphing calculator. a. { 6 or } b. { 8} a. or b. and 6 /sol Lesson 6- Solving Compound Inequalities 0 0 Skills Practice, p. 6 and NAME DATE PERID 6- Practice (Average) Practice, Solving Compound p. 6 Inequalities (shown) Graph the solution set of each compound inequalit.. e. 0 or 6 0. g or g. p 0 Write a compound inequalit for each graph or or Solve each compound inequalit. Then graph the solution set. 9. k 7 or k 8 0. n or n {k k or k } {n n } 0. h. c 6 and c {h h } {c c } Define a variable, write an inequalit, and solve each problem. Then check our solution.. Sample answer: Let n the number.. Two times a number plus one is greater than five and less than seven. n 7; {n n }. A number minus one is at most nine, or two times the number is at least twent-four. n 9 or n ; {n n 0 or n } METERLGY For Eercises and 6, use the following information. Strong winds called the prevailing westerlies blow from west to east in a belt from 0 to 60 latitude in both the Northern and Southern Hemispheres.. Write an inequalit to represent the latitude of the prevailing westerlies. {w 0 w 60} 6. Write an inequalit to represent the latitudes where the prevailing westerlies are not located. {w w 0 or w 60} 7. NUTRITIN A cookie contains 9 grams of fat. If ou eat no fewer than and no more than 7 cookies, how man grams of fat will ou consume? between 6 g and 6 g inclusive Reading to Learn Mathematics, p NAME DATE PERID Reading to Learn Mathematics Solving Compound Inequalities Pre-Activit How are compound inequalities used in ta tables? Read the introduction to Lesson 6- at the top of page 9 in our tetbook. Eplain wh it is possible that Mr. Kell s income is $,70. $,70 is greater than or equal to $,0 and less than $,00. Eplain wh it is not possible that Mr. Kell s income is $,00. $,00 is not less than $,00. Reading the Lesson. When is a compound inequalit containing and true? It is true when both inequalities are true. ELL. The graph of a compound inequalit containing and is the intersection of the graphs of the two inequalities. NAME DATE PERID 6- Enrichment, p. 66 Some Properties of Inequalities The two epressions on either side of an inequalit smbol are sometimes called the first and second members of the inequalit. If the inequalit smbols of two inequalities point in the same direction, the inequalities have the same sense. For eample, a b and c d have the same sense; a b and c d have opposite senses. In the problems on this page, ou will eplore some properties of inequalities. Three of the four statements below are true for all numbers a and b (or a, b, c, and d). Write each statement in algebraic form. If the statement is true for all numbers, prove it. If it is not true, give an eample to show that it is false.. Given an inequalit, a new and equivalent inequalit can be created b interchanging the members and reversing the sense. If a b, then b a. a b, a b 0, b a, ()(b) ()(a), b a. When is a compound inequalit containing or true? It is true when one or both of the inequalities is true.. The graph of a compound inequalit containing or is the union of the graphs of the two inequalities.. Suppose ou use ellow to show the graph of Inequalit # on the number line. You use blue to show the graph of Inequalit #. Write and or or in each blank to complete the sentence. a. The part that is green is the graph of Inequalit # and Inequalit #. b. All colored parts form the graph of Inequalit # or Inequalit #. Helping You Remember 6. ne wa to remember something is to connect it to something that is familiar to ou. Write two true compound statements about ourself, one using the word and and the other using the word or. Sample answer: I am and I am a freshman in high school. After school, I will go to football practice or I will go home. Lesson 6- Lesson 6- Solving Compound Inequalities

35 Assess pen-ended Assessment Writing Have students write a paragraph comparing and contrasting compound inequalities containing and with inequalities containing or. The paragraph should contain eamples of the different tpes of inequalities and their graphs. Getting Read for Lesson 6- PREREQUISITE SKILL Students will learn how to solve open sentences involving absolute value in Lesson 6-. A good grasp of what absolute value means will help them understand how this concept applies to epressions in open sentences. Use Eercises 7 80 to determine our students familiarit with finding absolute values. Assessment ptions Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons 6- and 6-. Lesson numbers are given to the right of the eercises or instruction lines so students can review concepts not et mastered. Answers Maintain Your Skills Mied Review 6. {(6, 0), (, ), (, ), (, )}; {,, 6}; {, 0,, }; {(0, 6), (, ), (, ), (, )} 66. {(, ), (, ), (, ), (, 7)}; {,,, }; {,, 7}; {(, ), (, ), (, ), (7, )} 67. {(, ), (, ), (, 9), (, ) (, 8), (7, )}; {7,,, }; {,, 8, 9}; {(, ), (, ), (9, ), (, ), (8, ), (, 7)} Getting Read for the Net Lesson P ractice Quiz Chapter 6 Solving Linear Inequalities 8. FUND-RAISING A universit is running a drive to raise mone. A corporation has promised to match 0% of whatever the universit can raise from other sources. How much must the school raise from other sources to have a total of at least $800,000 after the corporation s donation? (Lesson 6-) at least $7,8.7 Solve each inequalit. Then check our solution. (Lesson 6-) t 9. 8d v b 9 8 {d d } {v v } {t t 69} {b b } Solve. Assume that varies directl as. (Lesson -) 6. If 8 when, find when If. when 0., find when Epress the relation shown in each mapping as a set of ordered pairs. Then state the domain, range, and inverse. (Lesson -) 6. X Y 66. X Y 67. X Y 6 0 Find the odds of each outcome if a die is rolled. (Lesson -6) 68. a number greater than : 69. not a : Find each product. (Lesson -) (.7) PREREQUISITE SKILL Find each value. (To review absolute value, see Lesson -.) Solve each inequalit. Then check our solution. (Lesson 6-). b {b b 7}. n {n n 6}. (t 6) 9 {t t }. 9 0 { }. m m {m m } 6. a a {a a } Solve each compound inequalit. Then graph the solution set. (Lesson 6-) 7 0. See margin for graphs and { 9} 8. b or b {b b or b 0} 9. m 7 or m 9 0. a and a {a a } {m m or m } Lessons 6- and 6- Chapter 6 Solving Linear Inequalities

36 Solving pen Sentences Involving Absolute Value Lesson Notes Virginia SL STANDARD A. The student will solve multistep linear equations and inequalities in one variable, solve literal equations (formulas) for a given variable, and appl these skills to solve practical problems. Graphing calculators will be used to confirm algebraic solutions. Stud Tip Look Back To review absolute value, see Lesson -. Solve absolute value equations. Solve absolute value inequalities. is absolute value used in election polls? Voters in Hamilton will vote on a new ta lev in the net election. A poll conducted before the election found that 7% of the voters surveed were for the ta lev, % were against the ta lev, and 8% were undecided. The poll has a -point margin of error. Ta Lev Poll 60 Voters in Hamilton (%) The margin of error means that the result ma be percentage points higher or lower. So, the number of people in favor of the ta lev ma be as high as 0% or as low as %. This can be written as an inequalit using absolute value. 7 7% For % Against The difference between the actual number and 7 is within points. ABSLUTE VALUE EQUATINS that can involve absolute value. 8% Undecided There are three tpes of open sentences n n n Consider the case of n. means the distance between 0 and is units. units units BALLTS If, then or. The solution set is {, }. When solving equations that involve absolute value, there are two cases to consider. Case The value inside the absolute value smbols is positive. Case The value inside the absolute value smbols is negative. Equations involving absolute value can be solved b graphing them on a number line or b writing them as a compound sentence and solving it. Lesson 6- Solving pen Sentences Involving Absolute Value Focus -Minute Check Transparenc 6- Use as a quiz or a review of Lesson 6-. Mathematical Background notes are available for this lesson on p. 6D. is absolute value used in election polls? Ask students: With a -point margin of error, the percent of people who are against the ta lev could be how high and how low? 8% and % How would ou represent the percent of people,, who are against the ta lev, with an absolute value inequalit? Wh is it necessar to represent the percent of people who ma be for the ta lev with the inequalit 7? The percent of people who are for the ta lev could be less than 7. Assume that 6% actuall vote for the lev. Substituting 6 for in the inequalit would ield 6 7. While this inequalit is true, because, the margin of error cannot be negative. So adding the absolute value smbols makes the margin of error positive. Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 69 Practice, p. 70 Reading to Learn Mathematics, p. 7 Enrichment, p. 7 Assessment, p. 9 Graphing Calculator and Spreadsheet Masters, p. Parent and Student Stud Guide Workbook, p. 0 Prerequisite Skills Workbook, pp , 8 8 Transparencies -Minute Check Transparenc 6- Answer Ke Transparencies Technolog Interactive Chalkboard Lesson - Lesson Title

