Chapter 9 Rational Expressions And Equations Lesson 9-1 .
Chapter 9 Rational Expressions and EquationsLesson 9-1 Multiplying and Dividing Rational ExpressionsPages 476–478461. Sample answer: ,4(x 1 2)6(x 1 2)2. To multiply rational numbersor rational expressions, youmultiply the numerators andmultiply the denominators. Todivide rational numbers orrational expressions, youmultiply by the reciprocal ofthe divisor. In either case, youcan reduce your answer bydividing the numerator andthe denominator of the resultsby any common factors.3. Never; solving the equationusing cross products leads to15 5 10, which is never true.4.9m4n45.1a2b6.3y 2y147.3c20b8.512x9.6510.p15p1111. cd 2x12.2y(y 2 2)3(y 1 2)13. D14.5c2bn27m16. 23x 4y15. 217.s318.5t1119.1220.y123y 2 121.a112a 1 122.3x 22yGlencoe/McGraw-Hill243Algebra 2Chapter 9
4bc27a23. 224. 2f25. 22p 226.xz8y27.b3x 2y 228. 329.4330.2331. 132.5(x 2 3)2(x 1 1)3(r 1 4)r1333.w23w2434.35.2(a 1 5)(a 2 2)(a 1 2)36. 23nm37. 22p38.m1nm2 1 n239.2x 1 y2x 2 y40. y 1 141.4342. d 5 22, 21 or 243. a 5 2b or b45.44.6827 1 m13,129 1 a682713,12946. 2x 1 1 units1a2247. (2x 2 1 x 2 15)m 248.49. A rational expression can beused to express the fractionof a nut mixture that ispeanuts. Answers shouldinclude the following. The rational expression50. C81x13 1 xis in simplest formbecause the numeratorand the denominator haveno common factors. Glencoe/McGraw-Hill244Algebra 2Chapter 9
81x13 1 x 1 y Sample answer:could be used to representthe fraction that is peanutsif x pounds of peanuts andy pounds of cashews wereadded to the originalmixture.51. A52. (21, 64), (5, 62)53. (6217, 6222)54. x 513(y 1 3)2 1 1; parabolayOxx 5 13 (y 1 3)2 1 155.(x 2 7)2 (y 2 2)2915 1;56. even; 2hyperbola8y4xO4248(x 2 7)921222(y 2 2)51128AAC A BLACK57. odd; 358. even; 059. 21, 460. 2 ,61. 0, 562. 4.99 3 102 s or about 8 min19 s63. [64.1 16 319112465. 21 Glencoe/McGraw-Hill3 19,2 1666. 21245Algebra 2Chapter 9
415111667. 168. 2111869. 270.Lesson 9-216Adding and Subtracting Rational EpressionsPages 481–4841. Catalina; you need acommon denominator, not acommon numerator, tosubtract two rationalexpressions.2. Sample answer: d 2 2 d, d 1 13a. Always; since a, b, and c arefactors of abc, abc is alwaysa common denominator of4. 12x 2y 21111 1 .acb3b. Sometimes; if a, b, and chave no common factors,then abc is the LCD of1111 1 .acb3c. Sometimes; if a and b haveno common factors and c isa factor of ab, then ab is the111LCD of 1 1 .abc3d. Sometimes; if a and c arefactors of b, then b is the111LCD of 1 1 .ab3e. Always; sincec1111 1 5acbacabbc11, the sumabcabcabcbc 1 ac 1 ab.alwaysabcis5. 80ab3c7.6. x(x 2 2)(x 1 2)2 2 x3x 2y Glencoe/McGraw-Hill8.24642a 2 1 5b 290ab 2Algebra 2Chapter 9
3742m10.5d 1 16(d 1 2) 211.3a 2 10(a 2 5)(a 1 4)12.8513.13x 2 1 4x 2 92x (x 2 1)(x 1 1)9.14. 70s 2t 2units15. 180x 2yz16. 420a3b3c 317. 36p 3q 418. 4(w 2 3)19. x 2(x 2 y)(x 1 y)20. (2t 1 3)(t 2 1)(t 1 1)21. (n 2 4)(n 2 3)(n 1 2)22.6 1 8bab23.3112v24.5 1 7rr25.2x 1 15y3y26.9x 2 2 2y 312x 2y27.25b 2 7a35a 2b 228. 229.110w 2 42390w30.13y2831.a13a2432.5m 2 43(m 1 2)(m 2 2)33.y (y 2 9)(y 1 3)(y 2 3)34.7x 1 382(x 2 7)(x 1 4)35.28d 1 20(d 2 4)(d 1 4)(d 2 2)36.24h 1 15(h 2 4)(h 2 5)237.x2 2 6(x 1 2)2(x 1 3)38. 039.2y 2 1 y 2 4(y 2 1)(y 2 2)40.1b1142.2s 2 12s 1 1a17a1244.3x 2 42x (x 2 2)45. 12 ohms46.24hx320q41. 2143. Glencoe/McGraw-Hill247Algebra 2Chapter 9
