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Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Chapter 11DATEBefore you begin Chapter 11Rational Expressions and EquationsAnticipation GuidePERIODA15yAfter you complete Chapter 1110. Extraneous solutions are solutions that can be eliminatedbecause they are extremely high or low.9. Did any of your opinions about the statements change from the first column?Chapter 11Glencoe Algebra 1Answers3Glencoe Algebra 1 For those statements that you mark with a D, use a piece of paper to write anexample of why you disagree.DDA8. A complex fraction contains a fraction in its numeratoror denominator.( ab )accan be rewritten as .The fraction bd( dc )ADADADDSTEP 2A or D7. The least common denominator of two rational expressionswill be the least common multiple of the denominators.numerators and then the denominators.2a56. To find the sum of and , first add the(3a - 4)(3a - 4)5. To divide (4x2 12x) by 2x, divide 4x2 by 2x and 12x by 2x.4. When solving problems involving units of measure,dimensional analysis is the process of determining the unitsof the final answer so that the units can be ignored whileperforming calculations.multiply the numerators and the denominators.3c2xy23c 23. To multiply two rational expressions, such as and ,2. A rational expression is an algebraic fraction that containsa radical.k1. Since a direct variation can be written as y kx, an inversexvariation can be written as y .Statement Reread each statement and complete the last column by entering an A or a D.Step 2STEP 1A, D, or NS Write A or D in the first column OR if you are not sure whether you agree ordisagree, write NS (Not Sure). Decide whether you Agree (A) or Disagree (D) with the statement. Read each statement.Step 111NAMECopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.DATEPERIODx1 y1 x2 y2xySimplify.3Divide each side by 18.212348 18x28 x2184 x Simplify.Cross multiply.x1 4, y1 12, y2 18Proportion for inverse variation162410512638yxdirect variation;of the formy kxdirect variation;of the formy kx2. y 6x3. xy 15inversevariation; of theformxy k7. If y -16 when x 4,find x when y 32. xy -64;-25. If y 8 when x -2,find y when x 4. xy -16; -4Chapter 115Glencoe Algebra 110. GEOMETRY For a rectangle with given area, the width of the rectangle varies inverselyas the length. If the width of the rectangle is 40 meters when the length is 5 meters, findthe width of the rectangle when the length is 20 meters. 10 m9. DRIVING The Gerardi family can travel to Oshkosh, Wisconsin, from Chicago, Illinois,in 4 hours if they drive an average of 45 miles per hour. How long would it take them ifthey increased their average speed to 50 miles per hour? 3.6 h8. If y -7.5 when x 25, find y when x 5. xy -187.5; -37.56. If y 100 when x 120,find x when y 20. xy 12,000; 6004. If y 10 when x 5,find y when x 2. xy 50; 25Assume that y varies inversely as x. Write an inverse variation equation thatrelates x and y. Then solve.1.Determine whether each table or equation represents an inverse or a directvariation. Explain.Exercises8Both methods show that x2 when y 18.188 x2348 x2y21 x2 y1xMethod 2 Use a proportion.If y varies inversely as x and y 12 when x 4, find x when y 18.Method 1 Use the product rule.x1 y1 x2 y2Product rule for inverse variation4 12 x2 18x1 4, y1 12, y2 18Example11From the product rule, you can form the proportion x2 y2 .Product Rule f or Inverse VariationAn inverse variation is an equation in thekform of y x or xy k. If two points (x1, y1) and (x2, y2) are solutions of an inverse variation,then x1 y1 k and x2 y2 k.Inverse VariationStudy Guide and InterventionIdentify and Use Inverse Variations11-1NAMEAnswers (Anticipation Guide and Lesson 11-1)Lesson 11-1Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Chapter Resources