37 Teach ABSLUTE VALUE EQUATINS In-Class Eamples Solve b 6. {, } Power Point Write an equation involving absolute value for the graph Stud Tip Absolute Value Recall that a means a or a. The second equation can be written as a. So, a means a or (a ). These can be written as a or a. Eample Solve an Absolute Value Equation Solve a. Method Graphing a means that the distance between a and is units. To find a on the number line, start at and move units in either direction. 0 units units 7 8 The solution set is {, 7}. The distance from to is units. The distance from to 7 is units. Method Compound Sentence Write a as a or a. Case Case a a a Add to each side. a Add to each side. a 7 Simplif. a Simplif. The solution set is {, 7}. 6 Eample Write an Absolute Value Equation Write an equation involving absolute value for the graph Find the point that is the same distance from as the distance from 9. The midpoint between and 9 is 6. units units The distance from 6 to is units. The distance from 6 to 9 is units. So, an equation is 6. CHECK Substitute and 9 into ABSLUTE VALUE INEQUALITIES Consider the inequalit n. means that the distance from 0 to is less than units. units units Therefore, and. The solution set is { }. 6 Chapter 6 Solving Linear Inequalities Differentiated Instruction Logical Some students ma respond better to rewriting absolute value equations b appling the two situations (positive and negative) to the epression within the absolute value smbols. For eample, can be written as or, which ields. Eample can be written as a or (a ). Then students can solve each equation. 6 Chapter 6 Solving Linear Inequalities

38 . A negative error indicates that the time guessed was less than min. A positive error indicates that the time guessed was more than min.. estimate greater than s and less than 66 s Stud Tip Less Than When an absolute value is on the left and the inequalit smbol is or, the compound sentence uses and. The Algebra Activit eplores an inequalit of the form n. Absolute Value Collect the Data Work in pairs. ne person is the timekeeper. Start timing. The other person tells the timekeeper to stop timing after he or she thinks that one minute has elapsed. Write down the time in seconds. Switch places. Make a table that includes the results of the entire class. Analze the Data. See students work.. Determine the error b subtracting 60 seconds from each student s time.. What does a negative error represent? a positive error?. The absolute error is the absolute value of the error. Since absolute value cannot be negative, the absolute error is positive. If the absolute error is 6seconds, write two possibilities for a student s estimated time of one minute. s or 66 s. What estimates would have an absolute error less than 6 seconds?. Graph the responses and highlight all values such that How man guesses were within 6 seconds? See students work. When solving inequalities of the form n, find the intersection of these two cases. Case The value inside the absolute value smbols is less than the positive value of n. Case The value inside the absolute value smbols is greater than the negative value of n. Eample Solve an Absolute Value Inequalit () Solve t 9. Then graph the solution set. Write t 9 as t 9 and t 9. Case t 9 t 9 Subtract from each side. t Simplif. Case t 9 t 9 Subtract from each side. t Simplif. The solution set is {t t } ABSLUTE VALUE INEQUALITIES Building on Prior Knowledge Before presenting Eample, remind students that the two cases presented represent a compound inequalit involving and, which the learned to solve in Lesson 6-. In-Class Eample Teaching Tip Students ma be confused about wh t 9 is rewritten as t 9 and t 9. An alternative method is to rewrite the inequalit as t 9 and (t ) 9. Then multipl each side of the second inequalit b to ield t 9. This method makes the switch of the direction of the inequalit more obvious, as students must make the switch when the divide each side b. Solve s. Then graph the solution set. {s 9 s } Power Point Consider the inequalit n. means that the distance from 0 to is greater than units. units units Therefore, or. The solution set is { or }. Lesson 6- Solving pen Sentences Involving Absolute Value 7 Algebra Activit Materials: clock or watch that displas seconds Eplain to students that the purpose of this activit is not to see which student can guess closest to the length of a minute, but to collect data for the rest of the activit. If students do not understand wh the error cannot be negative, refer them to the eample involving election poll results at the beginning of the lesson. Lesson 6- Solving pen Sentences Involving Absolute Value 7

39 In-Class Eample Teaching Tip Point out to students that when the graph absolute value inequalities, the circles that indicate the points on the graph will either be both open or both closed. Solve 9. Then graph the solution set. { or } 0 Concept Check Absolute Value Ask students what the values inside the absolute value smbols are compared to in absolute value inequalities. The values inside the inequalities are compared to n and n. Practice/Appl Stud Notebook Power Point Have students record the three rules to remember when solving equations and inequalities involving absolute values. include an other item(s) that the find helpful in mastering the skills in this lesson. Stud Tip Greater Than When the absolute value is on the left and the inequalit smbol is or, the compound sentence uses or. Concept Check When solving inequalities of the form n, find the union of these two cases. Case The value inside the absolute value smbols is greater than the positive value of n. Case The value inside the absolute value smbols is less than the negative value of n. Eample Solve an Absolute Value Inequalit () Solve 8 6. Then graph the solution set. Write 8 6 as 8 6 or 8 6. Case If n, then n or n. If n, then n and n. If n, then n or n. Subtract 8 from each side. Simplif. Divide each side b. Simplif. Case Subtract 8 from each side. Simplif. Divide each side b. 7 Simplif. The solution set is { 7 or } In general, there are three rules to remember when solving equations and inequalities involving absolute value. Absolute Value Equations and Inequalities These properties are also true when or is replaced with or.. Compare and contrast the solution of 6 and the solution of 6.. See margin.. PEN ENDED Write an absolute value inequalit and graph its solution set.. FIND THE ERRR Leslie and Holl are solving. FIND THE ERRR Suggest that students determine whether Leslie and Holl are considering the correct two cases for each absolute value. Students should notice that Holl s second case, is incorrect. It should also be noted that if Holl s second case equation was, she would come up with the correct solution. 8 Chapter 6 Solving Linear Inequalities Leslie + = or + = + = + = = = Holl += or = + = += + = = Who is correct? Eplain our reasoning. Leslie; see margin for eplanation. Answers. The solution of 6 includes all values that are less than or greater than 8. The solution of 6 includes all values that are greater than and less than 8.. Sample answer: 0.You need to consider the case when the value inside the absolute value smbols is positive and the case when the value inside the absolute value smbols is negative. So or. 8 Chapter 6 Solving Linear Inequalities

