To solve some rational equations, you need to factor a quadratic expression. Multiply each side by the LCD, x 2. 1 x2 Use the Distributive Property.
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1 . Pla Objectives To solve ratioal equatios To solve proportios Eamples Solvig Equatios Wit Ratioal Epressios Solvig b Factorig WorkRate Problem Solvig a Ratioal Proportio Ceckig to Fid a Etraeous Solutio Wat You ll Lear To solve ratioal equatios To solve proportios... Ad W To fid te time it will take two people to do a job, as i Eample Solvig Ratioal Equatios Ceck Skills You ll Need GO for Help Lessos ad Solve eac proportio.... m 7 t m 9, 9 Fid te LCD of eac group of epressios New Vocabular ratioal equatio Mat Backgroud Some ratioal fuctios ma ave etraeous solutios. For eample, solutios ma make a deomiator equal we substituted ito te epressio. More Mat Backgroud: p. D Lesso Plaig ad Resources See p. E for a list of te resources tat support tis lesso. PowerPoit Bell Riger Practice Ceck Skills You ll Need For itervetio, direct studets to: Ratios ad Proportios Lesso : Eamples, Etra Skills ad Word Problem Practice, C. Addig ad Subtractig Ratioal Epressios Lesso : Eample Etra Skills ad Word Problem Practice, C. Part Solvig Ratioal Equatios lie Visit: PHScool.com Web Code: ate77 A ratioal equatio cotais oe or more ratioal epressios. Oe metod for solvig ratioal equatios is similar to te metod ou leared i Capter for solvig equatios wit ratioal umbers. Te deomiators are,, ad. Te LCD is. Q R Q R Multipl eac side b. Q R Q R Q R Use te Distributive Propert. No fractios! Tis equatio is easier to solve. Solvig Equatios Wit Ratioal Epressios Solve. Ceck te solutio. Te deomiators are,, ad. Te LCD is. Q R Q R Multipl eac side b. Q R Q R Q R Use te Distributive Propert. No ratioal epressios! Now ou ca solve. Ceck See if makes true. () () Subtract from eac side. Divide eac side b, ad te simplif. Solve eac equatio. Ceck our solutio. b. c c 9 Capter Ratioal Epressios ad Fuctios 9 Special Needs L I Eample, ave studets make a table or draw a diagram wit two clock faces to clarif te problem. Help studets fid alterate was to solve te problem. Ceck teir processes ad solutios. learig stle: visual Below Level L Empasize to studets tat, if a perso ca complete a job i das, te te rate of tat perso job per d Te work doe rate time. learig stle: verbal
2 RealWorld Coectio Ma studet groups old car wases to raise moe. To solve some ratioal equatios, ou eed to factor a quadratic epressio. Solvig b Factorig Solve. Ceck te solutio. Q Q R Multipl eac side b te LCD,. R Collect terms o oe side. ( )( ) Factor te quadratic epressio. ( ) or ( ) Use te ZeroProduct Propert. or Solve. Ceck Solve eac equatio. Ceck our solutio., b. t 8 t t, m m You ca solve a work problem b fidig te part of te job eac perso does i oe uit of time (our or miute). Fid te part of te job eac perso does, ad te write a equatio. Q R Q R () Use te Distributive Propert. Simplif. WorkRate Problem Voluteerism Ma ca was ad wa a car i mi. His older sister Kala ca do te same job i mi. How log will te car take tem if te work togeter? Defie Let te time to complete te job if te work togeter (i miutes). Relate fractio of job fractio of job fractio of Ma ca do Kala ca do job completed i miute i miute i miute Write 8Q 8Q R Multipl eac side b te LCD, 8. 8 Use te Distributive Propert. 7 8 Simplif. 