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Answers for the lesson “Find Square Roots and Compare Real Numbers” LESSON 2.1 } rational number, Ï144 : real number, rational number, integer, } whole number; 211.5, 2Ï 121 , Skill Practice 1. real numbers 2. If it is not a perfect square, the square root is irrational. 3. 2 4. 27 5. 23 6. 61 7. 14 8. 611 9. 650 10. 216 11. 215 12. 19 13. 613 14. 240 15. 3 16. 24 17. 22 18. 12 19. 29 20. 6 21. 14 22. 28 23. B } Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 24. Ï 49 : real number, rational number, integer, whole number, 8: real number, rational number, } integer, whole number, 2Ï4 : real number, rational number, integer, 23: real number,}rational number, } integer; 23, 2Ï4 , Ï49 , 8 } 25. 2Ï 12 : real number, irrational number, 23.7: real}number, rational number, Ï9 : real number, rational number, integer, whole number, 2.9: real number, rational } } number; 23.7, 2Ï12 , 2.9, Ï 9 26. 211.5: real number, rational } number, 2Ï 121 : real number, rational number, integer, 210: real number, rational number, } 25 210, Ï 144 , } 2 } 27. Ï 8 : real number, irrational 2 number, 2}5: real number, rational number, 21: real number, rational number, integer, 0.6: real } number, rational number, Ï 6 : real number, irrational number; 21, 2 } } 2}5, 0.6, Ï6 , Ï 8 8 28. 2}: real number, rational 3 } number, 2Ï5 : real number, irrational number, 2.6: real number, rational number, 21.5: real number, rational number, } Ï 5 : real number, irrational 8 } } number; 2}3, 2Ï5 , 21.5, Ï 5 , 2.6 29. 28.3: real number, rational } number, 2Ï80 : real number, 17 irrational number, 2} : real 2 number, rational number, 28.25: real} number, rational number, 2Ï 100 : real number,} rational} number, integer; 2Ï100 , 2Ï 80 , 17 2} , 28.3, 28.25 2 25 integer, } : real number, 2 Algebra 1 Answer Transparencies for Checking Homework 26 30. If a number is a whole number, then it is a real number; true. 41. B } 31. If a number is a real number, then it is an irrational number; false. Sample answer: 3 is a real number and a rational number. 42. For 0 , x , 1, x , Ï x ; for 1 } 1 example, if x 5 }4 , Ïx 5 }2 , and 1 1 } } , }. For x . 1, x . Ï x ; for 2 4 } example, if x 5 9, Ïx 5 3, and 9 . 3. 32. If a number is a perfect square, then it is not a whole number; false. Sample answer: 9 is a perfect square and a whole number. 1 43. 4, 64, 324, 1024, 2500; take } of 2 an even perfect square, and then square it. 44. 31; the square root of 1000 is 33. If a number is an irrational between 31 and 32, so there are 31 perfect squares less than 1000. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. number, then it is not a whole number; true. 34. 6 35. 2 36. 28 Problem Solving 37. 242 38. 240 39. 63 45. 60 in. 40. Set Whole numbers Integers Rational numbers Real numbers Additive identity? 46. 40 yd Additive Multiplicative Multiplicative inverses? identity? inverses? Yes No (except 0) Yes No (except 1) Yes Yes Yes No (except 1) Yes Yes Yes Yes (except 0) Yes Yes Yes Yes (except 0) Algebra 1 Answer Transparencies for Checking Homework 27 47. 35 ft; 24 ft; 48 ft; 30 ft; they are all rational numbers. 48. No; each side is a little more than 38 feet. 38 feet times 4 is 152 feet, which is more than 150 feet. 49. 2.2 feet 50. a. 290 K b. 342 m/sec 51. a. 144 tiles Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. b. 16 ft. Sample answer: If the homeowner can buy 144 tiles that are each 256 square inches, then the total area is (144 tiles) + (256 square inches per tile) 5 36,864 square inches. Divide 36,864 square inches by 144 square inches to find the number of square feet, 256 square feet. If the area of the square is 256 square feet, take the square root of 256 to find the side length, 16 feet. 52. a. Quotient Quotient Quotient of area of side of length of pyramids of bases bases }} Khafre Menafaure 3.9 2.0 }} Khufu Menafaure 4.6 2.1 Khufu Khafre 1.2 1.1 } The quotient of the areas is the square of the quotient of the side lengths. } b. q 5 Ï r 53. * 5 Ï } A 6 } Mixed Review of Problem Solving 1. a. 1104 points b. 354 points 2. a. T 5 15 1 (20.0065)e b. 256.58C Algebra 1 Answer Transparencies for Checking Homework 28 3. 