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# Class Note for MATH 1300 at UH 2

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SECTION 51 Simplz ing Rational Expressions Chapter 5 Rational Expressions Equations and Functions SCCtiOIl 512 Simplifying Rational Expressions gt Rational Expressions Rational Expressions De nition A rational expression is the ratio oftwo polynomials X2 3 is the ratio of two monomials The rational expression x2 4y2 2x4y The rational expression is the ratio of two binomial s x2 4x 5 X2X 2 The rational expression is the ratio oftwo trinomials Note The value of the polynomial in the denominator of a rational expression cannot be equal to 0 since division by U undefined For example in the rational 2x expression cannot be equal to 5 x b Simplifying A rational expression is not simplified ifthe numerator and denominator share comm on factors To simplify a rational expression we can factor the numerator MATH 1300 Fundamentals of M athematics 297 CHAPTER 5 RananalExpressmns Equatums and Functmns and factor the denominator and dunde out common factors that appear 1n boun numerator and denominator Example 51mp11 y each rat ona1 expmmon 3x2 a 3 12m 74 b 2x4y 2 x 4x75 E 7 x2x72 Snlntinn 1 1 1 a 3x y if x noomomnmnnomnnnonnnnmn my 2 2 f y y mannvmaonWoonmonnonomnnoon 1 L 5 4y uni6y Factaxthe difference onwo squares mLhe numeratm Foam autLhe ch mLhe denammatm Dwxde autLhe camman bmaxmalfactm afx 21 51mm poem the mnamul 1n the numeratm Factaxthe tnnm39mahnthe denammatm Dwxde autLhe camman bmaxmalfactm afx 71 51mm 298 Umversxty afHaustan Department anaLhemancs SECTION 5 1 11111an Razmmz Expressans Additinnzl Example 1 1mp11ry each 11101111 expressmn a 1819 24W 2 7 b 728 72 7x 2 smug 1 1 a 189 2 hambamnumm m1 11mm m1 24W y 11m 111m 11111111 cammanmanamalfactms 1 Swapth 1 1 28 7 41271 Factax 11111111 GCFanhe 111111111 1nd b 7 g 7L 11cm 1 11mm 11m 11m 11 7182 7 x x l 111 cammanmanamals factms 1 1 41271 7 7 51111th gt3 AdditjnnzlExamplel 31mp11 y each rat m1 expression 17 21 a smug 1 r 2 Pasta nut a 711nm numeratax Factax the a 1 22 M difference arm 51111111511111 denammatax 4x 1 211 W Thendwxdeaunhecammanbmamulfactm 51mm MATH 1300 Fundamenmls anathemancs 299 CHAPTER 5 RananalExpressmns Equatums and Functmns Factaxthe difference arm cubes mLhe numeratm39 Factaxthe difference arm squares mLhe denammatax Then amt gum camman bxnaxmalfactax 5mm AdditinnzlExampleS hmphfy each rational expression a x274x712 4730 b 2x23x75 3x2 512 Snlntinn 1 7 12 LUV zmmxfsssgzsngwz x7 30 x5 W mm mm camman bxnm39malfaclm 1 5mm Factaxthe mnamuls um appearmthe numeratm39 and the denammatax Then amt gum camman bxnaxmalfactax 5mm Addi nnzlExampleA lmpley each rational expressmn 4x2 716 I m W75x3y715 300 Umversxty aHaustan Department anaLhemancs Snllltinn 4x246 4amp2 3x273x76 327x72 a iww 39 3x1 Zj 7 4x2 7 3x1 b W75x3y715 7 xy753y75 W7 x4y7 MATH 1300 Fundamenmls anathemancs SECTION 5 1 sampwng Razmmz Expressans mm am the m m bum numeratm39 and denammatax Factaxthe difference arm squares m the numeratm39 andfactaxthe mnamuhn the denammatax Then amt gum camman bmamul mm 5mm mm by gaupmg m bum numeratm39 and denammatax Us the dnsmbuuve pmpmymbam numeratm39 and denammatax Then amt gum camman bxnaxmalfactax 5mm 301 Exercise Set 51 Simplifying Rational Expressions Simplify the following