New Vocabulary equivalent inequalities. x 1 4, 7 and x, 3 are equivalent inequalities.

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1 -. Plan - Solving Inequalities Using Addition and Subtraction Objectives To use addition to solve To use subtraction to solve Eamples Using the Addition Property of Inequality Solving and Checking Solutions Using the Subtraction Property of Inequality Real-World Problem Solving Math Background What You ll Learn To use addition to solve To use subtraction to solve... And Why To solve a problem involving safe loads, as in Eample Check Skills You ll Need GO for Help Lesson - and Review page 8 Complete each statement with R,, or S. S R S Solve each equation.. 9. n 6. t 9 7. k 6 6 New Vocabulary equivalent Part Using Addition to Solve Inequalities Equivalent are with the same solutions. For eample,, 7 and, are equivalent. Students may have used a balance scale to model equations. Here it may help them to solve if they refer to an unbalanced scale. More Math Background: p. 98C Lesson Planning and Resources See p. 98E for a list of the resources that support this lesson. PowerPoint Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Adding Rational Numbers Lesson -: Eample Etra Skills and Word Problems Practice, Ch. Solving One-Step Equations Review p. 8: Eample You can add the same value to each side of an inequality, just as you did with equations. Using the Addition Property of Inequality Solve,. Graph the solution., Add to each side., 8 + < 7 < Key Concepts Property Addition Property of Inequality For every real number a, b, and c, if a. b, then a c. b c; if a, b, then a c, b c. Eamples., so..,, so,. This property is also true for $ and #. Simplify Solve m 6.. Graph your solution. m S ; 0 06 Chapter Solving Inequalities 06 Special Needs L To help students understand infinity, use the number line in Eample. Ask: What number is to the right of 0? of? Then ask: Will there ever be a number that does not have yet another number to its right? No learning style: visual Below Level L Present students with several pairs of and their solutions. Have students take turns eplaining what was done to each inequality to produce its solution. learning style: verbal

2 Vocabulary Tip The word infinite indicates that the number of solutions is unlimited. The solutions cannot be listed. An inequality has an infinite number of solutions, so it is not possible to check all the solutions. You can check your computations and the direction of the inequality symbol. The steps below show how to check that < 8 describes the solutions to Eample. Step Check the computation. See if 8 is the solution to the equation. 8 0 Substitute 8 for. Step Check the inequality symbol. Choose any number less than 8 and substitute it into,. In this case, use 7. < 7 < Substitute 7 for. < Since the computation and the direction of the inequality symbol are correct,, and, 8 are equivalent. So the solution of, is, 8. Solving and Checking Solutions Solve #. Graph and check your solution. # Add to each side. 7 # Simplify.. Teach Guided Instruction Tactile Learners Let students use tiles to model the inequality. Have students write the inequality sign on an inde card. Instruct them to place the card between the two sets of tiles. Then have them solve the inequality. When the solution is complete, students can check by substituting tiles for a value of in the original model. Special Needs L Remind students that the dot and arrow in Eample means that 7 and all numbers to the right of 7 on the number line are solutions to the inequality For: Linear Inequalities Activity Use: Interactive Tetbook, - Check Check the computation. 0 7 Substitute 7 for. # Check the direction of the inequality. # 8 Substitute 8 for. # Solve n 7 #. Graph and check your solution. n K ; Part Using Subtraction to Solve Inequalities You can subtract the same number from each side of an inequality to create an equivalent inequality. Key Concepts Property Subtraction Property of Inequality For every real number a, b, and c, if a. b, then a c. b c; if a, b, then a c, b c. Eamples., so.,, so, This property is also true for $ and #. Lesson - Solving Inequalities Using Addition and Subtraction 07 Advanced Learners L Have students discuss whether is equivalent to - - and eplain each step of their reasoning. learning style: verbal English Language Learners ELL Be sure students understand how the solution of an equation differs from the solution of an addition or subtraction inequality. Ask them to compare and to give a real-world eample of each. learning style: verbal 07