A2Glencoe Algebra 1Inverse VariationStudy Guide and 04060xx32 x-100 -50 O16-32 -16 O-24 -12 O24 x501612Chapter 11-18-36-10-20-36 -18 O-20 -10 O20 x1810103620y6y1836 x24 xy50x100-7.2-3.6-7.2 -3.6 O3.67.23.67.2xGlencoe Algebra 1y6. y 4.8 when x 5.4-100-32-245. y -18 when x -9-50-16-124. y 4 when x 5123. y -25 when x 51224100y2. y 12 when x 4O12y2432y1. y 9 when x -3612618 18 23 12 32 6y 6Graph each variation if y varies inversely as x.Exercises54010206.7201030yxThe equation xy 200 can be used torepresent the situation. Use various speedsto make a table.Solve for k.xy kInverse variation equation12(3) kx 12 and y 336 kSimplify.Choose values for x and y, which have aproduct of 36.Example 2Graph an inversevariation in which y varies inversely asx and y 3 when x 12.an equation of the form xy k, where k 0Example 1Suppose you drive200 miles without stopping. The timeit takes to travel a distance variesinversely as the rate at which youtravel. Let x speed in miles per hourand y time in hours. Graph thevariation.Inverse Variation EquationCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Chapter 11Graph Inverse Variations Situations in which the values of y decrease as the valuesof x increase are examples of inverse variation. We say that y varies inversely as x, or y isinversely proportional to x.11-1NAMEInverse VariationSkills PracticeDATEPERIOD421124inverse, xy 4y8x0.52inverse, xy 3322. xy direct, y 2x3. -2x y 0-48-8O816 xxy 48816 xO10y1020 xxy 45Chapter 1127Glencoe Algebra 113. If y 48 when x -4,find y when x 6. xy -192; -3212. If y 15 when x 1,find x when y -3. xy 15; -5114. If y -4 when x , find x when y 2. xy -2; -111. If y -24 when x -3,find x when y -6. xy 72; -129. If y -7 when x 3,find y when x -3. xy -21; 7-20-10-20 -1020xy 3610. If y -6 when x -2,find y when x 4. xy 12; 38. If y 4 when x 8,find y when x 2. xy 32; 16Solve. Assume that y varies inversely as x.-16-16 -816yy7. y 15 when x 3-16xy 10-88x-84-16 -8 O8165. y -6 when x -6-4O46. y -4 when x -12-88y4. y 2 when x 5Assume that y varies inversely as x. Write an inverse variation equation thatrelates x and y. Then graph the equation.1.Determine whether each table or equation represents an inverse or a directvariation. Explain.11-1NAMEAnswers (Lesson 11-1)Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Lesson 11-1Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 11Inverse VariationPracticeDATEPERIOD2051.250.528inverse; xy ky40x0.252.80-8-16024direct; y kxyx-2ydirect;y kx3. x -3inverse;xy k74. y xA3-241224 xxy 5Oyx7. y 2.5 when x 2Chapter 11Glencoe Algebra 1Answers8between 3 h 20 min and 3 h 45 minGlencoe Algebra 113. TRAVEL Jesse and Joaquin can drive to their grandparents’ home in 3 hours if theyaverage 50 miles per hour. Since the road between the homes is winding andmountainous, their parents prefer they average between 40 and 45 miles per hour.How long will it take to drive to the grandparents’ home at the reduced speed?12. EMPLOYMENT The manager of a lumber store schedules 6 employees to take inventoryin an 8-hour work period. The manager assumes all employees work at the same rate.a. Suppose 2 employees call in sick. How many hours will 4 employees need to takeinventory? 