rational expressions If the expression cannot be simplified any further then simply rewrite the original expression 10 11 12 13 14 15 302 g 25 E 36 64 E 39 60x2 y5 48x5 y3 7 49114129 561171710 7 5x3 x y7 10x5 ery3 2aibcid 6b7a 4x8 x73 5x715 x5 x2725 16 7 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 49702 c279c18 x2711x10 1004c2 x272x715 x210x21 m27m720 m2m730 x25x6 x2x712 x27x12 x277x730 x278x12 x2713x42 x277x10 x27x10 x2736 x212x36 x278x16 x2716 9x36 x2 4x 7x2714x x72 10x2 730x 5x2 le University of H ouston Department of M athemattcs Exercise Set 51 Simplifying Rational Expressions 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 6x2 7 8x 9x3 42x2 x2 7x6 8x28x 4x2 7 20x x274x75 6x224x18 4x278x760 5x210x740 10x2730x20 4x217x15 5x213x76 4x278x721 8x2724x714 6x275x74 10x29x2 15x24x74 5x2722x8 8x230x7 16x24 9x245 6x2713x5 m3m2m1 m3mm2nn axiaybxiby axiay2x72y xyi3x2y76 yzi3275y15 ab75a2b710 all 4127 5a2 720 x378 x72 MATH 1300 Fundamentals of M athematz39 cs x5 x3125 x3 27 49 7 x273x9 x 71 2 x x1 303 CHAPTER 5 Rational Expressions Equations analFunctions SCCtiOIl 522 Multiplying and Dividing Rational Expressions gt Multiplication and Division Multiplication and Division Multiplication of Rational Expressions Recall the rule for multiplication of fractions To multiply two fractions place the product of the numerators over the product of the denominators If air C and d are real numbers and c i U and d t 0 then Wife apply the rule for multiplication of fractions to find the product of rati on 31 express sic11 3 Example Find the 011 owing product x22x EE xl x2 1 X 4 Solution 2 x2 2x 8 xl x 27 8x1J Write the product ofthe numerators over the 2 39 2 product ofthe denominators x 1 x4 x 1x4 1 1 F th a1 11 d actor e trinorni int e numerator an W X 2 ilk7 factor the difference oftwo squares in the denominator Then divide out the common 15quot T x I M binomial factors 1 1 x 2 I Simplify x l 304 University of H ouston Department of M athematics SECTION 52 Multiplying anal Dividing Rational Expressions Division of Rational Expressions Recall the rule for di 39 39 n of Haiti on 3 To find the quotient of two fractions multiply the first fraction by the reciprocal of the second fracti on Ifa b c andd are real numbers and b 0 c at CI andd 0 then calcb aba39d Two numbers are reciprocals of each otheriftheir product is 1 For example 5 8 and E are reciprocals since EEl 1N Epplj the rule for division of fractions to find the quotient of rational expresssmns Example Find the following quotient n7 quot 9 Solution x3 x2 9 x3x2 25 x S 39 x2 25X5 22 9 x3lx2 25 paw 9 z nintwswidi Lazawaiix zt x5 x 3 MATH 1300 Fundamentals of M athemati cs Multiply the first rational expression by the reciprocal of the second rational expression Write the product of the numerators over the product of the denominators Factor the di erence oftwo squares in the numerator and factor the di erence of39 two squares inthe denominator Then divide out the common binomial factors Simplify 305 CHAPTER 5 RananalExpressmns Equatums and Functums Additinnzl Example 1 Perform the followlng operaoons All results should be ln slmpllfled form a 2 3x6 2x710 b 3X Snllltinn a g szntethe rstexpxessmnln acuan farm 3x 6 Wm lhc pmduct af ne numerath overlhc pmduct af ne denamlnatms Facial ounhc ch lnLhe denamlnatax Then dlvlde