8,or 7 7 Simplif. It will take te two of tem about miutes to was te car workig togeter.. Teac Guided Istructio Alterative Metod Suggest to studets tat te grap Y o teir grapig calculators ad fid a itercepts. Teacig Tip Remid studets tat oce te ave foud a solutio, te sould substitute te value back ito te origial equatio to ceck teir work. Remid tem to ceck for values tat are ecluded from te domai. PowerPoit Additioal Eamples Solve 8. Ceck te solutio. 8 Solve. Ceck te solutio., Reee ca mow te law i miutes. Joae ca do te same job i miutes. How log will it take tem if te work togeter? mi 8 8 Ceck Ma will do 7? 7 of te job, ad Kala will do 7? 7 of te job. Togeter, te will do 7 7, or te wole job. Pegg ca pick a busel of apples i mi. Peter ca pick a busel of apples i 7 mi. How log will it take tem to pick a busel if te work togeter? 88 mi Lesso Solvig Ratioal Equatios 9 Advaced Learers L Callege studets to solve. # real, learig stle: verbal Eglis Laguage Learers ELL Some studets ma ot uderstad te meaig of a busel of apples (i te problem followig Eample ). Have tem fid ad discuss defiitios for tis word. If possible, sow several differetsize produce baskets ad relate tem to a busel. learig stle: verbal 9
3 Mat Tip Some studets ma tik tere is a error if a equatio produces o solutio. O te board write. Ask studets to stud te equatio ad suggest values for tat solve te equatio. Oce studets recogize tat o suc values eist, use tis to elp tem remember tat equatios wit o solutios are possible. PowerPoit Additioal Eamples Solve. Ceck te 8 solutio., 8 Solve. o solutio Resources Dail Notetakig Guide L Dail Notetakig Guide Adapted Istructio L Closure Ask studets to describe ow to solve a equatio cotaiig a ratioal epressio. Elimiate te fractios b multiplig eac side of te equatio b te LCD, or, if it is a proportio, b writig cross products. Collect like terms o oe side ad solve te equatio. If ecessar, factor ad use te ZeroProduct Propert. Fiall, ceck te solutios to rule out a etraeous solutios. Part Solvig Solvig Rroportios Proportios Some ratioal equatios are proportios. You ca solve tem b usig cross products. Solvig a Ratioal Proportio Solve. Ceck te solutio. ( ) () Write cross products. 8 Ceck Solve for. Use te Distributive Propert. Solve eac equatio. Ceck our solutio. b. a a, Te process of cross multiplig or multiplig b te LCD ma give a etraeous solutio to a equatio, or a solutio tat makes a deomiator i te origial equatio equal zero. A etraeous solutio solves a ew equatio but ot te origial oe. So ou must ceck our solutios. Ceckig to Fid a Etraeous Solutio Solve. ( ) ( )( ) Write cross products. 8 8 Simplif eac side of te equatio. Collect terms o oe side. ( ) Factor. or Use te ZeroProduct Propert. Solve. Ceck Udefied! Te equatio as oe solutio,. Solve eac equatio. Ceck our solutios. o solutio b. w w w c c 9 Capter Ratioal Epressios ad Fuctios 9
4 EXERCISES Practice ad Problem Solvig A GO B Practice b Eample Eamples, (pages 9, 9) Eample (page 9) Eamples, (page 9) Appl Your Skills for Help For a guide to solvig Eercise, see p. 98. GO GO for Help lie Homework Help Visit: PHScool.com Web Code: ate pages 9 97 Eercises b. Aswers ma var. Sample: Crossmultiplig I tik it s quicker. c. No it ol works for ratioal equatios tat For more eercises, see Etra Skill ad Word Problem Practice. Solve eac equatio. Ceck our solutios p p a a.,. 9. t s s t 8, 7., 8. p 9. 8 c c p p,. v. c. a v v (c ) c a a z a a. a a z z. Gardeig Maria ca weed a garde i ours. Robi ca weed te same garde i ours. How log will te weedig take tem if te work togeter? 