12 hb. If the district supervisor calls in and says she needs the inventory finished in 6 hours,how many employees should the manager assign to take inventory? 811. If y 0.6 when x 7.5, find y when x -1.25. xy 4.5; -3.610. If y 3.2 when x -5.5, find y when x 6.4. xy -17.6; -2.759. If y -8.5 when x 6, find y when x -2.5. xy -51; 20.48. If y 124 when x 12, find y when x -24. xy 1488; -62Write an inverse variation equation that relates x and y. Assume that y variesinversely as x. Then solve.xy 30-12-8-161216 x-24 -12 O8824y6. y -6 when x -5Oxy 24-16 -816y5. y -2 when x -12Asssume that y varies inversely as x. Write an inverse variation equation thatrelates x and y. Then graph the equation.1.Determine whether each table or equation represents an inverse or a directvariation. Explain.11-1NAMEInverse Variation5671210.29Chapter 113. ELECTRICITY The resistance, in ohms,of a certain length of electric wire variesinversely as the square of the diameterof the wire. If a wire 0.04 centimeter indiameter has a resistance of 0.60 ohm,what is the resistance of a wire of thesame length and material that is 0.08centimeters in diameter? 0.15 ohm414.4Annual InterestRate(percent)18Yearsto DoubleMoney2. MONEY A formula called the Rule of72 approximates how fast money willdouble in a savings account. It is basedon the relation that the number of yearsit takes for money to double variesinversely as the annual interest rate.Use the information in the table to writethe Rule of 72 formula. yr 7231.25 foot-candlesFind the illumination produced 8 feetfrom the same source.9DATEPERIODy1020Units Produced(thousands)x3025,000,48Glencoe Algebra 1150 cycles per secondc. Suppose a piano string 2 feet longvibrates 300 cycles per second. Whatwould be the frequency of a string4 feet long?The shorter string vibratesmore quickly than the longerstring.b. If you have two different lengthstrings, which one vibrates morequickly (that is, which string has agreater frequency)? a. Write an equation that representsthe relationship between frequency fand length . Use k for the constantkof variation. f ℓ k or f 5. SOUND The sound produced by a stringinside a piano depends on its length. Thefrequency of a vibrating string variesinversely as its length.01002003004. BUSINESS In the manufacturing of acertain digital camera, the cost ofproducing the camera varies inversely asthe number produced. If 15,000 camerasare produced, the cost is 80 per unit.Graph the relationship and label thepoint that represents the cost per unit toproduce 25,000 cameras. 48Word Problem Practice1. PHYSICAL SCIENCE The illumination Iproduced by a light source variesinversely as the square of the distance dfrom the source. The illuminationproduced 5 feet from the light source is80 foot-candles.Id2 k80(5)2 k2000 k11-1NAMEPrice per Unit ( )Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Answers (Lesson 11-1)Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Lesson 11-1Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A4Glencoe Algebra 1EnrichmentPERIOD41648A324843264 12841622 cup32 cup33 cup1 cup205RateHours5024100251001100100For a 100-mile car trip, the time thetrip takes varies inversely as theaverage rate of speed the car travels.100MilesThe number of servings of oat brandirectlyvariesas the numberof cups of oat bran.11 cupWaterServings1 cupOat Bran3For a set of rectangles with a widthdirectlyof 4, the area variesas the length.42 WChapter 119. x varies8. z varies7. x variesinverselyinverselydirectlyas z.as y.as y.Use the table at the 530106604032yz1.51x30422452063Glencoe Algebra 