ounhc cornrnon blnamlal facial Simphfv sznte lhc seconu expreslen rnrrocuon farm Muluply lhc rst rouonol expreslen by lhc xeclpmcal af ne seconurouonol expreslen Wm lhc pmduct af ne numerath overlhc pmduct af ne denamlnatms Facial ounhc ch lnLhe numeratax Then dlvlde ounhc cornrnon blnamlal facial l Slm My 7 3x p AddidnnzlExampleZ Perform the followlng operaoons All results shoulol be ln slmpllfledform 2x 5x20 1 O x 73x728 6x2 18 21 b r23relo 306 Umverslty aHaustan Department anaLhemancs SECTION 5 2 Muzapzymg and wadmg Razramz Expressions Snllltinn 2 5H20 M WmLhepmductafthenumenlms avexthe 3X48 6X2 6X2 X2 28 pmductaflhe denammatms a Factm bath numeratm39 and denammatm Then divide aunhe camman manamul saints and me camman bxnm39mal factm Swarm b Mu uply are rstnuaml expressmn by are xecxpmcal athe secandnuanal expressmn 2 X 4 Weeeepreerewreenrmerroe mduct athe denammatms 2xx23x710 P Factm bath numeratm39 and denammatm Then divide aunhe camman manamul factm and me camman bxnarmalfactm Swarm AdditinnzlExample 3 Perform the followmg operations All results should be m srrrrplrfred form 4730 x2476 x27x10 x2713x42 a b 6x223x7 3x2 21 1 4x24x735 2x27x710 MATH 1300 Fundamenmls anathemancs 307 CHAPTER 5 RananalExpressmns Equatums and Functmns Snllltinn 4731 7x76 x27x10 x2713x42 7 x27x730x27x76 7 x2 7x 10x2 713x42 7 MWMUJ Factaxmmnamalsmnumentm and denammatax Then amt am 7 MWWX7 Lhecammanbxnamalfactms 5mm 0 Wm the mam mm numeratms avexthe mam mm denaxmnatms 6x2 23x7 b 4x24x735 2x27x710 5X223X7 2X2 Mu uplythe rslnuanalexpxessmn 27 2 bythexecxpmcalaflhesecand 4x 4x7 35 3x 72x71 nuanalexpxessmn 39 4X2 Hz 735 3X2 7 2X 71 Dvexthe mam mm denaxmnatms 9me hammmmmm and denammatax Then amt am 39 wwwhq mecammbmmmm 12 7 x71 5mm 308 Umversxty aHaustan Department anaLhemancs Exercise Set 52 Multiplying and Dividing Rational Expressions Multiply the following rational expressions and simplify No answers should contain negative exponents 1 EH 7 18 2 E 9 32 3 7102 5 4 127 7111 707d3 5 csdgi 116129 5 6 6 xy wz W32 x10 9 msnz nAIS p612 7 717315 39 7 5 397m3n7 x3y4 agbl x2y3a4 8 abz 397x77397b5 y y 9 72x2 is 6x 10 6x 3 2x 11 x5I x73 x73 x710 12 x6I 75 x71 x6 x75 13 x72 x72 14 muL3 xl 75x 15 77x x77 MATH 1300 Fundamentals of M athematz39 cs 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 3 27x 2 3x2 x75 57x 2xl x73 3 3 x74 5x720 72 4x28 x7 3x74 3x12 2x8 3x73 2x72 4x4 6x7124x12 x3 3x76 x7 6x724 2x78 5x35 6x710 3 5x 1579x 2x 4x76 6x79 x2x x2x76 I x276x5 x23x74 x272x715 x27x72 x27x712 x2 8x15 x2 79xl4 x23x710i2x274x 6x2724x x5 6x2730xx24x72l x27x76 40x78x2 x4I x279 37x x216 309 Exercise Set 52 Multiplying and Dividing Rational Expressions 32 33 34 35 36 Divide the following rational expressions and simplify x2725x212x36 x6 x5 2x2 9x10x2 7x12 x25x6 2x23x75 x2 2x78 3x2714x75 3x216x5 x27x720 axibxayiby Iax7x2al4 ax7x73a72 akibx2a72b a072adibc2bdi 027d2 acadibcibd 3aci3adbcibd No answers should contain negative exponents 37 38 39 40 41 42 43 44 45 46 310 LE 8 39 32 Exj 2539 5 7107 j 2 FE 39 7 2 76 4 77 740 5 lt gt x xquotz3 yzz7T ys 11307 bsc9 b4 a2 115126 7 sdz old5 3 Z 1257ng w z 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 x 71 x71 x6 393x18 x L 5x x24ix2 75 i 10 167x2 x74 x274 27x x5 725ml x29i x3 x271 I x272xl 4x279 i2x73 x2710x25 x75 x273x710i x27x76 x273x728 39 x2x712 x24x4 ix278x720 