3 in.; 7. Sample answer: A stock increases 3 in value by $8.90 one week, decreases by $2.34 the next week and decreases $1.15 the following week. The expression 8.90 1 (22.34) 1 (21.15) 3 }} represents the stock’s average change over 3 weeks; $1.80. 4. a. 40 cm; 35 cm Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. b. The yellow bin. Sample answer: For each bin, divide the cost by the surface area; the yellow bin costs less per cubic centimeter. 5. a. c 5 23.29x; 2$13.16 b. No; it depends on the amount of change for each item. 6. a. C 5 4x 1 5600 b. $6000 c. Yes; the cost will be $6200. Algebra 1 Answer Transparencies for Checking Homework 29 LESSON 2.2 Answers for the lesson “Solve One-Step Equations” Skill Practice 49. 54 5 12x; 4.5 in. 1. inverse operations 2. Division property of equality; to solve 14x 5 35, you need to divide each side by 14. 3. 3 4. 27 5. 5 6. 6 7. 23 8. 28 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 9. 7 10. 12 11. 17 12. 13 13. 4 14. 22 15. C 16. A 17. 4 18. 213 19. 6 20. 212 21. 215 22. 211 23. 15 24. 28 25. 48 26. 249 27. 22 28. 242 29. The student multiplied x by 100 to produce a number with a decimal part identical to the decimal part of x. When the student subtracted, the result was a whole number. 7 30. a. } 9 31. 3.5 2 b. } 11 32. 24.2 34. 28.3 35. 22.05 36. 26 5 37. } 8 38. 211.5 39. 0.06 40. 42 41. 96 42. 235 43. 12 1 46. } 6 44. 35 3 47. } 5 45. 256 1 50. 72 5 } (16)x; 9 cm 2 Problem Solving 53. 1046.6 ft 54. 2.5 mm 55. 11 ft 56. 245 lb 4 57. a. } x 5 200 7 b. Plants; if you solve the equation in part (a) you find the there are 350 species of birds. 58. Sample answer: Each member of the drama club needs to sell tickets to the upcoming play. If there are 15 members and you want to sell 135 tickets, how many tickets should each member sell? x 5 9; each member should sell 9 tickets. 59. a. 33. 21.1 52. 49 51. 20.5 t D 1 6.5 2 13 3 19.5 4 26 5 32.5 48. 22 Algebra 1 Answer Transparencies for Checking Homework 31 59. b. " 35 30 4 sec 25 20 15 10 5 0 0 1 2 3 4 5 ! c. 26 5 6.5t; 4 sec 60. a. 18 ft 2 b. 6 ft 61. a. 171 hits b. 215 hits Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. c. No; if Mueller had fewer hits than Wells but had a higher batting average, he must have had fewer at bats than Wells. 62. 47.5 ft 2 63. $2.60 Algebra 1 Answer Transparencies for Checking Homework 32 LESSON 2.3 Answers for the lesson “Solve Two-Step Equations” Skill Practice 36. 1, 2, 3; 4; the output of each 1. like terms 2. First, subtract 7 from each side to get 4x 5 8, then divide each side by 4 to get x 5 2. Problem Solving 3. 4 4. 3 5. 2 37. 28 classes 6. 8 7. 23 8. 1 39. 5 half-side advertisements 38. 1.5 h 10. 20 11. 40 40. B 12. 212 13. 18 14. 21 41. Yes; the equation 15. 4 16. 6 17. 9 18. 6 19. 24 20. 22 9. 6 21. D Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. equation is increased by 3 because x is multiplied by 3, each solution increases by 1. 22. Unlike terms were combined; 5 23x 5 5, x 5 2}3 23. The division of 22x 1 x by 22 is done incorrectly. Sample answer: If like terms are combined as the first step, the second line would be 2x 5 10 and the final result would be x 5 210. 24. y 5 3x 1 7; 25 $542 5 $50 1 6x gives the monthly cost of a guitar that costs $542. Solving the equation gives x 5 $82 per month, so you can afford the guitar. 