7 7. David ca uload a deliver truck i mi. Allie ca uload a deliver truck i mi. How log will te uloadig take tem if te work togeter? N.7 mi Solve eac equatio. Ceck our solutios. If tere is o solutio, write o solutio c,. m c c m m,... o solutio Solve eac equatio. Ceck our solutios.. r. r r b b,. r r 7. r r s, s 8. d d 9. u d d d u u. s s s. u u s, s s u, s u u u. Two pipes fill a storage tak i 9 ours. Te smaller pipe takes times as log to fill te tak as te larger pipe. How log would it take te larger pipe to fill te tak aloe?. A teacer assiged te equatio. Carlos studied te equatio ad said, I ll start b fidig te LCD. Igrid studied te equatio ad said, I ll start b cross multiplig. Solve te equatio usig Carlos s metod ad te Igrid s metod. b. Writig Wic metod do ou prefer? Eplai w. b c. See margi. c. Critical Tikig Will Igrid s metod work for all ratioal equatios? Eplai. a 7 7a b 9. Fid te value of eac variable. c 9 c d 7 a, b 7, c, d c d. Write two fuctios usig te epressios o te two sides of te equatio ( 7). Grap te fuctios. a b. See margi. b. Fid te coordiates of te poits of itersectio. c. Are te values of te poits of itersectio solutios to te equatio? Eplai. Yes te values are solutios to te origial equatio sice bot sides are equal. are proportios. Xmi Xma Xscl Ymi Yma Yscl Lesso Solvig Ratioal Equatios 9 ±, ( 7) b. ( 9.,.7), (.,.), (.,.7), (.8,.). Practice Assigmet Guide A B 7,, 9 A B 8,, C Callege 8 Test Prep 9 Mied Review 7 Homework To ceck studets uderstadig of ke skills ad cocepts, go over Eercises,, 8,, 8. Error Prevetio! Eercises Studets ma tr to elimiate te deomiator we simplifig. Remid studets to be sure to appl te correct properties of equatios. Eercise Ask studets wic equatios from Eercises ca be solved usig Igrid s metod ad w. Eercise it is a ratioal proportio. GPS Ericmet Guided Problem Solvig Reteacig Adapted Practice Practice Pearso Educatio, Ic. All rigts reserved. Name Class Date Practice 7 Solve eac equatio. Ceck our solutio d. 9 d... m m 8 m b b t t t. David ad Fioa ave a ouse paitig busiess. It takes Fioa das to pait a certai ouse. David could pait te same ouse i das. How log would it take tem to pait te ouse if David ad Fioa worked togeter?. Suppose te Williams Sprig Water Compa as two macies tat bottle te sprig water. Macie X fills te bottles twice as fast as Macie Y.Workig togeter, it takes tem mi to fill bottles. How log would it take eac macie workig aloe to fill te bottles?. Cao, wo is a eperieced arcitect, ca draw a certai set of plas i. It takes Carl, wo is a ew arcitect, to draw te same set of plas. How log would it take tem workig togeter to draw te set of plas?. For eercise, Josep likes to walk ad Vicet likes to ride is bike. Vicet rides is bike km/ faster ta Josep walks. Josep walks km i te same amout of time tat Vicet rides km. Fid te rate tat eac of tem travels.. Te Ra Publisig Compa as two pritig presses. It takes te ew pritig press mi to prit, fliers. Togeter te two presses ca prit te, fliers i mi. How log does it take te older pritig press b itself to prit te, fliers? m m Solvig Ratioal Equatios s s s 7 (a ) a r r r k k 9 k m m L L L L L 9