x276x716 x279x8 6x2xil i3x22xil 6x25xl 39 3x24xl 10x2 717x6i 6x2 5x74 5x2 4x712 3x2 72x78 amianbmibni amian73bm3bn amanibmibn I aman73bmi3bn 0x72dxcy72dyi cxcy5dx5dy x2x73xy73y I cx5d x5d University of H ouston Department of M athematz39cs SECTION 53 Adding and Subtracting Rational Expressions SCCtiOIl 532 Adding and Subtracting Rational Expressions gt Addition and Subtraction Addition and Subtraction Addition and Subtraction of Rational Expressions with Like Denominators Recall the rule for a1liti n and subtrac ti on of fractions with like denorninatcrs To add or subtract two fractions whose denominators are the same add or subtract the numerators and keep the common denominator If cab and C are real numbers and c 7 0 then a E ab c c c and aba 2 c 6 ENE apply the rule for addition subtraction of fractions with like denominators to find the sum and clilTerence rational expressmns With like denominators Example Perform the following operations All results should be in simpli ed form x 2 quot P x 4 x L l a x24 x2 4 Q bl MATH 1300 Fundamentals ofMathematics 311 CHAPTER 5 RananalExpressmns Equations and Functions Snllltinn x 2 x 2 Wute the sum orthe humemtots aver the commu denammatm a Factor the difference ortwo squares m the denammatm end then utvtue out the commu hmomul factor sm X 7 2 phfy x 2 x e 2 Wute the meme otthe numerath b aver the commu denammatm Factor the difference ortwo squares m the denammatm end then utvtue out 3 the somehomomuhsto stmpltry x 2 Addition and Subtraction of Rational Expressions with Unlike Denominators To add or subuaet two thaeuohs whose denomlnators are hot the same we must heWhte each fraction so that they have a eommoh denomlnator The smallest suah denommotot ls calledthe least common denommutot LCD The method omhdthg the LCM of the denomlnatol39s wtll produce the LCD See Seettoh l 3 for amethod omholmgthe LCM least eommoh multtple We apply the techmques rot addmon and subtractlon omaeuohs thh uhltke denommatol s to hdthe sum and difference of tatlohal expresslohs wtth uhllke denominators Example Fmd the followlng sum The result should be m slmphtled form 2 5 WW 312 Umversxty afHaustan Department anaLhemancs SECTION 53 Adding and Subtracting Rational Expressions Solution We must rewrite the rational expressions so they 11 10111111011 deninminatcar 1 u gt 392 x y milling Find the least common denominator of the den ominatcr 3x2y3xxvy and 9xy23437yy 3xxy 3x 323 The least comm on denominator is 3x x y a 3 y 9x2y2 t 2 2 Express each rational expression as an equivalent one With a denominator of 9x y For the first rational expression 2 we need to multiply the denominator 3x y 3x2 by By since 3x2y3y 9x2y2 We also need to multiply the numerator 2 by 3y 5 For the second rational expression 979v 2 we need to multiply the denominator 9022 by 1 since 9193 x 9x2y2 We also need to multiply the numerator 5 by x 2 5 2 3y 5 x 2 2 2 2 3xy 9292 3xy3y 9292 x 6y 5X Peform the multiplications in the 2 2 2 2 numerators and the denominators 9x y 9x y 6y it Write the sum ofthe numerators over 972 2 the common denominator k y Additional Example 1 Perform the following operations All results should be in simpli ed form a 2X 2 x23x2 x23x2 5x 15 b 2x2 x 15 279 r 15 MATH 1300 Fundamentals ofMathematics 313 CHAPTER 5 RananalExpressmns Equatums and Functxzms Snllltinn 2x