42. a. y 5 2,896,112 1 1600x b. 1163 days c. Sample answer: Estimate the number of days to be about 1200; 3,000,000 1 1600(1200) 5 3,000,000 1 1,920,000 5 4,920,000 so the solution makes sense. 25. y 5 2x 1 4; 27 26. y 5 10x 2 9; 2 27. 4 28. 8 29. 5 30. 10 31. 0.5 32. 1.5 33. 15.9 34. 3.5 35. 6.9 Algebra 1 Answer Transparencies for Checking Homework 34 43. a. y 5 12x b. x(h) Marissa Ryan 1 5 7 12 2 10 14 24 3 15 21 36 4 20 28 48 5 25 35 60 c. about 4.5 h " 60 48 36 24 12 0 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. Total 0 1 2 3 4 5 ! 44. 5 days; the restaurant serves 400 1 120 5 520 customers each day, use the equation 2600 5 520x to find x 5 5. 45. 200 sec Algebra 1 Answer Transparencies for Checking Homework 35 LESSON 2.4 Answers for the lesson “Solve Multi-Step Equations” Skill Practice 36. You should divide each side of the 5 1. } 3 2. Use the distributive property to get 12y 2 21 5 6, then add 21 to each side to get 12y 5 27, divide each side by 12 to get y 5 2.25. a. 12 b. 2.5 3. 3 4. 3 5. 6 37. 16, 18, 20 6. 25 7. 22 8. 25 Problem Solving 9. 28 10. 27 11. 28 38. 6 tickets 12. 2 13. 4 14. 3 39. 0.75 ft 15. 29 16. 5 17. 219 40. 8 min 41. a. 34 months 18. C Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. equation by the number outside the parentheses when the number is a factor of the number on the other side of the equal sign or it is a fraction. 19. 12 20. 1 21. 22 b. 307 ft per month 22. 5 23. 29 24. 22 c. After the work crews merged; 25. 23 times 26 is 18, not 218; 5x 2 3x 1 18 5 2, 2x 1 18 5 2, 2x 5 216, x 5 28. 1 26. Multiply each side by 2, not }; 2 2x 2 10 5 8, 2x 5 18, x 5 9. 27. 2 28. 0.5 29. 3 30. 0.1 31. 25 32. 1.2 33. 2 1 34. 5 } 3 35. 9.5 in., 6 in.; if you use the perimeter formula P 5 2* 1 2w and substitute 3.5 1 w for *, the solution is w 5 6. before the work crews merged they were working at a rate of 115 1 137 5 252 feet per month, and after merging at a rate of 307 feet per month. 42. a. y 5 74.5x 1 750; 21 squares b. Squares Cost (dollars) 5 1122.50 10 1495 15 1867.50 20 2240 25 2612.50 Algebra 1 Answer Transparencies for Checking Homework 37 43. Let q be the number of quarters, then q 1 8 is the number of dimes. The equation 25q 1 10 (q 1 8) 5 280 represents the situation, but the solution of the 5 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. equation is 5}7 . The number of quarters must be a whole number, so Jan must be mistaken. Algebra 1 Answer Transparencies for Checking Homework 38 LESSON 2.5 Answers for the lesson “Solve Equations with Variables on Both Sides” Skill Practice 28. When the equation is 0 5 0, it 1. identity 2. A number plus 3 can’t be equal to itself plus 1. If you solve the equation, you get 0 5 2. 3. 22 4. 1 5. 24 6. 3 7. 27 8. 26 9. 8 12. 5 10. 3 11. 24 13. 23 14. 7 15. C 16. D Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 17. Sample answer: Distribute the 3 to get 6z 2 15 5 2z 1 13, then subtract 2z from each side to get 4z 2 15 5 13, next add 15 to each side to get 4z 5 28, finally divide each side by 4 to get z 5 7. 18. no solution 19. 2 20. no solution 21. 27 22. 233 23. no solution 24. identity 25. no solution 26. identity means that it is true for all values of y, not just 0; the equation is an identity. 29. Sample answer: 5x 1 4 5 5x; the number 5x cannot be equal to 4 more than itself. 1 30. 2} 2 31. 2 32. no solution 33. 24 34. 10 35. 6 37. identity 39. 2 36. 6 38. 3 40. 224 41. 10 42. 24 43. identity 44. 120 45. 60 46. 136 47. 4 48. 5 Problem Solving 49. 9 nights 50. 4 months 51. about 4 yr 27. The 3 was not distributed to both terms; 3x 1 15 5 3x 1 15, 15 5 15, so the equation is an identity. Algebra 1 Answer Transparencies for Checking Homework 40 52. a. 360 1 4x 5 16x; 30 visits 2. 