5 . Assess & Reteac PowerPoit Lesso Quiz. Solve.. Solve.. Solve ( ).. Solve.,. Juaita ca was te car i miutes. Gabe ca was te car i miutes. Workig togeter, ow log will it take? 77 mi Alterative Assessmet Divide te class ito small groups. Tell te groups to write a quiz for te lesso tat cosists of tree problems oe problem modelig eac of Eamples,, ad. Te ave eac group ecage quizzes wit aoter group ad solve te problems, icludig ceckig te solutios. Circuit coected i series. Circuit coected i parallel. RealWorld Coectio Te resistace of a coductor is te oppositio it gives to te flow of electrical curret troug it. GPS Electricit Lamps ca be coected to a batter i a circuit i series or i parallel. You ca calculate te total resistace R T i a circuit if ou kow te resistace i eac lamp. Resistace is measured i oms ( ). Circuit Coected i Series Circuit Coected i Parallel R T R R R T R R. Fid R T.. V 7. R T V fid R. V R R R R 8. Fid R T.(Hit: Te lamps i tis 9. R T V fid R. V circuit are coected i series ad i parallel.).7 V R R R R 9 R R R R R R. Aswers ma var. b Sample: b b b C Callege 9 Capter Ratioal Epressios ad Fuctios. OpeEded Write a ratioal equatio tat as as a solutio. See left.. Travel A plae flies mi/. It ca travel 98 miles wit a wid i te same amout of time as it travels 8 mi agaist te wid. Solve te equatio 98 8 s s to fid te speed s of te wid. mi/ Solve eac equatio. Be sure to ceck our aswers.. 9.,. r r. t t r r r r t t t. It takes Jo 7 mi to pait a room. It takes Jeff mi ad Jackie 8 mi eac to pait te same room. How log will te paitig take if all tree work togeter? N. mi 9
6 RealWorld Coectio Ma cemistr classes iclude a lab o testig acidit. 7. Cemistr A cemist as oe solutio tat is 8% acid ad a secod solutio tat is % acid. Te cemist eeds to mi some of bot solutios to make liters of a solutio tat is % acid. Let s te umber of liters of te 8% solutio used i te miture..8s Write a epressio for te amout of acid i s liters of te 8% solutio. b. Write a epressio for te umber of liters of te % acid used i te miture. s.( s) c. Write a epressio for te amout of acid i a % acid solutio. d. Write a equatio tat combies te amout of acid i eac solutio to make te total amout of acid i liters of % acid solutio. e. Solve te equatio ou wrote i part (d). d..8s ±.( s) (.)() f. How ma liters of eac solutio will te cemist eed to make liters of % acid solutio? L of 8% solutio ad 8 L of % solutio 8. Sumi ca was te widows of a office buildig i te time it takes er appretice. Oe da te worked o a buildig togeter for mi, ad te Sumi cotiued aloe. It took mi more to complete te job. How log would it take er appretice to was all te widows aloe? Test Prep Resources For additioal practice wit a variet of test item formats: Stadardized Test Prep, p. 79 TestTakig Strategies, p. 7 TestTakig Strategies wit Trasparecies Error Prevetio! Eercise 9 Remid studets to multipl eac term b te LCD. Some studets forget to multipl a costat b te LCD. Stadardized Test Prep Test Prep Multiple Coice Sort Respose 9. Wic is a solutio of? B A. B. C. D.. Wic iequalit cotais bot solutios of? G F.,, G., # H. #, J. # #. Wat is te least commo deomiator of,, ad? C A. B. C. D.. You are trig to fid te umber ou would add to bot te umerator ad deomiator of to make a fractio equal to. Write a ratioal equatio tat ca be used to fid tis umber. Te solve te equatio to fid te umber ad ceck our solutio. See margi.. f() O 7. 8 O Mied Review GO for Help Lesso Add or subtract. t k (t )(t ) (k )(k ). 8.. k z z z t 8 t k k k 8. f() O 8. 8,. 8, Lesso Grap eac fuctio eiter b traslatig te grap of! or b makig a table of values.. See margi.. f()! 7.! 7 8. f()! 8 9.!..!.! 9. O.,. 8,., 7. 9, Lesso Solve eac equatio b factorig. 7. See left lesso quiz, PHScool.com, Web Code: ata Lesso Solvig Ratioal Equatios 97 O pages 9 97 Eercises. [] ( ± ) ± ± ± [] appropriate metod, but wit oe computatioal error. O 97
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