40 Guided Practice GUIDED PRACTICE KEY Eercises Eamples, 6, 7 0,,, 7 0. See margin for graphs... 8 Application. {d.99 d.0} indicates increased difficult Practice and Appl For See Eercises Eamples 9,,, 9, Etra Practice See page 8.. Which graph represents the solution of k? a a. b. 0 c. d. 0. Which graph represents the solution of? c a. b c. d Epress the statement in terms of an inequalit involving absolute value. Do not solve. g 8 6 Ajar contains 8 gumballs. Amanda s guess was within 6 pieces. Solve each open sentence. Then graph the solution set. 7. r 0 {, 7} 8. c 6 {c c 8} 9. 0 w {w w or w }0. g 7 {g g 6 or g } For each graph, write an open sentence involving absolute value MANUFACTURING Amanufacturer produces bolts which must have a diameter within 0.00 centimeter of. centimeters. What are the acceptable measurements for the diameter of the bolts?. cm greatest acceptable diameter least acceptable diameter Match each open sentence with the graph of its solution set.. c a f b a c. 7. b d. 8. e e About the Eercises rganization b bjective Absolute Value Equations: 6, 9, 7, 8, 0,, 7 Absolute Value Inequalities:,, 7, 8, 0, 8 7, 9,, 8 dd/even Assignments Eercises and 8 are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: odd, odd, 6 7, 9,, 79 Average: odd, 6 7,, 79 Advanced: even, 8 even, 7 (optional: 7 79) Answers d f p t 8.. s 98 6 Epress each statement using an inequalit involving absolute value. Do not solve. 0. The ph of a buffered ee solution must be within 0.00 of a ph of 7... The temperature inside a refrigerator should be within. degrees of 8 F.. Ramona s bowling score was within 6 points of her average score of 98.. The cruise control of a car set at miles per hour should keep the speed within miles per hour of. s Lesson 6- Solving pen Sentences Involving Absolute Value 9 Teacher to Teacher Laurie Newton Crossler M.S., Salem, R To help students understand what their solution set to an absolute inequalit represents, I have students check their work b testing numbers in all of the regions of the number line prescribed b the inequalit. In Eample, I have them test a number less than, itself, a number greater than and less than 7, 7 itself, and a number greater than 7. Lesson 6- Solving pen Sentences Involving Absolute Value 9

41 Stud Stud Guide and and Intervention Intervention, p. 67 (shown) and p NAME DATE PERID Solving pen Sentences Involving Absolute Value Absolute Value Equations When solving equations that involve absolute value, there are two cases to consider. Case : The value inside the absolute value smbols is positive. Case : The value inside the absolute value smbols is negative. Eample Solve. Then Eample Write an inequalit graph the solution set. involving absolute value for the graph. Write as or. 0 or Find the point that is the same distance from as it is from. units units The solution set is {, }. 0 The graph is shown below. The distance from to is units. The distance from to is units So,. Eercises Solve each open sentence. Then graph the solution set.. {, }. {0, 8}. {, } 0. b {, }. w {, 7} 6. t {6, } {, } 8. 7, 9. p {0., 0.7} 0 0. d 00 0 {0, 0}. {, 6}. 6, For each graph, write an open sentence involving absolute value Skills Practice Practice, p. 69 and (Average) Practice, p. 70 (shown) 6- NAME DATE PERID Solving pen Sentences Involving Absolute Value Match each open sentence with the graph of its solution set.. 7 c a. 0. a b d c b d. 0 Epress each statement using an inequalit involving absolute value. Do not solve.. The height of the plant must be within inches of the standard -inch show size. h 6. The majorit of grades in Sean s English class are within points of 8. g 8 Solve each open sentence. Then graph the solution set. 7. z 9 {z z } 8. r 7 {r r or r } 0 9. t 6 9 {t t } 0. g 9 {g g or g 7} For each graph, write an open sentence involving absolute value FITNESS Taisha uses the elliptical cross-trainer at the gm. Her general goal is to burn 80 Calories per workout, but she varies b as much as Calories from this amount on an given da. What is the range of the number of Calories burned for Taisha s crosstrainer workout? {c c 0} 6. TEMPERATURE A thermometer is guaranteed to give a temperature no more than. F from the actual temperature. If the thermometer reads 8 F, what is the range for the actual temperature? {t 6.8 t 9.} Reading to Learn Mathematics, p NAME DATE PERID Reading to Learn Mathematics Solving pen Sentences Involving Absolute Value Pre-Activit How is absolute value used in election polls? Read the introduction to Lesson 6- at the top of page in our tetbook. What does the phrase margin of error mean to ou? Reading the Lesson Sample answer: The number of points a reported result ma be off from the eact result. In this poll, the number of people opposed to the ta lev ma be as high as 8% or as low as %. This can be written as the inequalit. Complete each compound sentence b writing and or or in the blank. Use the result to help ou graph the absolute value sentence. ELL 6 6 Lesson 6-9. See margin for graphs Tire Pressure Alwas inflate our tires to the pressure that is recommended b the manufacturer. The pressure stamped on the tire is the maimum pressure and should onl be used under certain circumstances. Source: %. {p 8 p } 0 Chapter 6 Solving Linear Inequalities Solve each open sentence. Then graph the solution set. 6. d d 8. 8 {, }. b 9 {, 7} 6. p 7 {7, 0} 7. c 8 {0.8, } 8. z {z z 7} 9. t 8 {t 0 t 6} 0. v {v v or v }. w 6 {w w or w 9}. s 7 {s s is a real number.}. k 8 k k or k. n 9 {n n }. 6r (d ) 7. 8 (w ) 9 {w 0 w 8} 8. h 6 8, 8 9. or For each graph, write an open sentence involving absolute value HEALTH For Eercises 6 and 7, use the following information. The average length of a human pregnanc is 80 das. However, a health, full-term pregnanc can be das longer or shorter. 6. d Write an absolute value inequalit for the length of a full-term pregnanc. 7. Solve the inequalit for the length of a full-term pregnanc. {d 66 d 9} 8. FIRE SAFETY The pressure of a tpical fire etinguisher should be within pounds per square inch (psi) of 9 psi. Write the range of pressures for safe fire etinguishers. {p 70 p 0} 9. HEATING Athermostat with a -degree differential will keep the temperature within degrees Fahrenheit of the temperature set point. Suppose our home has a thermostat with a -degree differential. If ou set the thermostat at 68 F, what is the range of temperatures in the house? {t 6 t 7} 0. ENERGY Use the margin of error indicated in the graph at the right to find the range of the percent of people who sa protection of the environment should have priorit over developing energ supplies.. TIRE PRESSURE Tire pressure is measured in pounds per square inch (psi). Tires should be kept within psi of the manufacturer s recommended tire pressure. If the recommended inflation pressure for a tire is 0 psi, what is the range of acceptable pressures?. CRITICAL THINKING State whether each open sentence is alwas, sometimes, or never true. a. never b. 6 alwas c. 0 sometimes USA TDAY Snapshots Environment first Americans sa protecting the environment should be given priorit over developing U.S. energ supplies. Preferences: Protection of environment % Neither/ other % No opinion % Equall important 6% Source: Gallup Poll of,060 adults; March -7, 00. Margin of error: plus or minus percentage points. Development of energ supplies 6% B Marc E. Mullins, USA TDAY Absolute Value Sentence Compound Sentence Graph. 8 8 or 8. and. or NAME DATE PERID 6- Enrichment, p. 7. How would ou write the compound sentence 7 or 7 as an absolute value sentence? 7 Helping You Remember. Recall that tells ou how man units the number is from zero on the number line. Eplain the meaning of n, n, and n b using the idea of the distance from to zero. n means is eactl n units from zero. n means is less than n units from zero. n means is more than n units from zero. Precision of Measurement The precision of a measurement depends both on our accurac in measuring and the number of divisions on the ruler ou use. Suppose ou measured a length of wood to the nearest one-eighth of an inch and got a length of 6 in The drawing shows that the actual measurement lies somewhere 9 between 6 in. and 6 in. This measurement can be written using 6 the smbol, which is read plus or minus. It can also be written as a compound inequalit. 9 in. 6 m 6 in. 6 0 Chapter 6 Solving Linear Inequalities