47.49.24hx242md(d 2 L)2(d 1 L) 22md2(d 2 L2 ) 248.or50. Sample answer:11,x11 x2251. Subtraction of rationalexpressions can be used todetermine the distancebetween the lens and thefilm if the focal length of thelens and the distancebetween the lens and theobject are known. Answersshould include the following. To subtract rationalexpressions, first find acommon denominator.Then, write each fractionas an equivalent fractionwith the commondenominator. Subtract thenumerators and place thedifference over thecommon denominator. Ifpossible, reduce theanswer. 11125q106048(x 2 2)hx(x 2 4)52. Bcould be usedto determine the distancebetween the lens and thefilm if the focal length ofthe lens is 10 cm and thedistance between the lensand the object is 60 cm.53. C Glencoe/McGraw-Hill54.248415xyz 2Algebra 2Chapter 9
55.a(a 1 2)a1156.y8x 2 1 y 2 5 168xO28289x 2 1 y 2 5 8157.58. 2.5 fty2(y 2 3) 5 x 1 2xOx25 y1 459.60.yy15610228O58x22210 25 Ox 226 y 2 5 12 2016510x25210y2 x22152514961.108642262225yO2 4 6x2422(x 1 2)(y 2 5)512 281625 Glencoe/McGraw-Hill249Algebra 2Chapter 9
Chapter 9Practice Quiz 1Page 484t12t232.c6b 23. 2y2324.725. (w 1 4)(3w 1 4)6. x 2 11.7.4a 1 1a1b9.n 2 29(n 1 6)(n 2 1)8.Lesson 9-310.14Graphing Rational FunctionsPages 488–4901. Sample answer:f(x) 56ax 1 20by15a 2b32. Each of the graphs is astraight line passing through(25, 0) and (0, 5). However,1(x 1 5)(x 2 2)the graph of f (x) 5(x 2 1)(x 1 5)x21has a hole at (1, 6), and thegraph of g(x ) 5 x 1 5 doesnot have a hole.3. x 5 2 and y 5 0 areasymptotes of the graph. They-intercept is 0.5 and there isno x-intercept because y 5 0is an asymptote.4. asymptote: x 5 25. asymptote: x 5 25; hole:x516.f (x )Oxxf (x ) 5x 11 Glencoe/McGraw-Hill250Algebra 2Chapter 9
7.8.f (x )4f (x )10262822f (x ) 5x 2 2 258x4O2426(x 2 2)(x 1 3)24 O10.f (x )4228O2210x4( x 2 1)28x4f (x ) 5x 25x 1124OC11.6f (x )f (x ) 524224249.f (x ) 5 x 2 5xC12. 100 mgf (x)Oxf(x) 513.x 12x2 2 x 2 614. y 5 212, C 5 1; 0; 0C106C5yy 1 12816 y2O216 282415. y . 0 and 0 , C , 116. asymptotes: x 5 2, hole:x5317. asymptotes: x 5 24, x 5 218. asymptotes: x 5 24, hole:x 5 2319. asymptotes: x 5 21, hole:x5520. hole: x 5 4 Glencoe/McGraw-Hill251Algebra 2Chapter 9
21. hole: x 5 122.f (x )1f (x ) 5 xxO23.24.f (x )f (x )f (x ) 51x12xOxO3f (x ) 5 x25.26.f (x )6f (x ) 525x 11228xO24f (x )4xO824f (x ) 5xx 232827.8C28.f (x )f (x ) 5f (x )Cf (x ) 55xx 1123( x 2 2)2xO42824O48x2429.30.f (x )f (x ) 51( x 1 3)2f (x )f (x ) 5x 14x 21622824 O48x24O Glencoe/McGraw-Hillx28252Algebra 2Chapter 9
31.32.f (x )428xOf (x ) 5f (x) 5f (x)x424 Ox 2 2 36 24x 16x 21x 232821233.34.f (x)f (x )2f (x) 5x 21x 21OxOxf (x ) 535.36.f (x )f (x )f (x ) 5Ox21( x 1 2)( x 2 3)37.38.f (x )f (x ) 5xx2 2 1xOf (x ) 53( x 2 1)( x 1 5)x 21x2 2 4f (x )f (x ) 5OxO Glencoe/McGraw-Hill6( x 2 6 )2253xAlgebra 2Chapter 9
39.40.f (x )f (x )1f (x ) 5( x 1 2 )2f (x ) 564x 2 1 16xOxO41. The graph is bell-shapedwith a horizontal asymptoteat f(x) 5 0.642645 2a 2b,x 1 16x 1 16264graph of f (x) 5 2x 1 1642. Sincethe2would be a reflection of thegraph of f (x) 564x 2 1 16overthe x-axis.43.44. m1 5 27; 7; 25Vf20124Vf 5m1 2 7m1 1 75O8 m1216 28 2446. Sample answers:45. about 20.83 m/sf(x) 5f (x) 5f (x) 547.8P (x ) 521228x12,(x 1 2)(x 2 3)2(x 1 2),(x 1 2)(x 2 3)5(x 1 2)(x 1 2)(x 2 3)48. the part in the first quadrantP (x )61x10 14 x24 O4x2428Glencoe/McGraw-Hill254Algebra 2Chapter 9