12.5For a set of right triangles with a heightdirectlyof 10, the area variesas the base.1025102010hA543b816A job requires 128 hours of work. Thenumber of hours each person worksvaries inversely as the numberof people working.ServingsPeople WorkingHours of Work12833055For a car traveling at 55 mi/h, thedirectlydistance covered variesas the hours driven.DistanceSpeed2. HoursFill in each table below. Then write inversely, or directly to complete eachconclusion.Direct or Indirect Variation11-1DATECopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Chapter 11Rational FunctionsStudy Guide and InterventionDATEPERIODx -77x - 35x15. y x 53x - 414. y x 4x -4x-212. y x -11x 11x 19. y x 0xx-511. y x 06x5x 10x-18. y x -23x56. y - x 0x 1x-33. y x -1Chapter 11048121620242832361345611Number of People278Glencoe Algebra 116. DINING Mya and her friends are eating at a restaurant. The total bill of 36 is split36among x friends. The amount each person pays y is given by y x , where x is thenumber of people. Graph the function.713. y 3x 212x 8x-710. y x -4x 33x - 27. y x -3x 2x5. y 2x - 444. y x-2x 212. y x 421. y x x 0x-4State the excluded value for each function.Exercisesx-5x-5 0Set the denominator equal to 0.x 5Add 5 to each side.The excluded value is x 5.4b. y State the excluded value for each function.3a. y xThe denominator cannot equal zero.The excluded value is x 0.ExampleIdentify Excluded Values10The function y x is an example of a rational function.Because division by zero is undefined, any value of a variable that results in a denominatorof zero must be excluded from the domain of that variable. These are called excludedvalues of the rational function.11-2NAMEBill per Person ( )NAMEAnswers (Lesson 11-1 and Lesson 11-2)Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Lesson 11-2Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 11DATERational FunctionsStudy Guide and pter 110y24. y x -30yxx 0; y 3x31. y x x 0; y 0Glencoe Algebra 1x12xx 1; y 0yAnswers025. y x 10y-22. y x x 0; y 00y x-1 2yx 1x0-26. y x-30yxGlencoe Algebra 1yx 3; y 0x43. y x 1 x 0; y 1Identify the asymptotes of each function. Then graph the function.Exercises–1xStep 2 Make a table of values and plot the points.Then connect them.y 21Identify the asymptotes of y 2 . Then graph the function.Step 1 Identify and graph the asymptotes using dashed lines.vertical asymptote: x 1horizontal asymptote: y 2Examplemakes the denominator equal zero, x b. It has a horizontal asymptote at y c.x-ba c has a vertical asymptote at the x-value thatA rational function in the form y Because excluded vales are undefined, they affectthe graph of the function. An asymptote is a line that the graph of a function approaches.Identify and Use Asymptotes11-2NAMECopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Rational FunctionsSkills Practicex-18. y x 4x7. y x -79x - 365x 40Chapter 110yx 2, y 0x-2313. y 0yx 0, y 0110. y xxxx013yxx -1, y -1x 1214. y -10yx 0, y 0311. y xx 1x-20yxGlencoe Algebra 1x 2, y 3115. y 30yx -1, y 0212. y x99. y x -82x - 14-56. y x 7x 6x3. y x -6PERIODIdentify the asymptotes of each function. Then graph the function.3x 21x 83x - 55. y x -8x 4x-34. y x -4x-222. y x 2DATE61. y x x 0State the excluded value for each function.11-2NAMEAnswers (Lesson 11-2)Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Lesson 11-2Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Rational FunctionsPractice325. y x - x 12x 3x 532. y x -55x - 2xxChapter 1114where x is the total flight time. Graph the function. plane