We the sum am mum m ltagt HM thummdrmmm rrctarbammemmm magnum 5K th b 5x 7 15 5x715 whtethemereneeermenumerrtm 72 X 15 72 X 15 72 HS mu camaneemmm r 51537 Factmbaththenumentm mw magnum r 5 Sxmphfy 2x5 Addi nnzl Example Perform the addmon Grve the result m srmphfred form We mustrewrrte the rational expressrons so they have a common denomrhator End the least common dehormnator ofthe denomrhators 18x and By 18x2 3 3 x and 6y2 3y 233K 23 y The least common denomrhatorrs 2 3 3 x yaw Express each amonal exprerrroh as an eqmvalem one wrth a denommator olexy 314 Umversxty afHaustan Department aMaLhemmzcs SECTION 5 3 Addmg and Subvamng Rat sz Expressions 1 Forthe fxrstrauonal expressxon K we needle nnu1np1y me denommator x 18x byy smce 18xy18xy We also needle muluplythe numerator 1 byy 7 Forthe secondranonal expressxon 67 we needle nnu1np1y me denommator y 6y by 3x smce 6y3x18xy We also needle nnu1l1p1y me numerator 7 by 3x 11 1 y 7 3x 18x 6y 18x y 6y Er y 21x Pefarm une mmuphceuens mlhe numeratms and denaxmnalms 18W 18W 7 y 21x Wnle lne sum enne nnnenlen 5 my avexthecamman denammatax AdditinnzlExample Perform lne subtracuon Gwe lne resultm snnpb ed form xy7x7y 5x 15y Snllltinn We mustrewnte lne rational expressxons so lney have a eennnnen denommator Fmd lhe least eemmen denommator ofthe denommators 5x and 15y 5x5 x and 15y3 5 y 5x 5 3y Tne1easl eennnnen denommatons 5 x 3 y15w Express each mend expressxon as an eqmvalent ene wlln a denommator ofley 5 y we needle multiply lne denommator x 5x by 3y smce 5x3y 15w We alsoneedtomuluply lne numerator x For me rst rauonal expressx en xy by 3y MATH 1300 Fundamenmls anathemancs 315 CHAPTER 5 RanamzlExpressmns Equatums and Functmns Forthe seeohdrahoha1 expressroh 15 y we heedto multiply the dehorhrhator y 15y by2 srhee 15yx15xy We a1so heedto multiply the numerator 2yby2 xy7x7y My 3yxiy X 52 15y 52 3y 15y 2 3v3y2 2 W Pefarm the mu txphcauans mm 715 W 15W hemerrtors one eerrorrmrtors 2 2 32 3y x i W whte the merehee orthe numeratms avexthe camman eerrormrrrtor 1529 329 3y2 e 2y 1r the numenwt Add the rst Forymmrto the negrtwe of 1529 the secandpalynamul 2 7 2 1r the numeralal xemave the M were heposraorhre 1529 terms together 429 3y2 e 1r the numenwt eorhhrre 7 he terms 152W Additinnzl Example Perform the subtraetroh Grve the result m slmph ed form 2 1 22 32 2 2 2 Snlutinn Ne must rewnte the rauoha1 expressxons so they have a common denommator Fmd the 1east eommoh denommator of the denommatots 22 32 2 and 22 2 2232 2122 and 222 21 21 22 21 2 The 1east eorhrhor denomrhatorrs 21222 2122 Express each rahoha1 expresstoh as an equtva1eht ohe wnh a dehommator of 2 2 1 2 2 316 Umversxty afHaustan Department anaLhemancs SECTION 5 3 Addmg and Subvac ng Rat sz Expressions x x Forthe firstrauonal expression 27 7 x 3x2 xlx2 we needle muleply me denominator xlx2 by x We also needle muleply me numerator xbyx Le 2xx1 denominator xxl by x2 We also needle muleply me numerator 1 by x2 Forthe secondrauonal expression we needle muleply me x 1 x 1 Perfarm une muhtphcauans m numerath end denammatms Wnle une difference erlne numerath eyenune camman denammalax 1n une numenwl edd me rst palynamul le une negelwe erlne seeend ene Remeye une pnenlneses Factax une numeralm39 end lnen dmde eul une camman bmamul factm39 Swapth Additinnzl Example 5 Perform lne feueynng