42 . {a. a.} SL/EC Practice Standardized Test Practice Maintain Your Skills Mied Review z e Getting Read for the Net Lesson. PHYSICAL SCIENCE Li-Cheng must add.0 milliliters of sodium chloride to a solution. The sodium chloride must be within 0. milliliter of the required amount. How much sodium chloride can she add and obtain the correct results?. ENTERTAINMENT Luis Gomez is a contestant on a television game show. He must guess within $00 of the actual price of a car without going over to win the car. The actual price of the car is $8,000. What is the range of guesses in which Luis can win the vehicle? {p 6,00 p 8,000}. CRITICAL THINKING The smbol means plus or minus. a. If., what are the values of?.8,. b. Write. as an epression involving absolute value.. 6. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How is absolute value used in election polls? Include the following in our answer: an eplanation of how to solve the inequalit describing the percent of people who are against the ta lev, and a prediction of whether ou think the ta lev will pass and wh. 7. Choose the replacement set that makes true. B A {, } B {, 7} C {, } D {, 7} 8. What can ou conclude about if 6 6? C A 0 B 0 C 6 D 6 9. FITNESS To achieve the maimum benefits from aerobic activit, our heart rate should be in our target zone. Your target zone is the range between 60% and 80% of our maimum heart rate. If Rafael s maimum heart rate is 90 beats per minute, what is his target zone? (Lesson 6-) between and beats per min Solve each inequalit. Then check our solution. (Lesson 6-) 60. m w 7 {w w } {m m } Find the slope and -intercept of each equation. (Lesson -) 6. ; 6. ; 6. 0 ; 0 Solve each equation or formula for the variable specified. (Lesson -8) I 66. I prt, for r r 67. e z, for 68. a 7, for p t a Find each sum or difference. (Lesson -) (8.9). Name the propert illustrated b each statement. (Lesson -6) ( ) 7. ( )a 7 a 7 Distributive Propert Substitution Propert PREREQUISITE SKILL Graph each equation. (To review graphing linear equations, see Lesson -.) See pp. 6A 6D ( ) 0 nline Lesson Plans Lesson 6- Solving pen Sentences Involving Absolute Value USA TDAY Education s nline site offers resources and interactive features connected to each da s newspaper. Eperience TDAY, USA TDAY s dail lesson plan, is available on the site and delivered dail to subscribers. This plan provides instruction for integrating USA TDAY graphics and ke editorial features into our mathematics classroom. Log on to Assess pen-ended Assessment Modeling Draw a number line on the chalkboard or overhead projector. Have a student volunteer use our number line to create the graph of an absolute value equation or inequalit. Then have the rest of the class write the equation or inequalit that the graph models. Getting Read for Lesson 6-6 PREREQUISITE SKILL To graph inequalities in two variables, ou must first graph the line that will be the boundar between shaded and nonshaded regions. Graphing that line is a prerequisite skill for Lesson 6-6. Use Eercises 7 79 to determine our students familiarit with graphing linear equations. Assessment ptions Quiz (Lessons 6- and 6-) is available on p. 9 of the Chapter 6 Resource Masters. Answer 6. Inequalities involving absolute value are used to represent margin of error. Answers should include the following. The inequalit representing the people who are against the ta lev is. To solve this inequalit, find the intersection of and. To solve these inequalities, add to each side of each inequalit. The solution set is { 8}. The votes for the ta lev can be between % and 0%. The votes against the ta lev can be between % and 8%. Depending on where the actual votes are in each range, it could either pass or fail. Lesson 6- Solving pen Sentences Involving Absolute Value

43 Lesson Notes Focus -Minute Check Transparenc 6-6 Use as a quiz or a review of Lesson 6-. Mathematical Background notes are available for this lesson on p. 6D. Building on Prior Knowledge In Chapter, students learned how to graph equations on the coordinate plane. In this lesson, the will graph a line and then decide which side of the line represents an inequalit. are inequalities used in budgets? Ask students: What does the in the quantit represent? her average cost of a cafeteria lunch, which is $ What does the in the quantit represent? her average cost of restaurant lunches, which is $ Wh can t this problem be represented with an inequalit containing onl one variable? Because the amount Hannah spends on cafeteria lunches and the amount she spends on restaurant lunches are not the same. Vocabular half-plane boundar Graphing Inequalities in Two Variables Graph inequalities on the coordinate plane. Solve real-world problems involving linear inequalities. Hannah budgets $0 a month for lunch. n most das, she brings her lunch. She can also bu lunch at the cafeteria or at a fast-food restaurant. She spends an average of $ for lunch at the cafeteria and an average of $ for lunch at a restaurant. How man times a month can Hannah bu her lunch and remain within her budget? Let represent the number of das she bus lunch at the cafeteria, and let represent the number of das she bus lunch at a restaurant. Then the following inequalit can be used to represent the situation. There are man solutions of this inequalit. GRAPH LINEAR INEQUALITIES The solution set of an inequalit in two variables is the set of all ordered pairs that satisf the inequalit. Like a linear equation in two variables, the solution set is graphed on a coordinate plane. Eample are inequalities used in budgets? M Monthl Budget Lunch (school das) $0 Entertainment $ Clothes $0 Fuel $60 The cost of eating the cost of eating is less than in the cafeteria plus in a restaurant or equal to $0. 0 rdered Pairs that Satisf an Inequalit From the set {(, 6), (, 0), (, ), (, )}, which ordered pairs are part of the solution set for? Use a table to substitute the and values of each ordered pair into the inequalit. True or False () (6) 6 false 0 () (0) true 9 () () true 0 () () false 8 Virginia SL STANDARD A.6 The student will select, justif, and appl an appropriate technique to graph linear functions and linear inequalities in two variables. Techniques will include slope-intercepts, - and -intercepts, graphing b transformation, and the use of the graphing calculator. The ordered pairs {(, 0), (, )} are part of the solution set of. In the graph, notice the location of the two ordered pairs that are solutions for in relation to the line. (, ) (, 6) (, ) (, 0) Chapter 6 Solving Linear Inequalities Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters Stud Guide and Intervention, pp. 7 7 Skills Practice, p. 7 Practice, p. 76 Reading to Learn Mathematics, p. 77 Enrichment, p. 78 Assessment, p. 9 Graphing Calculator and Spreadsheet Masters, p. Parent and Student Stud Guide Workbook, p. Transparencies -Minute Check Transparenc 6-6 Answer Ke Transparencies Technolog Interactive Chalkboard

44 Stud Tip Dashed Line Like a circle on a number line, a dashed line on a coordinate plane indicates that the boundar is not part of the solution set. Solid Line Like a dot on a number line, a solid line on a coordinate plane indicates that the boundar is included. The solution set for an inequalit in two variables contains man ordered pairs when the domain and range are the set of real numbers. The graphs of all of these ordered pairs fill a region on the coordinate plane called a half-plane. An equation defines the boundar or edge for each half-plane. Words Model Half-Planes and Boundaries An line in the plane divides the plane into two regions called half-planes. The line is called the boundar of each of the two half-planes. Half-Plane Boundar Half-Plane Consider the graph of. First determine the boundar b graphing, the equation ou obtain b replacing the inequalit sign with an equals sign. Since the inequalit involves -values greater than, but not equal to, the line should be dashed. The boundar divides the coordinate plane into two half-planes. To determine which half-plane contains the solution, choose a point from each half-plane and test it in the inequalit. Tr (, 0). Tr (, 6) false 6 true The half-plane that contains (, 6) contains the solution. Shade that half-plane. (, 0) (, 6) Teach GRAPH LINEAR INEQUALITIES In-Class Eamples From the set {(, ), (0, ), (, ), (, 0)}, which ordered pairs are part of the solution set for 8? {(, ), (, )} Teaching Tip Students ma need a quick refresher on slopeintercept form before the graph inequalities. Remind students that slope-intercept form is m b. Graph 6. Power Point Eample Graph. Graph an Inequalit Step Solve for in terms of. riginal inequalit Add to each side. Simplif. Step Graph. Since means or, the boundar is included in the solution set. The boundar should be drawn as a solid line. (continued on the net page) Lesson 6-6 Graphing Inequalities in Two Variables Differentiated Instruction Intrapersonal Before students work Eample, suggest that the first eplore the problem and tr to write a mathematicall correct answer. Students will likel write an inequalit in one variable or the ma create a table of possible answers. Then work through Eample as a class, so students can appreciate how the solution is described b the inequalit graph. Afterward, give students time to compare their original reasoning to the method shown in Eample. Lesson 6-6 Graphing Inequalities in Two Variables