traveling between the two cities is given by y 1000x ,x-1x020040060080010000y1234y 1000x5Glencoe Algebra 1Total Flight Timexx -1, y -1x 1212. y -10yx 1, y 029. y 1000 miles from Indianapolis, Indiana. The average speed of a13. AIR TRAVEL Denver, Colorado, is located approximately00xyx 3, y 2x-3111. y 2yx -2, y 0x 2210. y 00xyx 0, y 038. y xyx 0, y 017. y x2516. y x x-52x3. y x 5PERIODIdentify the asymptotes of each function. Then graph the function.12x 36x-14. y x -3-11. y x x 0State the excluded value for each function.11-2DATEAverage Speed (mph)A6Glencoe Algebra 1Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Chapter 11Rational Functions23Time (hours)1y 515x4Chapter 11should be x 3, y –6.made in his graph. The asymptotesx – 6. Explain the error that Nicolasgraph with asymptotes at y 3 andx 320the equation y - 6 and draws a3. ERROR ANALYSIS Nicolas is graphingx -0.5, y 0are the asymptotes of the function?x is the total time spent in the car. Whatx 0.51540, wherePeter’s trip is given by y stop for lunch. The average speed ofDuring the trip, Peter makes a 30-minutegrandparents’ house 40 miles away.2. DRIVING Peter is driving to his050100150200300250DATEPERIODxGlencoe Algebra 1d. What are the asymptotes for the newequation? x 1, y 8x -158y 8 c. Now assume that the family wants tolet a long-lost cousin attend for free.Rewrite the equation to find the newcost per paying person y.b. What are the asymptotes of theequation? x 0, y 8a. Write an equation showing the costper person y if x people attend the50reunion. y 8 5. FAMILY REUNION The Gaudet familyis holding their annual reunion atWatkins Park. It costs 50 to get apermit to hold the reunion at the park,and the family is spending 8 per personon food. The Gaudets have agreed tosplit the cost of the event evenly amongall those attending.x 0 would give y, the cost of the car,an infinite value when the car is brandnew. The asymptote needs to belocated at x a, where a 0.the car. What are the asymptotes ofthe function? Explain why x 0cannot be an asymptote.x -1.2, y 100; An asymptote aty 100, where x is the age of17,900x 1.24. USED CARS While researching cars topurchase online, Ms. Jacobs found thatthe value of a used car is inverselyproportional to the age of the car. Theaverage price of a used car is given byWord Problem Practice1. BULLET TRAINS The Shinkansen, orJapanese bullet train network, provideshigh-speed ground transportationthroughout Japan. Trains regularlyoperate at speeds in excess of 200kilometers per hour. The average speedof a bullet train traveling between Tokyo515and Kyoto is given by y x , where x isthe total travel time in hours. Graphthe function.11-2NAMEAverage Speed (km/h)NAMEAnswers (Lesson 11-2)Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Lesson 11-2Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 11PERIODInequalities involving Rational FunctionsEnrichmentDATE1Graph y x.A7Chapter 11011. y x 1yxGraph each inequality.ExercisesGlencoe Algebra 10y16Answers22. y xxStep 4 Shade the regions where the inequality is true.Step 3 Begin testing values. A value must be tested betweeneach set of lines, including asymptotes.Region 1 Test (–1, 1). This returns a true value forthe inequality.Region 2 Test (–1, –0.5). This returns a true valuefor the inequality.Region 3 Test (–1, –2). This returns a false value forthe inequality.Region 4 Test (1, 2). This returns a true value forthe inequality.Region 5 Test (1, 0.5). This returns a false value forthe inequality.Region 6 Test (1, –1). This returns