operations Gwe all results m sxmplexedform 3 a Key yen 1 b 477 MATH 1300 Fundamenmls anathemancs 317 CHAPTER 5 RananalExpressmns Equatums and Functmns Snlutinn a The denommators are negatives ofeach other We can muluplythe numerator and denommator ofthe second rational cxpressmn by 1to obtam a common denommator 3 72 Wm are sum rm Hummers X y werure camman eerrerrrrrrmr 1 Snapth xiy 1 4 1 4777i b x3 1 x3 74x3 1 1x3 x3 74x127 1 x3 x3 12 1 wme me mererree mm numeralms X 3 aver are camman eerrerrrrrrmr 4 11 X Smphfy x3 318 Umversxty afHaustan Department anaLhemancs Exercise Set 53 Adding and Subtracting Rational Expressions Perform the indicated operations and simplify Whenever possible write both the numerator and denominator of the answer in factored form 10 11 12 13 14 15 31 5 7 114127 115124 x8x7 x5 x5 3x2 7 2x6 5x720 5x720 2x310x79 4x73 4x73 x4 x7 xl x2 MATH 1300 Fundamentals of M athematz39 cs 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 l 7 2 x72 x2 319 Exercise Set 53 Adding and Subtracting Rational Expressions 320 32 33 34 35 36 37 38 39 5 40 41 42 43 44 x1 x72 x73 x71 x57x72 x4 x73 8x12 6x76 5 7 2 12x76 10x40 6 3 8 777 x x71 x23x718 x26x x273x x 4 2 x2 10x24 x2 12x327x2 14x48 x 2 3 777 x277x12 x274x3 x2 75x4 University of H ouston Department of M athematz39cs SECTION 54 Complex Fractions Section 542 Complex Fractions gt Simplifying Complex Fractions Simplifying Complex Fractions De nition A complex fraction is a fraction that contains a fraction in its numerator or in its denominator or both Here are some examples of complex fractions E i 2 2 3 6 22v 14 and 72 413 6y E 5 10 Simplifying We simplify a complex fraction by eliminating the fractions that appear in the numerator 811de or denominator There are several methods of simplifying a complex fraction One method is to work in the numerator and the denominator of the given complex fraction separately and rewrite each of them as a single fraction if necessary and then perform the division and give the result in simplified form Another method is to multiply the numerator and denominator of the given complex fraction by the least comm on denominator of all the denominators that appear in all of the fractions in the numerator and denominator and then give the result in simplified form MATH 1300 Fundamentals ofMathematics 321 CHAPTER 5 Rational Expressions Equations andFunctions Example Simplify the following mmpl x fram on 2X x Solution Method 1 Ci btzun a smgl E fraction in the num Br ator Note that the denominstrbr 1s EdI39Eaquot 1quotf a single fractizm To obtain a singl 3 fraction in the num erator add the fi39actions end The least common c1311c ninamr is 6 l 3 a 39 39 2 quot 2 5 3 6 3 2 2 2 x 12 12 4x x 6 6 X2 12 4xx 322 University of H ouston Department of M athematics SECTION 5 4 Complex Fmenans 5 2 1 765 0 1 x Method 2 Multaply the numerator and denommator of the glven complex fraetrorr by the least eommoh denommator of all the denommators that appearm all ofthe fraetlohs m the numerator and denommator These denommators are 3 6 and 12 The least eommoh denommator ls 12 MATH 1300 Fundamenmls aMazhemanes 323 CHAPTER 5 RananalExpressmns Equatums and Functmns Addidnnzl Example 1 51mp11 y1he following complex mum 342 6x2y 25 smug 213 5 6x2y 25 2196 25 5 6x2y 219 25 5 6x2y 1 1 1 1 Z X y y 5 5 5 Z 3 X x 1 1 1 