45 SLVE REAL-WRLD PRBLEMS In-Class Eample Journalism Lee Cooper writes and edits short articles for a local newspaper. It generall takes her an hour to write an article and about a half-hour to edit an article. If Lee works up to 8 hours a da, how man articles can she write and edit in one da? Power Point ne solution is (, ), meaning she could write two articles and edit three articles. Stud Tip rigin as the Test Point Use the origin as a standard test point because the values are eas to substitute into the inequalit. Step Select a point in one of the half-planes and test it. Let s use (0, 0). riginal inequalit 0 (0) 0, 0 0 false Since the statement is false, the half-plane containing the origin is not part of the solution. Shade the other half-plane. CHECK Test a point in the other half plane, for eample, (, ). riginal inequalit (), Since the statement is true, the half-plane containing (, ) should be shaded. The graph of the solution is correct. SLVE REAL-WRLD PRBLEMS When solving real-world inequalities, the domain and range of the inequalit are often restricted to nonnegative numbers or whole numbers. Eample Write and Solve an Inequalit (0, 0) ADVERTISING Rosa Padilla sells radio advertising in 0-second and 60-second time slots. During ever hour, there are up to minutes available for commercials. How man commercial slots can she sell for one hour of broadcasting? Step Let equal the number of 0-second commercials. Let equal the number of 60-second or -minute commercials. Write an open sentence representing this situation. the number of the number of is min times 0-s commercials plus -min commercials up to min. Answers (p. ). The graph of is a line. The graph of does not include the boundar, and it includes all ordered pairs in the half-plane that contains the origin.. Sample answer: Advertising A tpical one-hour program on television contains 0 minutes of the program and 0 minutes of commercials. During peak periods, a 0-second commercial can cost an average of $. million. Source: Step Solve for in terms of. Step riginal inequalit Subtract from each side. Simplif. Since the open sentence includes the equation, graph as a solid line. Test a point in one of the half-planes, for eample (0, 0). Shade the half-plane containing (0, 0) since 0 (0) is true Chapter 6 Solving Linear Inequalities If the test point results in a true statement, shade the half-plane that contains the point. If the test point results in a false statement, shade the other half-plane. Chapter 6 Solving Linear Inequalities

46 Concept Check. See margin. Guided Practice GUIDED PRACTICE KEY Eercises Eamples, 6 0 Application Step Eamine the solution. Rosa cannot sell a negative number of commercials. Therefore, the domain and range contain onl nonnegative numbers. She also cannot sell half of a commercial. Thus, onl points in the shaded half-plane whose - and -coordinates are whole numbers are possible solutions. ne solution is (, 8). This represents twelve 0-second commercials and eight 60-second commercials in a one hour period.. Compare and contrast the graph of and the graph of.. PEN ENDED Write an inequalit in two variables and graph it.. Eplain wh it is usuall onl necessar to test one point when graphing an inequalit. Determine which ordered pairs are part of the solution set for each inequalit.., {(, 0), (, ), (, ), (, )} {(, 0), (, )}., {(, 6), (0, ), (, ), (, )} {(, 6)} 6. Which graph represents? b a. b. c. Graph each inequalit See margin ENTERTAINMENT Coach Rile wants to take her softball team out for pizza and soft drinks after the last game of the season. She doesn t want to spend more than $60. Write an inequalit that represents this situation and graph the solution set. 60; See margin for graph (, 8) Practice/Appl Stud Notebook Have students complete the definitions/eamples for the remaining terms on their Vocabular Builder worksheets for Chapter 6. include an other item(s) that the find helpful in mastering the skills in this lesson. About the Eercises rganization b bjective Graph Linear Inequalities: 7 Solve Real-World Problems: 8 dd/even Assignments Eercises 7 are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: 7 odd, odd, 8 9, 6 Average: 7 odd, 8, 6 Advanced: 6 even, 0 6 Lesson 6-6 Graphing Inequalities in Two Variables Lesson 6-6 Graphing Inequalities in Two Variables

47 Stud Stud Guide and and Intervention Intervention, p. 7 (shown) and p NAME DATE PERID Graphing Inequalities in Two Variables Graph Linear Inequalities The solution set of an inequalit that involves two variables is graphed b graphing a related linear equation that forms a boundar of a half-plane.the graph of the ordered pairs that make up the solution set of the inequalit fill a region of the coordinate plane on one side of the half-plane. Eample Graph. Graph. Since is the same as and, the boundar is included in the solution set and the graph should be drawn as a solid line. Select a point in each half plane and test it. Choose (0, 0) and (, ). 0 (0) () 0 is false. 6 is true. The half-plane that contains (, ) contains the solution. Shade that half-plane. Eercises Graph each inequalit.... Lesson 6-6 indicates increased difficult Practice and Appl For Eercises See Eamples Etra Practice See page 86. Determine which ordered pairs are part of the solution set for each inequalit.., {(0, ), (, ), (6, 8), (, )} {(, ), (, )}., {(, ), (, ), (, ), (, )} {(, ), (, )}., {(, 7), (, 0), (, ), (6, )} {(, 0), (, ), (6, )}. 6, {(, ), (, ), (6, ), (, )} {(, ), (, )} , {(, ), (0, ), (, ), (, 0)} {(0, ), (, )} 7. 7, {(, ), (, ), (, ), (, )} {(, ), (, )} 8., {(6, ), (, 8), (, ), (, 7)} {(, )} 9., {(, ), (, ), (6, 7), (0, 0)} {(6, 7)} Match each inequalit with its graph c a. b.. a d Skills Practice Practice, p. 7 and (Average) Practice, p. 76 (shown) 6-6 NAME DATE PERID Graphing Inequalities in Two Variables Determine which ordered pairs are part of the solution set for each inequalit.. 6, {(, ), (, ), (, ), (, )} {(, ), (, )}., {(6, ), (, ), (, ), (, )} {(, )}. 6 b c. d.., {(, ), (, ), (, ), (, )} {(, ), (, )} Match each inequalit with its graph.. 0 d a. b.. 9 c 6. b 7. 6 a c. d.. Is the point A(, ) on, above, or below the graph of? on. Is the point B(0, ) on, above, or below the graph of? above Graph each inequalit MVING A moving van has an interior height of 7 feet (8 inches). You have boes in inch and inch heights, and want to stack them as high as possible to fit. Write an inequalit that represents this situation. 8 BUDGETING For Eercises and, use the following information. Satchi found a used bookstore that sells pre-owned videos and CDs. Videos cost $9 each, and CDs cost $7 each. Satchi can spend no more than $.. Write an inequalit that represents this situation Does Satchi have enough mone to bu videos and CDs? No, the purchases will be $9, which is greater than $. Reading to Learn Mathematics, p Reading to Learn Mathematics Graphing Inequalities in Two Variables NAME DATE PERID Pre-Activit How are inequalities used in budgets? Read the introduction to Lesson 6-6 at the top of page in our tetbook. What do and represent in the terms and? the average amount spent on a cafeteria lunch and a fast-food lunch Reading the Lesson. Complete the chart to show which tpe of line is needed for each smbol. Smbol Tpe of Line Boundar Part of Solution? dashed no dashed no solid es solid es. If a test point results in a false statement, what do ou know about the graph? The half-plane containing the test point is not part of the solution and is not shaded.. If a test point results in a true statement, what do ou know about the graph? The half-plane containing the test point is part of the solution and is shaded.. When can the origin not be used as a test point? The origin cannot be used as a test point when it is on the boundar. Helping You Remember. The two-variable inequalities in this lesson can be solved for in terms of to get a sentence in slope-intercept form. It looks much like a slope-intercept equation, but it has an inequalit smbol instead of an equals sign. For eample, can be written as.eplain how to graph an inequalit once it is written in slope-intercept form. Use the idea that greater can mean above and less can mean below. Draw the boundar line. If the inequalit smbol is or, make the boundar dashed. If the smbol is or, make the boundar line solid. If the smbol in the slope-intercept inequalit is or, shade below the boundar to indicate smaller values of. If the smbol is or, shade above the boundar to indicate greater values of. ELL NAME DATE PERID 6-6 Enrichment, p. 78 Using Equations: Ideal Weight You can find our ideal weight as follows. A woman should weigh 00 pounds for the first feet of height and additional pounds for each inch over feet ( feet 60 inches). A man should weigh 06 pounds for the first feet of height and 6 additional pounds for each inch over feet. These formulas appl to people with normal bone structures. To determine our bone structure, wrap our thumb and inde finger around the wrist of our other hand. If the thumb and finger just touch, ou have normal bone structure. If the overlap, ou are small-boned. If the don t overlap, ou are large-boned. Small-boned people should decrease their calculated ideal weight b 0%. Large-boned people should increase the value b 0%. Calculate the ideal weights of these people.. woman, ft in., normal-boned. man, ft in., large-boned 0 lb 89. lb Graph each inequalit See pp. 6A 6D ( ) 8 7. ( ) PSTAGE For Eercises 8 and 9, use the following information. The U.S. Postal Service limits the size of packages to those in which the length of the longest side plus the distance around the thickest part is less than or equal to 08 inches. 8. Write an inequalit that represents this situation. d Are there an restrictions on the domain or range? The solution set is limited to pairs of positive numbers. nline Research Data Update What are the current postage rates and regulations? Visit to learn more. SHIPPING For Eercises 0 and, use the following information. Adeliver truck is transporting televisions and microwaves to an appliance store. The weight limit for the truck is 000 pounds. The televisions weigh 77 pounds, and the microwaves weigh pounds. 0. Write an inequalit for this situation. 77t m 000. Will the truck be able to deliver televisions and microwaves at once? No, the weight will be greater than 000 pounds. 6 Chapter 6 Solving Linear Inequalities small-boned. ft tall 6 Chapter 6 Solving Linear Inequalities