49. It represents her original freethrow percentage of 60%.50. y 5 1; This represents 100%,which she cannot achievebecause she has alreadymissed 4 free throws.51. A rational function can beused to determine how mucheach person owes if the costof the gift is known and thenumber of people sharing thecost is s. Answers shouldinclude the following. c52. A10050s50c5150sO50 100 s2100 250250 c 5 02100 Only the portion in the firstquadrant is significant inthe real world becausethere cannot be a negativenumber of people nor anegative amount of moneyowed for the gift.53. B55.3x 2 16(x 1 3)(x 2 2)54.3m 1 4m1n56.5(w 2 2)(w 1 3) 258. (22, 0); 11357. (6, 2); 5yy2(x 2 6) 1 ( y 2 2)25 25OOxxx 2 1 y 2 1 4x 5 9Glencoe/McGraw-Hill255Algebra 2Chapter 9
59. 65,89260. 24 6 2i61. 212, 1062.63. 4.564. 1.465. 2066. 12Lesson 9-427 6 3 2132Direct, Joint, and Inverse VariationPages 495–4981a. inverse1b. direct2. Both are examples of directvariation. For y 5 5x, yincreases as x increases. Fory 5 25x, y decreases as xincreases.3. Sample answers: wages andhours worked, total cost andnumber of pounds of apples;distances traveled andamount of gas remaining inthe tank, distance of anobject and the size itappears4. inverse; 205. direct; 20.56. joint;7. 248. 2451210. P 5 0.43d9. 2811. 25.8 psi12. about 150 ft13.14. direct; 1.5pDepth(ft) Hill256Algebra 2Chapter 9
PP 5 0.43dOd15. joint; 516. inverse; 21817. direct; 318. inverse; 1219. direct; 2720. joint;21. inverse; 2.522. V 523. V 5 kt24. directly; 2p25. 118.5 km26. 6027. 2028. 21629. 6430. 2531. 432. 1.2533. 9.634. 212.635. 0.8336. 237.13kp141638. 30 mph39. 100.8 cm340. See students’ work.41. m 5 20sd42. joint43. 1860 lb44. / 515md45. joint46. See students’ work.47. I 5kd248.II 5 162dOGlencoe/McGraw-Hill257dAlgebra 2Chapter 9
10.02P1P249. The sound will be heard as4intensely.50. 0.02; C 551. about 127,572 calls52. about 601 mi53. no; d Þ 054. Sample answer: If the averagestudent spends 2.50 for lunchin the school cafeteria, writean equation to represent theamount s students will spendfor lunch in d days. How muchwill 30 students spend in aweek? a 5 2.50sd; 37555. A direct variation can beused to determine the totalcost when the cost per unitis known. Answers shouldinclude the following. Since the total cost T isthe cost per unit u timesthe number of units n orC 5 un, the relationship isa direct variation. In thisequation u is the constantof variation. Sample answer: The schoolstore sells pencils for 20 each. John wants to buy5 pencils. What is the totalcost of the pencils? ( 1.00)56. D57. C58. asymptote: x 5 1; holex 5 2159. asymptotes: x 5 24, x 5 360. hole: x 5 23t 2 2 2t 2 2(t 1 2)(t 2 2)61.xy2x62.63.m (m 1 1)m1564. 9.3 3 10765. 0.4; 1.266. 3; 73567. 2 ; 368. C69. A70. S Glencoe/McGraw-Hilld2258Algebra 2Chapter 9
71. P72. A73. CChapter 9Practice Quiz 2Page 4981.2.f (x )f (x )O1f (x ) 5 xx 224xxOf (x ) 5222x 2 6x 1 94. 4.43. 495. 112Lesson 9-5 Classes of FunctionsPages 501–5041. Sample answer:2. constant (y 5 1),direct variation (y 5 2x),identity (y 5 x)POdThis graph is a rationalfunction. It has an asymptoteat x 5 21.3. The equation is a greatestinteger function. The graphlooks like a series of steps.4. greatest integer5. inverse variation or rational6. constant Glencoe/McGraw-Hill259Algebra 2Chapter 9