3 PowerPoint Additional Eamples Solve p -. Graph the solution. p R 0 6 Using the Subtraction Property of Inequality Solve y,7. Graph the solution. y,7 Subtract from each side. y, Simplify. Solve 8 $ d -. Graph and check your solution. 0 L d, or d K Solve t $ 8. Graph and check your solution. t L ; You can use to model real-world situations. 0 6 Guided Instruction Real-World Problem Solving Diversity Some students may have lived in a shelter or may live in one now. Be sure other students are sensitive when discussing the question. PowerPoint Additional Eamples Solve c + 7. Graph the solution. c S 0 In order to receive a B in your literature class, you must earn more than 0 points of reading credits. Last week you earned 0 points. This week you earned 90 points. How many more points must you earn to receive a B? points Resources Daily Notetaking Guide - L Daily Notetaking Guide - Adapted Instruction L Closure Ask students to compare solving using addition and subtraction with solving equations using addition and subtraction. In both cases, you must add or subtract the same amount to or from both sides. A B C D E A B C D E A B C D E A B C D E A B C D E B C D E Test-Taking Tip You can test inequality choices by substituting a reasonable value for the variable. If the value is NOT a solution of the inequality you can eliminate that choice as the correct answer. EXERCISES Eamples, (pages 06, 07) 08 Chapter Solving Inequalities Multiple Choice The maimum safe load of a chairlift is 680 lb. In the spring, a cyclist and bicycle go to the top of the slope using the chairlift. The weight of the person is lb, and the weight of the bicycle is lb. Which inequality best describes how much additional weight w the chairlift could safely carry? Relate Define Let Practice and Problem Solving A GO Practice by Eample for Help w # 680 w $ 680 weight of a person and a bicycle w plus Write additional weight the amount of weight that can be added to the chairlift. The inequality w # 680 models the situation. So B is the correct answer. Your baseball team has a goal to collect at least 60 blankets for a shelter. Team members brought blankets on Monday and 6 blankets on Wednesday. Write an inequality to describe how many blankets the team must donate on Friday to make or eceed their goal. ± 6 ± b 60, or b State what number you would add to each side of the inequality to solve the inequality.. d $. 0, c 8 8. z. $.6. w w # 680 w $ 680 is at most Solve each inequality. Graph and check your solution. 6. See margin t, 6.. b 7. 7 # d 8. s $6 9. r 9 # , n. $ w., d. y #.. q. $ r 7... m.8 8. b, 9. n See back of book. # safe load 680 For more eercises, see Etra Skill and Word Problem Practice. 08 pages 08 Eercises. S 9 0. t R 0 6. b R d L s L 9. r K 9 0. n S