48 A linear inequalit can be used to represent trends in lmpic times. Visit com/webquest to continue work on our WebQuest project. SL/EC Practice Standardized Test Practice FALL DANCE For Eercises, use the following information. Tickets for the fall dance are $ per person or $8 for couples. In order to cover epenses, at least $00 worth of tickets must be sold.. Write an inequalit that represents this situation. s 8c 00. Graph the inequalit. See margin.. If 00 single tickets and couple tickets are sold, will the committee cover its epenses? es. CRITICAL THINKING Graph the intersection of the graphs of and. See margin. 6. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How are inequalities used in budgets? Include the following in our answer: an eplanation of the restrictions placed on the domain and range of the inequalit used to describe the number of times Hannah can bu her lunch, and three possible solutions of the inequalit. 7. Which ordered pair is not a solution of? D A (, ) B (, 8) C (, ) D (, 6) Assess pen-ended Assessment Speaking Have students eplain wh, when the use linear inequalities to solve real-world problems, the solution is often not the entire half-plane. The eplanations should include an eample. Assessment ptions Quiz (Lesson 6-6) is available on p. 9 of the Chapter 6 Resource Masters. 8. Which inequalit is represented b the graph at the right? B A B C D Maintain Your Skills Mied Review 9. See margin for graphs. Solve each open sentence. Then graph the solution set. (Lesson 6-) 9. t {7, } { } { or } Solve each compound inequalit. Then graph the solution. (Lesson 6-). 6 and. m or m { 7 6} {m m or m } State whether each percent of change is a percent of increase or decrease. Then find the percent of change. Round to the nearest whole percent. (Lesson -7). original: 00. original: original: new: 7 decrease; % new: increase; % new: 7 increase; % Solve each equation. (Lesson -) 7. d 7 8. n h 00. Simplif. (Lesson -) c 9 c 6. a b a 7b. c s 8c s Lesson 6-6 Graphing Inequalities in Two Variables 7 Answers 6. The amount of mone spent in each categor must be less than or equal to the budgeted amount. How much ou spend on individual items can var. Answers should include the following. The domain and range must be positive integers. Sample answers: Hannah could bu cafeteria lunches and restaurant lunches, cafeteria lunches and restaurant lunches, or 8 cafeteria lunches and restaurant lunch Lesson 6-6 Graphing Inequalities in Two Variables 7

49 Graphing Calculator Investigation A Follow-Up of Lesson 6-6 Getting Started Reset the Calculator Have students enter nd DRAW ENTER to clear an stored drawings. You ma also need students to reset the viewing windows. Have them enter ZM 6 to the graph in the standard viewing window. Teach Tell students that when using the DRAW function, the first boundar the enter is alwas the lower boundar. The comma separates the lower and upper boundaries. An alternative method to graphing inequalities is to enter the function in the Y= table at Y=. Then highlight the smbol in front of the Y= entr and press ENTER until either shading above or below appears. Assess Make sure students understand that the solutions for the inequalities can be anwhere in the shaded area of the graph, including the lines themselves. Answers. is shaded below the line. is shaded above the line. c. Graphing Inequalities You can use a TI-8 Plus graphing calculator to investigate the graphs of inequalities. Since graphing calculators onl shade between two functions, enter a lower boundar as well as an upper boundar for each inequalit. Graph two different inequalities on our graphing calculator. Graph. Clear all functions from the Y= list. KEYSTRKES: Graph in the standard window. KEYSTRKES: nd [DRAW] 7 ( ) 0, X,T,,n CLEAR The lower boundar is Ymin or 0. The upper boundar is. All ordered pairs for which is less than or equal to lie below or on the line and are solutions. ) ENTER A Follow-Up of Lesson 6-6 Virginia SL STANDARD A.6 The student will select, justif, and appl an appropriate technique to graph linear functions and linear inequalities in two variables. Techniques will include slope-intercepts, - and - intercepts, graphing b transformation, and the use of the graphing calculator. Graph. Clear the drawing that is currentl displaed. KEYSTRKES: nd [DRAW] Eercises b. Sample answer: {(0, ), (, 7), (, 6), (.,.)}. Compare and contrast the two graphs shown above. See margin. Rewrite as and graph it. KEYSTRKES: nd [DRAW] 7 X,T,,n, 0 ENTER This time, the lower boundar is. The upper boundar is Yma or 0. All ordered pairs for which is greater than or equal to lie above or on the line and are solutions.. Graph the inequalit in the standard viewing window. a. What functions do ou enter as the lower and upper boundaries? ; Yma or 0 b. Using our graph, name four solutions of the inequalit.. Suppose student movie tickets cost $ and adult movie tickets cost $8. You would like to bu at least 0 tickets, but spend no more than $80. a. Let number of student tickets and number of adult tickets. Write two inequalities, one representing the total number of tickets and the other representing the total cost of the tickets. 0; ; b. Which inequalities would ou use as the lower and upper boundaries? 0. 0 c. Graph the inequalities. Use the viewing window [0, 0] scl: b [0, 0] scl:. See margin. d. Name four possible combinations of student and adult tickets. Sample answer: {(8, ), (0, ), (, ), (0, 0)} 8 Chapter 6 Solving Linear Inequalities ) 8 Chapter 6 Solving Linear Inequalities