7. c8. b9. identity or direct variation10. quadraticyyy 5 2x 2 1 2xOy5xxO12. A 5 pr 2; quadratic; thegraph is a parabola11. absolute valueyy5 x12Ox13. absolute value14. square root15. rational16. direct variation17. quadratic18. constant19. b20. e21. g22. a23. constant24. direct variationyyy 5 2.5xxOOxy 5 21.5 Glencoe/McGraw-Hill260Algebra 2Chapter 9
25. square root26. inverse variation or rationalyyy 5 4xy 5 Ï9xxOxOAAC A BLACK27. rational28. greatest integeryy2y5x 21x21y 5 3[x ]xOxO29. absolute value30. quadraticyyy 5 2xOy 5 2x 2xOx31. C 5 4.5 m32. direct variation33. a line slanting to the right andpassing through the origin34. similar to a parabola Glencoe/McGraw-Hill261Algebra 2Chapter 9
35.36. The graph is similar to thegraph of the greatest integerfunction because both graphslook like a series of steps. Inthe graph of the postagerates, the solid dots are onthe right and the circles areon the left. However, in thegreatest integer function, thecircles are on the right andthe solid dots are on the left.yCost (cents)1601208040x0246Ounces81037a. absolute value37b. quadratic37c. greatest integer37d. square root38. A graph of the function thatrelates a person’s weight onEarth with his or her weighton a different planet can beused to determine a person’sweight on the other planet byfinding the point on thegraph that corresponds withthe weight on Earth anddetermining the value on theother planet’s axis. Answersshould include the following. The graph comparingweight on Earth and Marsrepresents a direct variationfunction because it is astraight line passing throughthe origin and is neitherhorizontal nor vertical. The equation V 5 0.9Ecompares a person’sweight on Earth with his orher weight on Venus.VVenus80604020E020 Glencoe/McGraw-Hill26240 60Earth80Algebra 2Chapter 9
39. C40. D41. 2242.f (x )3f (x ) 5 x 12xO43.44.f (x )f (x )2f (x ) 5 x 2 5x 1 4x24xOf (x ) 5 (8x 2 1)(x 1 3)46. a23 , 1b ; a22 , 1b;45. (8, 21); a8, 2 b ; x 5 8;y514121086422222781121 ; up;82xO14y 5 1; xunit1415 24 ;4right; 4 unitsyy11x 5 4 y2 2 2 y 2 31(y 1 1) 5 (x 2 8)22O2 4 6 Glencoe/McGraw-Hill10 12Oxx263Algebra 2Chapter 9
47.48. c(5, 24); a5 , 24b ; y 5 24;341422523 254d66 22657x 5 4 ; right; 3 unitsyxO3x 2 y 2 5 8y 1 3149. impossible50. (7, 25)51. a , 2b52. (2, 22)53. 154. 121317655. 256. 60a3b 2c 257. 45x 3y 358. 15(d 2 2)59. 3(x 2 y)(x 1 y)60. (a 2 3)(a 1 1)(a 1 2)61. (t 2 5)(t 1 6)(2t 1 1)Lesson 9-6Solving Rational Equations and InequalitiesPages 509–5111. Sample answer:1512a122. 2(x 1 4); 24513. Jeff; when Dustin multipliedby 3a, he forgot to multiplythe 2 by 3a.4. 35. 2, 66.7. 26, 228. 22 , c , 2 Glencoe/McGraw-Hill26423Algebra 2Chapter 9
291610. 2 h9. v , 0 or v . 14311. 212. 213. 26, 114. 23, 215. 21 , a , 016. 21 , m , 117. 1118. 319. t , 0 or t . 320. 0 , b , 121. 0 , y , 222. p , 0 or p . 223. 1424.25. [26. [27. 728.7330.1 6 2145429.23 6 3 2223231. 3232. 2 or 433. band, 80 members; chorale,50 members34. 4.8 cm/g35. 24 cm36. 15 km/h37. 5 mL38. 539. 6.1540.41. If something has a generalfee and cost per unit, rationalequations can be used todetermine how many units aperson must buy in order forthe actual unit price to be agiven number. Answersshould include the following.42. B To solve500 1 5xx12bbc 1 15 6,multiply each side of theequation by x to eliminatethe rational expression. Glencoe/McGraw-Hill265Algebra 2Chapter 9