a true value forthe inequality.Step 2 Plot the asymptotes, x 0 and y 0, as dashed lines.Step 1 Plot points and draw a smooth solid curve. Because theinequality involves a greater than or equal to sign,1solutions that satisfy y x will be a part of the graph.Example0xxxGlencoe Algebra 1y13. y -1x 10y0yInequalities involving rational functions can be graphed much like those involving linearfunctions.11-2NAMECopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Simplifying Rational Expressionsan algebraic fraction with numerator anddenominator that are polynomialsx2 1yExample: 2PERIOD2, -2 #8 , - #82x 2 4x 44x 11x - 3Chapter 11141716. -3, 215. 4, 162k 2 2k - 3k - 20k 642, 8214. -2, 222x 2 5x 112. x 2 - 10x 162m - m - 121 ,1m -110. - 2a 5a 6a-18. -3, -22y2 - y - 23y - 12-1, 52, -26. -4, 422x 18x - 162m - 8m2 - 44. 232 a12 - a2. -32Glencoe Algebra 1Exclude the values for which x2 - 9 0.x2 - 9 0 The denominator cannot equal 0.(x 3)(x - 3) 0 Factor.x 3 0 or x - 3 0 Zero Product Property -3 3Therefore, x cannot equal -3 or 3.2- 2n - 3 13. n-5, 12n 4n - 525 - n 211. n 2 - 4n - 59. -3, -12k - 2k 1k 4k 32x 4x 4x -47. -22n -42n - 125. -2, 22x 4x -23. 22b -82b1. 2State the excludedx2 1x -9Example 2.values of 2State the excluded values for each rational expression.ExercisesExclude the values for which m 2 0.m 2 0The denominator cannot equal 0.m 2 - 2 0 - 2 Subtract 2 from each side.m -2Simplify.Therefore, m cannot equal -2.m 2Example 1State the excluded4m - 8.value of Because a rational expression involves division, the denominator cannot equal zero. Anyvalue of the denominator that results in division by zero is called an excluded value ofthe denominator.RationalExpressionDATEStudy Guide and InterventionIdentify Excluded Values11-3NAMEAnswers (Lesson 11-2 and Lesson 11-3)Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Lesson 11-3Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A8Glencoe Algebra 1Simplifying Rational ExpressionsStudy Guide and InterventionDATE(6z)(9z 2)(6z)(4y)(6z)(9z 2)(6z)(4y)9z 2 4y13Chapter 1113.11.9.7.5.3.12ab1. a 2b 212 ; a 0; b 0abx 21 ; x -2 or 2 x2 - 4 x - 222n - 8 ; n -4 or 4 n 2 - 16 n 4x-2x2 - 4 ; x -2 x 2 4x 4 x 2k-1k2 - 1 ; k -3 or -1 k 2 4k 3 k 32n 5n - 25 ; n -1 or 5 n 2 - 4n - 5 n 12n 7n 12 n 3 ; n -4 or 2 n 2 2n - 8 n - 21814.12.10.8.6.4.Glencoe Algebra 11 5 ; n 03nm-2m2 - 4 ; m -4 or -2 m 2 6m 8 m 4x 2 2x 1 x 1 ; x -1 or 1 x2 - 1x-1a 2 3a 2 a 1 ; a -3 or -2 a 2 5a 6 a 321m - 2m 1m-1 ; m - or 1 2m 2 - m - 1 2m 122x 3x x-6 ; x -2 or 2 2x 2 - 82x 42y 1y -y-2 ; y 2 or 8 y 2 - 10y 16 y - 87n 32. 21n 8Simplify each expression. State the excluded values of the variables.ExercisesExclude the values for which x2 - 5x 6 0.x2 - 5x 6 0(x - 2)(x - 3) 0x 2orx 3Therefore, x 2 and x 3.3x - 9 2x - 5x 63(x - 3)Factor.(x - 2)(x - 3)x - 5x 63(x - 3) 1Divide by the GCF, x - 3. (x - 2)(x - 3) 13 Simplify.x-23x - 9. State the excluded values of x.Simplify 2Simplify.Divide the numerator and denominator by 6z.The GCF of the numerator and the denominator is 6z.24yz54z.Simplify 1Example 224yz54z 3Example 1Factoring polynomials is a useful tool for simplifying rationalexpressions. To simplify a rational expression, first factor the numerator and denominator.Then divide each by the greatest common factor.(continued)PERIODCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Chapter 11Simplify Expressions11-3NAMESimplifying Rational ExpressionsSkills PracticeDATE3x 15x - 25b 2 - 2b - 86. -5, -225. -6, 3249x y16x 3y 236x yChapter 11x 2 10x 24x - 2x - 24x 6x-6 ; -4, 6t 3t 42d-7 ; -5, 7 ; -4, -221. 