13 lb; b. 1 3 NonMembers’ members’ Visits cost cost (dollars) (dollars) 5 380 80 10 400 160 15 420 240 20 440 320 25 460 400 30 480 480 35 500 560 53. a. 23.4t 5 24(t 2 0.3); 12 sec 3. a. 4 games c. Yes; if Paul bowls only 3 games, he will have $3.75 left over. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. b. about 4.4 sec c. No; it would take 12 seconds for the sheepdog to catch up to the collie and it only takes 4.4 seconds for the collie to complete the last leg. 54. 20, 8 b. 3 games 4. 4; 4 55. 11, 5 Mixed Review of Problem Solving 1. a. C 5 0.25 1 0.07(t 2 1) b. 26 min Algebra 1 Answer Transparencies for Checking Homework 41 5. 30 min; if you make 40 cranes in an hour, it will take 2.5 hours to make 100 cranes, so you spent 30 minutes not making cranes. 6. 4 visits; use an equation to find when the cost of skiing with a pass is the same as skiing without a pass. Solve 90 1 22.50x 5 45x to find that x 5 4. 7. 9677 ft; if you add 1313 feet to the current elevation, you will get the elevation before the eruption. 8. a. 40 daffodil bulbs b. 9 in. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. c. 20 daffodil bulbs. Sample answer: If she plants bulbs 6 inches apart she will need 360 inches 4 6 inches 5 60, so she needs 60 2 40 5 20 more bulbs. 9. Sample answer: You want to buy a video game for $47, if you have $15 and plan on saving money from a paper route each week for 4 weeks, how much money should you save each week? x 5 8; you need to save $8 each week for 4 weeks to have enough money to buy the video game. Algebra 1 Answer Transparencies for Checking Homework 42 LESSON 2.6 Answers for the lesson “Write Ratios and Proportions” 4 3 43. Yes. Sample answer: } 5 } 8 6 Skill Practice 1. ratios 2 2. Sample answer: }, 2 : 5, 2 to 5 5 3. no; 7 to 9 4. yes 5. yes 6. no; 7 to 8 6 7. } 5 8. 64 10. 6 11. 48 12. 5 13. 15 14. 4 15. 40 16. 48 17. 12 18. 56 19. C 9. 22 Problem Solving 7 2 45. } 46. } 145 8 3 2 1 47. } 48. } 49. } 5 5 2 50. 21 pages 1.85 1 b. Width (ft) Length (ft) 20 37 25 46.25 30 55.5 35 64.75 40 74 side by 10 and the other side by 50 20; 10 + } 5 10 + } , m 5 25. 10 20 x 3 23. } 5 }; 12 32 8 8 x 25. } 5 }; 2 16 4 7 b 27. } 5 }; 35 2 10 d 12 29. } 5 }; 18 27 18 a 5 24. } 5 }; 35 49 7 y 9 26. } 5 }; 36 5 20 n 4 28. } 5 }; 1 3 12 40 t 30. } 5 }; 30 28 21 31. 1.8 32. 3.3 33. 2.4 34. 2 35. 4 36. 6 37. 4 38. 4 39. 2 40. 19 41. 3.5 42. 4 x 38 } 5 }, 70.3 ft 22. Multiply each side by 10, not one m 51. 45 goals 52. a. Sample answer: 20. B 1 21. Multiply each side by 6, not }; 6 3 x 1 } } } 6 + 4 5 6 + 6 , 4 2 5 x. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 44. Yes; multiply each side by b to get cb the equation a 5 } , then divide d b a each side by c to get }c 5 }d . 10 53. a. } 23 b. 110 lift tickets c. 40 snowboarders Algebra 1 Answer Transparencies for Checking Homework 44 x 10 54. a. Sample answer: } 5 } 11 3 x 5 b. Sample answer: } 5 } 14 2 c. Your friend; you will wait in 2 line for 36 }3 minutes and your friend will wait only 35 minutes. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 55. 37.5 mi, 22.5 mi Algebra 1 Answer Transparencies for Checking Homework 45 Answers for the lesson “Solve Proportions Using Cross Products” LESSON 2.7 Skill Practice 1. cross product 2. Measure the distance in centimeters in the drawing and then substitute the value in the distance in drawing 1 proportion, }3 5 }} . actual distance decrease. Problem Solving 3. 6 4. 7 5. 24 33. 5 c 34. 18 min 6. 6 7. 1 8. 6 35. 90 km 36. 48 km 10. 7 11. 2 37. 7.5 km 38. 70.5 km 13. 12 14. 26 39. 17.728 m 9. 249 12. 