Then subtract 5x from eachside. Therefore, 500 5 x.A person would need tomake 500 minutes of longdistance calls to makethe actual unit price 6 . Since the cost is 5 perminute plus 5.00 permonth, the actual cost perminute could never be 5 or less.44. quadratic43. Cyy 5 2x 2 1 1xO46. direct variation45. square rootyyxOy 5 0.8xy 5 2ÏxOx47. 3648. 33.7549. 2213050. 22552. 5x 0 x , 211 or x . 3651. 213753. 5x 0 0 # x # 46 Glencoe/McGraw-Hill54. e b 21 , b , 2 f12266Algebra 2Chapter 9
Lesson 9-1 Multiplying and Dividing Rational Expressions Pages 476–478 2. To multiply rational numbers or rational expressions, you multiply the numerators and multiply the denominators. To divide rational numbers or rational expressions, you multiply by the reciprocal of the divisor. In either case, you can reduce your answer by dividing the .
Multiplying and Dividing Rational Expressions Find the product of rational expressions Find the quotient of rational expressions Multiply or divide a rational expression more than two rational expressions 3.2 Add and Subtract Rational Expressions Adding and Subtracting Rational Expressions with a Common Denominator
Lesson 4: Introduction to Rational Expressions Define rational expressions. State restrictions on the variable values in a rational expression. Simplify rational expressions. Determine equivalence in rational expressions. Lesson 5: Multiplying and Dividing Rational Expressions Multiply and divide rational expressions.
64. Reduce rational expressions. 65. Multiply and divide rational expressions. 66. Find the least common multiple of polynomial expressions. 67. Add and subtract rational expressions. 68. Simplify complex rational expressions. 69. Solve rational equations. 70. Solve applied problems using rational equations, including proportions. Chapter 7 (7 .
Rational Rational Rational Irrational Irrational Rational 13. 2 13 14. 0.42̅̅̅̅ 15. 0.39 16. 100 17. 16 18. 43 Rational Rational Rational Rational Rational Irrational 19. If the number 0.77 is displayed on a calculator that can only display ten digits, do we know whether it is rational or irrational?
Rational Team Concert Rational DOORS NG Rational Collaborative Lifecycle Management Rational Developer for System z Worklight Studio Rational Quality Manager Rational Test Virtualization Server Rational Test Workbench Rational Test Workbench – Mobile Test Edition Rational Development and Test En
A) Graphing Simple Rational Functions B) Graphing More Complicated Rational Functions 4) Rational Expressions and Equations A) Adding and Subtracting Rational Expressions B) Multiplying and Dividing Rational Expressions C) Solving Rational Equations 4th 9 Weeks: 1) Radical Functions A) Inverses of Simple Quadratic and Cubic Functions
l. Rewrite rational expressions m. Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions n. Extend properties of exponents to rational exponents o.
The Asset Management Strategy supports our strategic priority to: To provide quality, well maintained homes that are fit for the future . Page 5 of 10 Asset Management Strategy 2018 The strategy supports our growth aspirations and development strategy. A key principle is that any development decision will complement and enhance our current asset portfolio. Our aim is that: We invest in our .