4 B Eample (page 08) Eample (page 08) Apply Your Skills 6c. Answers may vary. Sample: The sign means the sides are equal, so the two sides may be interchanged. The R sign does not indicate equality. One side cannot be both greater than and less than the other side. Real-World Connection More than 7,000 athletes compete in gymnastic programs in the United States.. w K. d S State what number you would subtract from each side of the inequality to solve the inequality. 0. w.. 8, r..7 $ k.. Solve each inequality. Graph and check your solution. 8. See back of book.. w # 9. m.., 8 b 6. $ a 7. r $ 8. k # , p. z $. 7. y,., m..7 $ a..9,. p 6. $ h 7.. d t, Vacation Budget Your brother has $000 saved for a vacation. His airplane ticket is $67. Write and solve an inequality to find how much he can spend for everything else. s ± 67 K 000, $6 0. Weekly Budget You have an allowance of $.00 per week. You are in a bowling league that costs $6.0 each week, and you save at least $.00 each week. Write and solve an inequality to show how much you have left to spend each week. s ± 6.0 ± K, $.0. Fund-Raising A school club is selling reflectors for Bicycle Safety Day. Each member is encouraged to sell at least 0 reflectors. You sell 7 on Monday and on Tuesday. How many reflectors do you need to sell on Wednesday to meet your goal? r ± 7 ± L 0, reflectors State what you must do to the first inequality in order to get the second.. 6 # y; 0# y. 9 b. ; b.. m, ; m, Add to each side. Subtract 9 from each side. Add to each side. Solve each inequality.. w $ 9 w L 6. c # c K 7. y 0.,.8 y R n n R 9. z., y 0.9 z R 9.7. t y R t S 6. # v 7 v L 6..6 k $. k L b 7, b R 6. m. #. 6. $ k k K m K. 7. h $ h L $ K w w R 60. Banking Your local bank offers free checking for accounts with a balance of at least $00. Suppose you have a balance of $6.6 and you write a check for $.96. How much must you deposit to avoid being charged a service fee? at least $.0 6. a. If 7 t, does t 7? yes b. If 7, r, is r, 7? no c. Discuss the differences between these two eamples. See above left. 6. Gymnastics Suppose your sister wants to qualify for a regional gymnastics GPS competition. At today s competition she must score at least.0 points. She scored 8.8 on the vault, 7.9 on the balance beam, and 8. on the uneven parallel bars. The event that remains is the floor eercise. a b. See back of book. a. Write and solve an inequality that models the information. b. Eplain what the solution means in terms of the original situation. c. Open-Ended Write three scores your sister could make that would allow her to qualify for the regional gymnastics competition. Answers may vary. Sample: 9., 9., 9. lesson quiz, PHSchool.com, Web Code: ata-00 Lesson - Solving Inequalities Using Addition and Subtraction y K. q R L. 6. r R Practice Assignment Guide A B -9, -, 67-7 A B 0-, -66, 7 C Challenge 7-78 Test Prep 79-8 Mied Review 8-97 Homework To check students understanding of key skills and concepts, go over Eercises 8, 9, 6, 6, 7. Error Prevention! Eercises 9 Some students may have difficulty deciding in which direction to draw the graph of a solution. Point out that when the solution has the variable on the left, the graph points in the same direction as the inequality symbol. When the solution has the variable on the right, the graph points in the opposite direction of the inequality symbol. Eercises 8 Suggest to students that they eliminate the fractions or decimals before solving. GPS Enrichment Guided Problem Solving Reteaching Adapted Practice Practice Pearson Education, Inc. All rights reserved. Name Class Date Practice - Each pair of figures is similar. Find the length of Use a proportion to solve nabc is similar to nxyz. The length AB is 0. The length BC is 7. Find the length XY if the length YZ is. 0. Marty has a scale model of a car. The scale is in. : in. If the model is 6.7 in. long, how long is the actual car?. A blueprint scale is in. : ft. The width of a building is 8 ft. What is the width of the building on the blueprint?. Angie is using similar triangles to find the height of a tree. A stick that is ft tall casts a shadow that is ft long. The tree casts a shadow that is ft long. How tall is the tree?. nabc is similar to nxyz. The length AC is 0. The length BC is 6. What is the length XZ if the length YZ is?. A map has a scale of in. : mi. Two cities are 7 mi apart. How far apart are they on the map? Proportions and Similar Figures 7 6. L L L L L 09