27 15. B 40. a. 213.75 ft 16. D 17. Use the cross products property to Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 32. Decreases; as k gets larger, the k value of } increases, so for the 14 5 value of } to increase, h must h multiply 4 by x and 16 by 3; 4 + x 5 3 + 16, 4x 5 48, x 5 12. 18. Distribute the 14 to both b and 2; 18b 5 14b 1 28, 4b 5 28, b 5 7. 19. 15 20. 2 21. 10 22. 22 23. 5.5 24. 22.5 25. 23.4 26. 5.2 27. 4.2 28. 7.4 29. 25.9 30. 27.1 31. a. Multiplication property of equality b. Multiply b. Shell diameter (in.) Burst diameter (ft) 2 90 3 135 4 180 5 225 6 270 41. 80 yd; find the actual length of the field by using the ratio 1 in. : 20 yd, then use the number to find the width of the soccer field by using the ratio 3 : 2. c. Simplify Algebra 1 Answer Transparencies for Checking Homework 47 42. a. 4.5 moles b. 3.4 moles c. Nearly equal. Sample answer: The ratio of moles of hydrogen to moles of carbon is about 1.33 to 1, the ratio of moles of hydrogen to moles of oxygen is about 1.32 to 1. The two ratios are nearly the same. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 43. Sample answer: I would assign 11 seats to the freshmen, 8 seats to the sophomores, 7 seats to the juniors, and 4 seats to the seniors. The actual numbers obtained from the proportions are 10.464, 8.4, 6.72, and 4.32. If I round each of those numbers in the traditional way, the sum is only 29. I rounded 10.464 up to 11 to get a total of 30 seats. I chose 10.464 because of the three numbers with decimal parts less than 0.5, its decimal part is the greatest. Algebra 1 Answer Transparencies for Checking Homework 48 LESSON 2.8 Answers for the lesson “Rewrite Equations and Formulas” Skill Practice 23. C 1. literal equation 2. Divide each side by pr to get I } 5 t. pr c 3. x 5 };22 b2a c 4. x 5 } 2 b; 3.5 a 5. x 5 bc 2 a; 9 ab 6. x 5 }; 3 c 29. Sample answer: You want to find how long it will take to drive 150 miles if you drive at an average rate of 55 miles per hour. 9. b should have been subtracted from both sides, not added; 2b ax 5 b, x 5 } . a 10. Both sides should have been divided by (a 2 b), not d12 32. a. n 5 } 4 b. 1.75 in., 3.5 in., 4.5 in., 5.5 in. C 2 25 33. a. x 5 } 12 b. 10 league nights; 13 league nights; 15 league nights c multiplied; x 5 } . a2b 11. y 5 7 2 2x 5 5 12. y 5 } 2 } x 2 4 13. 4 2 3x 5 y 14. y 5 213 1 9x 15. 17. 19. 21. by 31. a 5 } y 2 bx Problem Solving b1d 8. x 5 }; 2 c2a Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 5 1 25. y 5 18 2 5x 26. y 5 2} x 1 } 2 2 S 27. * 5 } pr 2 r; 13.03 cm A 28. p 5 }; 8.01 ft 4pw b1c 30. a 5 } bx 2 1 7. x 5 a(c 2 b); 28 6 2 1 }7 x 5 y 9 }x 2 6 5 y 5 1 1 y 5 }2 x 1 }3 S 2 2B h5} P 24. y 5 2.1x 2 8.4 16. 18. 20. 22. 5 y 5 x 2 }8 3 2 2 }2 x 5 y V w5} *h * f5} 24 P 2 2pr 34. a. x 5 } 2 b. r x 50 173 51 170 52 167 53 164 Algebra 1 Answer Transparencies for Checking Homework 50 34. c. " 173 161 ft 171 169 167 165 163 0 3 3 4. Solve Q 5 } for r to ger r 5 }; r Q 3 substitute } for r in the formula Q 6 d 5 2r to get d 5 } . Q 5. 140 attempts; 1 4 0 0 50 51 52 53 54 ! 35. Divide each side by the total bill, a b, to get } 5 p%. b C 36. a. r 5 }F 2p b. 1.1 ft; 1.3 ft; 1.4 ft C c. Solve for r using }F 5 r, 2p Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. then substitute r into the equation A 5 pr 2. 11t 37. g 5 } 4 Mixed Review of Problem Solving x 7 1. a. Sample answer: } 5 } 1209 31 b. 273 students 5 2. a. } 9 b. 120 females 3. $70. Sample answer: Use the x 7 } to find that proportion } 5 700 154 6. a. $12,480 b. $12,300 c. 0.6%; find that $1230 is 10% of $12,300, then find $1230 is what percent of $205,000. 7. Multiply each side by 2 and divide each side by d2 to get 2A d1 5 } . d 2 8. Sample answer: Increase the length, 16 feet by 4 feet. you will need about 35 gallons of gas. Multiply 35 by $2 per gallon. Algebra 1 Answer Transparencies for Checking Homework 51