50 Stud Guide and Review Vocabular and Concept Check Addition Propert of Inequalities (p. 8) boundar (p. ) compound inequalit (p. 9) Division Propert of Inequalities (p. 7) Choose the letter of the term that best matches each statement, algebraic epression, or algebraic sentence.. {w w } f. If, then. e. p and p 0 d. If a b, then a b. a. the graph on one side of a boundar c 6. If s t, then s 7 t 7. g 7. g 7 or g h 8. If m n, then m 7 n 7. b 6- See pages 8. Eamples Solving Inequalities b Addition and Subtraction Concept Summar If an number is added to each side of a true inequalit, the resulting inequalit is also true. If an number is subtracted from each side of a true inequalit, the resulting inequalit is also true. Solve each inequalit. f 9 v 9 6 f 9 riginal inequalit v 9 6 riginal inequalit f Subtract. v Add. f Simplif. v Simplif. The solution set is {f f }. The solution set is {v v }. Eercises Solve each inequalit. Then check our solution, and graph it on a number line. See Eamples on pages See pp. 6A 6D. 9. c 0. r 7. w. a n (0.0).. g (.). 7h 6h 6. b b 7. Define a variable, write an inequalit, and solve the problem. Then check our solution. Twent-one is no less than the sum of a number and negative two. Sample answer: Let n the number; n (); {n n }. For more information about Foldables, see Teaching Mathematics with Foldables. TM half-plane (p. ) intersection (p. 9) Multiplication Propert of Inequalities (p. ) set-builder notation (p. 9) Subtraction Propert of Inequalities (p. 9) union (p. 0) a. Addition Propert of Inequalities b. Division Propert of Inequalities c. half-plane d. intersection e. Multiplication Propert of Inequalities f. set-builder notation g. Subtraction Propert of Inequalities h. union Chapter 6 Stud Guide and Review 9 Have students look through the chapter to make sure the have included eamples in their Foldable journal for each tpe of inequalit the learned to solve. Encourage students to refer to their Foldable journal while completing the Stud Guide and Review and to use them in preparing for the Chapter Test. Vocabular and Concept Check This alphabetical list of vocabular terms in Chapter 6 includes a page reference where each term was introduced. Assessment Avocabular test/review for Chapter 6 is available on p. 9 of the Chapter 6 Resource Masters. Lesson-b-Lesson Review For each lesson, the main ideas are summarized, additional eamples review concepts, and practice eercises are provided. Vocabular PuzzleMaker ELL The Vocabular PuzzleMaker software improves students mathematics vocabular using four puzzle formats crossword, scramble, word search using a word list, and word search using clues. Students can work on a computer screen or from a printed handout. MindJogger Videoquizzes ELL MindJogger Videoquizzes provide an alternative review of concepts presented in this chapter. Students work in teams in a game show format to gain points for correct answers. The questions are presented in three rounds. Round Concepts ( questions) Round Skills ( questions) Round Problem Solving ( questions) Chapter 6 Stud Guide and Review 9

51 Stud Guide and Review Chapter 6 Stud Guide and Review 6- See pages. Eamples Solving Inequalities b Multiplication and Division Concept Summar If each side of a true inequalit is multiplied or divided b the same positive number, the resulting inequalit is also true. If each side of a true inequalit is multiplied or divided b the same negative number, the direction of the inequalit must be reversed. Solve each inequalit. g 6 d 8. {v v } 9. {r r 6} 0. {z z }. {m m }. {b b 6}. {d d 6}. {w w }. {p p } 6- See pages 7. Eample g 6 riginal inequalit d riginal inequalit g 6 g 9 Simplif. d 0 Simplif. Divide and change to. d Multipl each side b. The solution set is {g g 9}. The solution set is {d d 0}. Eercises Solve each inequalit. Then check our solution. See Eamples on pages v r 7 0. z 7. 9m 99 b.. d. w. p 6. Define a variable, write an inequalit, and solve the problem. Then check our solution. Eight percent of a number is greater than or equal to. Sample answer: Let n the number; 0.80n ; {n n 0}. Solving Multi-Step Inequalities Concept Summar Multi-step inequalities can be solved b undoing the operations. Remember to reverse the inequalit sign when multipling or dividing each side b a negative number. When solving equations that contain grouping smbols, first use the Distributive Propert to remove the grouping smbols. Solve (n ) 7n 8. (n ) 7n 8 riginal inequalit n 7n 8 Distributive Propert n 7n 7n 8 7n Subtract 7n from each side. n 8 Simplif. n 8 Add to each side. n Simplif. n Divide each side b and change < to >. n Simplif. The solution set is {n n }. 60 Chapter 6 Solving Linear Inequalities 60 Chapter 6 Solving Linear Inequalities

52 Chapter 6 Stud Guide and Review Stud Guide and Review 6- Solving Compound Inequalities 6- See pages 9. Eamples Eercises Solve each inequalit. Then check our solution. See Eamples on pages. 7. h n b 7b 60. (q ) q. 7(g 8) (g ) g. ( ) { }. 7n 0 {n n 7}. Define a variable, write an inequalit, and solve the problem. Then check our solution. Two thirds of a number decreased b 7 is at least 9. Sample answer: Let n the number; n 7 9; {n n }. 7. {h h } 8. {n n } 9. { } 0. {b b 9}. {q q 7} Concept Summar The solution of a compound inequalit containing and is the intersection of the graphs of the two inequalities. The solution of a compound inequalit containing or is the union of the graphs of the two inequalities. Graph the solution set of each compound inequalit. and 8 or Answers Find the Find the intersection union. The solution set is { }. The solution set is { 8}. Eercises Solve each compound inequalit. Then graph the solution set. See Eamples on pages See margin for graphs. 6. p 7. k {p p } {k k } 8. w 8 or{w w is a real w w number.} 9. a 8 or 0. m 8 and. 0 and a m 7 9 {a a or a 6} { } 6- Solving pen Sentences Involving Absolute Value See pages Concept Summar. If n, then n or n. n 0 n If n, then n and n. If n, then n or n. n 0 n n 0 n Chapter 6 Stud Guide and Review 6 Chapter 6 Stud Guide and Review 6

53 Stud Guide and Review Etra Practice, see pages 8 8. Mied Problem Solving, see page 88. Answers Eample. {, 0}. {7, }. {h h or h }. {w w 9 or w 7} 6-6 See pages 7. Eample Solve or The solution set is {, 9}. Eercises Solve each open sentence. Then graph the solution set. See Eamples,, and on pages See margin for graphs.. w 8. q. h 7. w 8 6. r 0 7. t d 8 {r r 7} {t 7 t } Graphing Inequalities in Two Variables Concept Summar To graph an inequalit in two variables: Step Determine the boundar and draw a dashed or solid line. Step Select a test point. Test that point. Step Shade the half-plane that contains the solution. Graph. Since the boundar is included in the solution, draw a solid line. Test the point (0, 0). riginal inequalit 0 0 0, 0 0 true The half plane that contains (0, 0) should be shaded. d d 6. Eercises Determine which ordered pairs are part of the solution set for each inequalit. See Eample on page. 0. 9, {(, ), (, ), (, 7), (, )} {(, 7)}., (, ),, 7, (, 6), (, 0) {(, ), (, 6)}. 6, {(, ), (, ), (, 8), (, )} {(, ), (, )}. 8, {(, 0), (, 6), (, 0), (, 6)} {(, 0), (, 6)} Graph each inequalit. See Eample on pages and. 7. See margin Chapter 6 Solving Linear Inequalities Chapter 6 Solving Linear Inequalities