2 x Addi nnzlExamplel 1mp11fy the 011 owmg complex mum E X y 2 4 2xy 8 smug 11 472 2 4 2 2 4 2xy 2xy 8 8 21 4 4 2xy 8 324 Umversxty aHaustan Department anaLhemancs Additinnal Example 3 Simplify the 011 owmg complex fraction Snllltinn x 7 y 3y 24 5x 24y 24 24 8 MM 111 H 7 3x78y 5x 3y MATH 1300 Fundamenmls anathemancs SECTION 5 4 Complex Fractions 325 CHAPTER 5 RananalExpressmns Equatums and Functmns Additinnzl Example 4 Stmphfy the 011 owmg complex mm 326 Umversxty aHaustan Department anaLhemancs SECTION 5 4 Complex Fractions Additinnzl Example 5 Rewrite une gwen expressxon so that 1f contams posmve exponents rather than neganve exponents and men snnphfy x3 Snlutinn xquot 2 xquot 3 MATH 1300 Fundamenmls anathemancs 327 Exercise Set 54 Complex Fractions Simplify the following No answers should contain 5 1 negative exponents 10 gig 39 3 2 l 3 E 12 1 7 E 5 8 E 11 6 12 7 2 3 2 g 11 7 12 1 E i y 3 x4 3 27 5y 13 1 if 2 12115 2 b 4 5 402 g b7 14 2 4 7 3 211122 5 03d xi 2 81130 15 5 SM g 10 3x4 6 x7 6 8w 2 8 9x2w 16 7 x71 42 12 3 7 4x72 ab 8 17 b b a y 11123 51230 2 2d x y lObd 18 xy xy 3 E 9 i 2 x2 7xl2 5 19 8 35 x2 7 x7 20 6x4 328 University of H ouston Department of M athematz39cs Exercise Set 54 Complex Fractions 9x5 6x f 2 HP 2 20 w 30 x 18x3 4 x2711x24 2 x 1 1 3 55 7 21 g 31 x33 x34 x x 77 E Z x3 xl 2 5 LE 71 2 x x7 22 M 3239 3 2 gt x72 x2 71 x2 33 x L 2339 1 1 x77E 2 2x x 5 1 x9E iii x 3x 3 34 24 737 x671 x x 2 25 a b For each ofthe following expressions 2 7 i a Rewrite the expression so that it contains a 1 positive exponents rather than negative exponents 7 2 b Simplify the expression x y 26 7 ii 35 x71 x y x 1 1 6 x 7 37x 1 x71 36 27 9 x4 x x 1 71 37 xifyil x7i x 7y 28 x 35 10 38 c Ier 1 x7 cizd72 2 3 2 2 77 x 29 x75 x5 39 71 2 x y MATH 1300 Fundamentals of M athematics 329 Exercise Set 54 Complex Fractions 40 41 42 43 44 45 46 47 48 330 1 1 71f1 If2 7 at 2 0 1 7 d 1 0 3 7 d 3 x 3 y 3 x 1 y 1 1 3 b 3 1 2 71f2 x 3 y 3 2x 1 University of H ouston Department of M athematz39cs SECTION 55 Solving Rational Equations SCCtiOIl 552 Solving Rational Equations gt Rational Equations Rational Equations De nition of a Rational Equation Equations that contain atleast one rational expression are calledrational equations Here are three examples of rational equations 315 x l x 2 4x Z 325 x3 3 2x6 x1 72 1 Solving a Rational Equation We will solve a rational equation by multiplying both sides of the equation by the least common denominator LCD of all the rational expressions that are contained in the equation This will clear the equation offractions and the resulting equation can be solved by applying known techniques for solving equations Example 3 l 5 Solve and cheek 2 X b 2 Solution We first note that it cannot be equal to 0 since this would give a U in the denominator ofboth Z and Thus ifx 0 then we can multiply both sides ofthe given it 2x equation by the 6x LCD to clear the equation of fractions MATH 1300 Fundamentals ofMathematics 331 CHAPTER 5 RananalExpressmns Equatums and Functmns 3 2x on ion leoi i x 6 2x 23 3 i o pi 5 1X 18x15 18x71815718 Fe Check Substitute 7 forx tn the ongmal equation The solution 15 x Example some and check i Snllltinn Rewnte the equation by faetonng the btnonnta1 2io L 1 K 743 4 2x3 We rst note thatx eannot be equa1toe3 stnee nus wouldgtve a 0 in a denomtnaton