2t 2 5t 6t 6t 819. 217. 22d 10d - 2d - 35z 1z -115. 21z-1y - 8; -813. ; -1, 113y 2 - 64y 8 ; -611. n 63n 18 9. ; 0, 05 324c28bc321bc ; 0, 07. 219b 2 - 6b 9b - 9b 1822. 2a 2 3a - 4a 2a - 8b-6b-3; 3, 6 a-1a-2h-43; 3, 4 x–4Glencoe Algebra 11 ; -6, 4 y 2; 9 ; -4, 220. 2h - 7h 123h - 918. 2x 6x 2x - 2416. 2y 2 - 7y - 18y-9x 1x – 1 ; -1 14. 4x 44x - 412. 25a340a b22rb ; 0, 08a b 10. 324mr12m 2r m ; 0, 08. 32Simplify each expression. State the excluded values of the variables.y2 - 9y 3y - 18b 7b 104. -5, 523. -2, 22k 2k -44n 1n 42. -4PERIOD1. 72pp-7State the excluded values for each rational expression.11-3NAMEAnswers (Lesson 11-3)Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Lesson 11-3Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 11Simplifying Rational ExpressionsPracticeDATEp 2 - 16p - 13p 362. 4, 92a 8a 1514a3422m 2; 68 cc d ; c: 0, -2, d: 0 3A9x 2 - 7x 10x - 2x - 15r2 r - 616. r 2 4r - 1214. 2r 2 - 7r 6r 6r - 712. 2r 3r 6 ; -6, 2r 7r–6; -7, 1 x 3x - 2 ; -3, 5 550d 550 fulcrumbttree stumpChapter 11Glencoe Algebra 1Answers20Glencoe Algebra 1c. If a force of 200 pounds is applied to the end of the lever, what is the force placed onthe tree stump? 2200 lbb. What is the mechanical advantage? 11ta. Write an equation that can be used to calculate theb-tmechanical advantage. MA 20. PHYSICAL SCIENCE Mr. Kaminksi plans to dislodge atree stump in his yard by using a 6-foot bar as a lever.He places the bar so that 0.5 foot extends from thefulcrum to the end of the bar under the tree stump. Inthe diagram, b represents the total length of the barand t represents the portion of the bar beyond thefulcrum.b. If d is 1650, what percent of the budget did the band account for? 25%a. Write an expression that represents the cost of the band as afraction of the total amount spent for the school dance.advertising for a dance. In addition, they hired a band for 550. ; -3, 41m-3 ; -3, 310. 2m 3m -98. p - 6; 2p 2 - 8p 12p-20, 0, 036k 3np 2 9k ;6. 20k 2np 5 5p 3y 42(x 6)2y 2 9y 41 3 ; - , 18. 4y 2 - 4y - 3 2y - 32 23(x - 4)19. ENTERTAINMENT Fairfield High spent d dollars for refreshments, decorations, and2x 2 18x 363x - 3x - 36t 9 ; 3, 9t-3t 2 - 8115. t 2 - 12t 2717. 2 ; 2, 72b - 1411. 22b - 9b 14 b - 2y 2 6y - 16 y 8 ; 213. y 2 - 4y 4 y - 2m-62- 4m - 12 9. m40cd 5c d5c d7. 24 248a12a 2 ; 04. 36xyz 3 2z 2 ;5. 3x 2y 2z xy0, 0, 02- 2a - 15 3. a-5, -32PERIODSimplify each expression. State the excluded values of the variables.n - 494n - 28-7, 71. 2State the excluded values for each rational expression.11-3NAMECopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.AFP (in)7-p448 lb, r 0r radius in feets speed in mphw weight in poundsf force in pounds0.0672wsf r23. AUTOMOBILES The force needed tokeep a car from skidding out of a turn ona particular road is given by the formulabelow. What force is required to keep a2000-pound car traveling at 50 miles perhour on a curve with radius of 750 feeton the road? What value of r is excluded?p-7r 3-3 - r or 2. GRAPHING Recall that the slope of aline is a ratio of the vertical change tothe horizontal change in coordinates fortwo given points. Write a rationalexpression that represents the slope ofthe line containing the points at (p, r)and (7, -3).in2.5 lb2 lbShelby’s: vs Gabe’s 22Gabe and Shelby each push open a doorwith one hand. In order to open, the doorrequires 20 pounds of force. The surfacearea of Gabe’s hand is 10 square inches,and the surface area of Shelby’s hand is8 square inches. Whose hand feels thegreater pressure?Chapter 11DATE21PERIODGlencoe Algebra 1b. How many students attended privateschool? 