54 Practice Test Vocabular and Concepts. Write the set of all numbers t such that t is greater than or equal to 7 in set-builder notation. {t t 7}. Show how to solve 6(a ) a 8. Justif our work. See pp. 6A 6D.. PEN ENDED Give an eample of a compound inequalit that is an Sample answers: intersection and an eample of a compound inequalit that is a union. 8; or 8. Compare and contrast the graphs of and. Both graphs have dots at and. The graph of is darkened between the two dots. The graph of is darkened to the right of the dot at and to the left of the dot at. Skills and Applications Solve each inequalit. Then check our solution.. g 6 {g g 7} 6. 9p 8p 8 {p p 8} 7. d d {d d 9} w {w w } 9. b 99 {b b.} 0. m 8m 7{m m.}. (k ) {k k }. f {f f }. 0.( ) 0.8(0. ) { 0}. REAL ESTATE A homeowner is selling her house. She must pa 7% of the selling price to her real estate agent after the house is sold. To the nearest dollar, what must be the selling price of her house to have at least $0,000 after the agent is paid? at least $8,80. Solve 6 r. 6. Solve d. {d d is a real number.} Solve each compound inequalit. Then graph the solution set. 7. See pp. 6A 6D for graphs. 7. r and r {r r } 8. n 7 or n {n n or n } 9. 9 p and 8p {p p } 0. a 7 {a a 6}. 7 s s s or s. 7 z {z z 0.8 or z } Define a variable, write an inequalit, and solve each problem. Then check our solution.. Sample answer: Let n the number.. ne fourth of a number is no less than. n ; {n n }. Three times a number subtracted from is less than two. n ; {n n }. Five less than twice a number is between and. n ; {n 9 n } 6. TRAVEL Megan s car gets between 8 and miles per gallon of gasoline. If her car s tank holds gallons, what is the range of distance that Megan can drive her car on one tank of gasoline? between 70 and mi Graph each inequalit See pp. 6A 6D STANDARDIZED TEST PRACTICE Which inequalit is represented b the graph? B SL/EC Practice A B C D Portfolio Suggestion Chapter 6 Practice Test 6 Introduction In mathematics, there is often more than one wa to solve a problem. In order to solve an inequalit, for eample, ou can write a solution set or graph the inequalit. Ask Students Find an inequalit from our work in this chapter and show two different was to solve it. Place our work in our portfolio. Assessment ptions Vocabular Test A vocabular test/review for Chapter 6 can be found on p. 9 of the Chapter 6 Resource Masters. Chapter Tests There are si Chapter 6 Tests and an pen- Ended Assessment task available in the Chapter 6 Resource Masters. Chapter 6 Tests Form Tpe Level Pages MC basic A MC average 8 8 B MC average 8 8 C FR average 8 86 D FR average FR advanced MC = multiple-choice questions FR = free-response questions pen-ended Assessment Performance tasks for Chapter 6 can be found on p. 9 of the Chapter 6 Resource Masters. A sample scoring rubric for these tasks appears on p. A. First Semester Test A test for Chapters 6 can be found on pp of the Chapter 6 Resource Masters. EamView Pro Use the networkable EamView Pro to: Create multiple versions of tests. Create modified tests for Inclusion students. Edit eisting questions and add our own questions. Use built-in state curriculum correlations to create tests aligned with state standards. Change English tests to Spanish and vice versa. Chapter 6 Test Practice 6

55 Standardized Test Practice These two pages contain practice questions in the various formats that can be found on the most frequentl given standardized tests. A practice answer sheet for these two pages can be found on p. A of the Chapter 6 Resource Masters. NAME DATE PERID 6Standardized Test Practice Practice Student Recording Sheet, p. A Student Record Sheet (Use with pages 6 6 of the Student Edition.) Part Multiple Choice Select the best answer from the choices given and fill in the corresponding oval. A B C D A B C D 7 A B C D A B C D 8 A B C D 6 A B C D 9 Part Short Response/Grid In A A A B B B C C C D D D SL/EC Practice Part Record our answers on the answer sheet provided b our teacher or on a sheet of paper.. Which of the following is a correct statement? (Lesson -) B A C (6)(7) (Lesson -) D A C Multiple Choice B D B D 6. The graph of the function is shown. If the graph is translated units up, which equation will best represent the new line? (Lesson -) A A C B D 7. The table shows a set of values for and. Which equation best represents this set of data? (Lesson -8) D Solve the problem and write our answer in the blank. For Questions and, also enter our answer b writing each number or smbol in a bo. Then fill in the corresponding oval for that number or smbol. 0 / / (grid in) (grid in) / / Part Etended Response Record our answers for Questions 9 on the back of this paper. Answers. Aclindrical can has a volume of 6 cubic centimeters. Its height is centimeters. What is the radius of the can? Use the formula V r h. (Lessons -8 and -8) C A C.8 cm B 7. cm cm D 7. cm. Afurnace repair service charged a customer $80 for parts and $6 per hour worked. The bill totaled $77.0. About how long did the repair technician work on the furnace? (Lessons - and -) B A C B 0 D 8. Ali s grade depends on test scores. n the first tests, she earned scores of 78, 8, and 7. She wants to average at least 80. Which inequalit can she use to find the score that she needs on the fourth test in order to earn a final grade of at least 80? (Lesson 6-) B A 80 Additional Practice See pp in the Chapter 6 Resource Masters for additional standardized test practice. A C 0. hour B. hours hours D hours. The formula P (0 A) determines the recommended maimum pulse rate P during eercise for a person who is A ears old. Cameron is ears old. What is his recommended maimum pulse rate during eercise? (Lesson -8) B B C D Which inequalit is represented b the graph? (Lesson 6-) C 0 A 6 B 6 A B C 7 D 6 C D 6 Chapter 6 Solving Linear Inequalities EamView Pro Special banks of standardized test questions similar to those on the SAT, ACT, TIMSS 8, NAEP 8, and Algebra End-of-Course tests can be found on this CD-RM. 6 Chapter 6 Solving Linear Inequalities

56 Preparing for Standardized Tests For test-taking strategies and more practice, see pages Part Short Response/Grid In Record our answers on the answer sheet provided b our teacher or on a sheet of paper. 0. A die is rolled. What are the odds of rolling a number less than? (Lesson -6) : or :. Acar is traveling at an average speed of miles per hour. How man minutes will it take the car to travel 7 miles? (Lesson -6) 0. The price of a tape plaer was cut from $8 to $6. What was the percent of decrease? (Lesson -7) %. Quadrilateral MNP has vertices M(0, ), N(, 8), (, ), and P(, 9). Find the coordinates of the vertices of the image if it is reflected over the -ais. (Lesson -) M (0, ), N (, 8), (, ), P (, 9). Write an equation in slope-intercept form that describes the graph. (Lesson -). A line is parallel to the graph of the equation. What is the slope of the parallel line? (Lessons - and -6) 7. Find all values of that make the inequalit true. (Lesson 6-) { or 8} 8. Graph the equation and indicate which region represents. (Lesson 6-6) See margin. Part Etended Response Record our answers on a sheet of paper. Show our work. 9. The Carlson famil is building a house on a lot that is 9 feet long and 8 feet wide. (Lessons 6-, 6-, and 6-) a. Town law states that the sides of a house cannot be closer than 0 feet to the edges of a lot. Write an inequalit for the possible lengths of the Carlson famil s house, and solve the inequalit. 9 0; 7 b. The Carlson famil wants their house to be at least 800 square feet and no more than 00 square feet. The also want their house to have the maimum possible length. Write an inequalit for the possible widths of their house, and solve the inequalit. Round our answer to the nearest whole number of feet w 00; 9 w 0. For the graph below, write an open sentence involving absolute value. (Lesson 6-) Sample answer: Solve (0 8) ( ) for.. Astreet vendor sells hot dogs for $ each (Lesson 6-) 8 and bratwurst for $ each. In order to cover d. The solution set is limited to nonnegative numbers his dail epenses, he must sell at least $00 because the vendor cannot sell less than zero product. worth of food. (Lesson 6-6) Test-Taking Tip Questions and Know the slope-intercept form of linear equations: m b. Understand the definition of slope. Recognize the relationships between the slopes of parallel lines and between the slopes of perpendicular lines. a. Write an inequalit that represents this situation. h b 00 b. If 68 hot dogs and 8 bratwursts are sold, will the street vendor cover his costs? no c. Find a number of hotdogs and bratwursts that could be sold and cover the dail costs. Sample answer: 00 hot dogs and bratwursts d. Are there an restrictions on the domain and range? Eplain. See margin. Chapter 6 Standardized Test Practice 6 Evaluating Etended Response Questions Etended Response questions are graded b using a multilevel rubric that guides ou in assessing a student s knowledge of a particular concept. Goal: Write inequalities to describe the possible dimensions of a house. Sample Scoring Rubric: The following rubric is a sample scoring device. You ma wish to add more detail to this sample to meet our individual scoring needs. Score Criteria A correct solution that is supported b well-developed, accurate eplanations A generall correct solution, but ma contain minor flaws in reasoning or computation A partiall correct interpretation and/or solution to the problem A correct solution with no supporting evidence or eplanation 0 An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given Answer 8. d. The solution set is limited to nonnegative numbers because the vendor cannot sell less than zero products. Chapter 6 Standardized Test Practice 6