5 . Assess & Reteach PowerPoint Lesson Quiz Solve each inequality. Graph the solutions.. p - 7 $ - p L 0. w - -9 w R S 0. $ 9 + h L h, or h K 0 Alternative Assessment Group students in pairs. Instruct each student to model an inequality using tiles and an inde card with the inequality sign written on it. Have students echange with their partners and find the solutions. Encourage students to check their solutions. 66b. Answers may vary. Sample: Substituting values does not work because there is always the possibility that the solution lies between values that make the inequality true and a value that does not. Real-World C Connection In 97, a computer chip could hold 00 transistors. In 00, a chip could hold 0,000,000 transistors. 7. not true; sample countereample: for a and b 6, ( 6) ±( 6) GO nline Homework Help Visit: PHSchool.com Web Code: ate not true; sample countereample: for a 0, b, and c, 0 R and 0 ± ( ) Challenge 7d. The length of the third side must be greater than the difference of the lengths of the other two sides. 6. Computers Suppose your computer has nearly 6 megabytes (MB) of memory. Its basic systems require.8 MB. How much memory is available for other programs and functions? nearly. MB 6. To earn an A in Ms. Orlando s math class, students must score a total of at least points on the three tests. On the first two tests, Amy s scores were 7 and 8. What is the minimum score she must get on the third test to earn an A? 0 points 6. a. Open-Ended Use each of the inequality symbols,, #,., and $ to write four addition or subtraction. a b. Check students work. b. Solve each of the in part (a) and graph your solution. 66. a. Sam says that he can solve z 8.6 $. by replacing z with,, and. When z, the inequality is false. When z and z, the inequality is true. So Sam says that the solution is z $. Is his reasoning correct? Justify your answer. No; the solution is z L.8, so z L is not correct. b. Critical Thinking Eplain why substituting values into the inequality does not guarantee that your solution is correct. See margin. Solve each inequality. 67. $ L 68. n 6n, n R 69. 7t (6t ) # t K 70. k (k ). 8 k S (a ) 7a # a K 7. (a ) (a ), a R 8 7. Geometry The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. Following are for sides of the triangle shown. a b. c b c. a a c. b a. Write an inequality using c b and a. a S c b b. Write an inequality using a c and b. b S a c c. Write an inequality using b a and c. c S b a d. Writing Write a generalization about the length of the third side and the difference of the lengths of the other two sides. See left. Reasoning Decide if each inequality is true for all real numbers. If the inequality is not true, give a countereample. 7. a b, a b See above left. 7. If a $ b, then a c $ b c. true 76. If c. d, then a c, a d. true 77. If a, b, then a, b c. See above left. 78. Reasoning Find real numbers, y, z, and w for which it is true that. y and z. w, but it is not true that z. y w. Answers may vary. Sample: For, y, z, and w, S and S, but. a c b Test Prep Multiple Choice 79. Solve,. B A.. 8 B., 8 C.. 8 D., Solve - n. 0. G F. n, G. n. H. n, 8 J. n. 8 0 Chapter Solving Inequalities 0

6 8. [] at least points; (9.) ± S (0) 68 ± S 00 S (OR equivalent eplanation) [] incorrect answer OR no work or eplanation Short Response 8. Which graph represents all real number solutions of $ 8? A A. B. 0 0 C. D Which of the following is a solution for, n 0.? J F..99 G..0 H..0 J.. 8. Hector is flying his plane. To avoid a storm, he climbs 00 ft without going above his plane s maimum safe altitude of,000 ft. The inequality a 00 #,000 represents his original altitude a in feet. Which of the following could have been the original altitude? D A. 0,00 ft B. 0,000 ft C. 9,70 ft D. 7,70 ft 8. The leading scorer in your high school s basketball division finished the season with a game average of 0 points for games. As the division s second leading scorer, you have a 9. point per game average for games. You have your last game yet to play. How many points must you score in the final game of the season to overtake the division s leading scorer? Show your work. See above left. Test Prep Resources For additional practice with a variety of test item formats: Standardized Test Prep, p. 7 Test-Taking Strategies, p. Test-Taking Strategies with Transparencies Eercise 8 Point out to students that they can quickly eliminate answer choices B and D, since these have open dots. Mied Review GO for Help Lesson - Lesson - Define a variable and write an inequality to model each situation. 8. An octopus can be up to 0 ft long. Let c length of octopus in feet. c K A hummingbird migrates more than 80 mi. Let h distance in miles a hummingbird migrates. h S Your average in algebra class must be 90 or greater for you to receive an A for the term. Let a average. a L You must read at least pages this weekend. Let p number of pages to read. p L Simplify ( ) (..) ? ( ) (6 ) Algebra at Work Marketing Director Marketing directors rely on equations and to predict the actions their companies must take to stay competitive. For eample, the marketing director of a manufacturing company determines how much the cost of raw materials can increase before the company must raise the price of its finished goods or services. The director also predicts the effect of price changes on the quantity of goods and services sold by his company. PHSchool.com For: Information about a career in marketing Web Code: atb-0 Lesson - Solving Inequalities Using Addition and Subtraction

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