6,400,000x 42,240,000 xa. Write a rational expression to expressthe ratio of public school students to xprivate school students.5. SCHOOL CHOICE During a recentschool year, the ratio of public schoolstudents to private school students inthe United States was approximately7.6 to 1. The students attending publicschool outnumbered those attendingprivate schools by 42,240,000.x 24. PACKAGING In order to safely ship anew electronic device, the distributionmanager at Data Products Companydetermines that the package mustcontain a certain amount of cushioningon each side of the device. The device isshaped like a cube with side length x,and some sides need more cushioningthan others because of the device’sdesign. The volume of a shippingcontainer is represented by theexpression (x2 6x 8)(x 6). Find thepolynomial that represents the area ofthe top of the box if the height of the boxis x 2. x 2 10x 24Simplifying Rational ExpressionsWord Problem Practice1. PHYSICAL SCIENCE Pressure is equal tothe magnitude of a force divided by thearea over which the force acts.11-3NAMEAnswers (Lesson 11-3)Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Lesson 11-3Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A10Glencoe Algebra 1EnrichmentPERIODCross multiply.Simplify.Replace with the values given.Original equationChapter 114225. flight time 18 seconds, vacant time 3 seconds,dwell time 2 seconds, number of hands 1124. flight time 16 seconds, vacant time 1 second,dwell time 2 seconds, number of hands 253. flight time 4 seconds, vacant time 1 second,dwell time 1 second, number of hands 232. flight time 13 seconds, vacant time 1 second,dwell time 5 seconds, number of hands 171. flight time 6 seconds, vacant time 1 second,dwell time 1 second, number of hands 2Glencoe Algebra 1Given the following information, determine the number of balls being juggled.6 bDivide.So, the number of balls being juggled is 6.24 4bb f dv dh9 3b 21 3b12 24where f is the flight time, or how long a ball is in the air, d is the dwell time, or how long aball is in a hand, v is the vacant time, or how long a hand is empty, b is the number of balls,and h is the number of hands (either 1 or 2 for a real-life situation, possibly more for acomputer simulation).So, given the values for f, d, v, and h, it is possible to determine the number of balls beingjuggled. If the flight time is 9 seconds, the dwell time is 3 seconds, the vacant time is 1second, and the number of hands is 2, how many balls are being juggled?f db v dhMathematicians look at various mathematical ways to represent juggling. One way theyhave found to represent juggling is Shannon’s Juggling Theorem. Shannon’s JugglingTheorem uses the rational equationDATECopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Chapter 11Shannon’s Juggling Theorem11-3NAMEDATEPERIOD3cf2x 8x 8x 16x 4x 2 - 16· .Find 2Simplify.Simplify.Multiply.b6ab 322x 22x - 3x 2Chapter 11(a 3)(a 5) (a - 2)(a 4)a 2 7a 12a 2a - 8a 2 3a - 10a 2a - 8n 6n 513. · 22n - 7n 10n -1n - 2511. · 2222x 9x 4 n-1n-2x 2 x-24x-1 x 6x 8x -x-19. · 2 2228x 8x - 2x 1n-4x-17
Lesson 11-1 Chapter 11 7 Glencoe Algebra 1 Skills Practice Inverse Variation Determine whether each table or equation represents an inverse or a direct variation. Explain. 1. x y 0.5 8 1 4 2 2 4 1 2. xy 2 3 3. -2 x y 0 Assume that y varies inversely as x. Write an inverse variation equation that relates x and y. Then graph the .