Thus in 73 then we ean multiply both states of the given equation by 4i3 LCD to eleattne equation offracuons 332 Umversxty afHaustan Department anaLhemancs SECTION 5 5 Salvmg Ranamzl Equanans 4xx32x 5x3 2x 5x372x2x72x Check Substitute 71 for x m the ongmal equation x 1 x x3 4 2x6 71 1 7 71 43 4 17 24 71 71 7 2 MATH 1300 Fundamenmls anathemancs 333 CHAPTER 5 RananalExpressmns Equatums and Functmns Extraneous Solutions In the two examples above we rnuluphed both states of the equatron by the LCD and obtatned aresulung equauon whose solutron saus ed the orgmal ratrona1 equauon Note that m both eases the LCD eontatnedthe varrable x However m sorne eases where we rnuluply both states ofarahonal equatron by an enpressron eontatnrng avanable we obtatn aresultrng equatron that we solve to obtatn what appears to be a solutron but we nd that thrs apparent solutron does not checkm the ongmal equauon beeause rt wru make a denornrnator equal to 0 1n thrs ease we rnust ducard thrs value as a soluuon smce rt does not sausfy the ongmal equauon We ean sueh avalue an extraneous solutron An extraneous soluuon xs avalue thans obtaned upon somng an equatton that does not satrsfy the ongmal equauon and thus eannotbe a soluuon to the equauon Example Solve and eheek Snllltin Rewnte the equauon by factonng the brnormal x7 71 We rst note thatx eannot be equal toel or 1 smce thrs would gwe a 0 m a denormnutor Thus xf x z 1 then we Ban multtply both males of the gtven equatron by x1x71 LCD to elearthe equauoh offraouons 4x 2 rd 2 2 x1xrl 2 WW1 334 Umversxty nHnnstnn Department anaLhemancs SECTION 5 5 Sulvmg Ranamzl Equanuns 2x7274x4x74x 72x720 e2xe2202 72x2 amp Check Note that 71 does not sausfy the ongmal equataon stnee substttuttng 71 for XWlll resultm a 0 tn the denomtnaton 71 is an extraneous solutton Wennust dASEard 71 as a solutaon ofthe equataon smcex eannotbe equal to 71 The gtven equatton has no solutaon Additinnzl Example 1 Solve and eheek 3731 x 3 3x Snllltinn We rst note thatx eannot be equal to 0 stnee the would gtve a 0 tn the denomtnaton ofboth 3 and 31 Thus 1fx 0 then we ean nnultaply both states ofthe gtven x x equatton by the 3x LCD to eleatthe equataon offracuons MATH 1300 Fundamenmls anathemancs 335 CHAPTER 5 Rational Expressions Equations andFunctions Cheek Substitute 4 for x in the original equation 52 EE E 5237 ZE w837 5 5 5 77 5 5 The solution is X 4 Additional Example 2 9 iolve and check x 1 x x5 3 2x10 Solution Rewrite the equation by factoring the binomial 2x10 x x5 2x5 2 3 We first note thatx cannot be equal to 5 since this would give a U in a denominator Thus ifx 5 then we can multiply both sides ofthe given equation by 6x5 LCD to clear the equation offractions x 2 x x5 3 2xlO x 2 x 6x 4x5 3x 6x 4x 20 3x 2x 20 3x 2x 20 3x 3x3x x 20 CI 336 University of H ouston Department of M athematics SECTION 5 5 Sulvmg Ranamzl Equanuns exe2o20 020 x20 x720 Check Substitute e 20 for x m the ongmal equauou x 72 x x5 3 2x10 720 27 720 e20573 272010 720 2 The soluuon x5 x 720 Addi nnzlExample 3 olve and check 1 37 x 2x1 Snllltinn 1 We rst note thatx cannot be equal to 0 are smee each of these would give a 0 m a denominator Thus xfx 0 x 7 men we can muluply both sudes ofthe gweu equauou by x2x1 LCD to clearthe equauou offracuons 3 6 x 2x1 mm1x2x1ex2x1 2 x1 x2x112x1 exM x2x132x176x 2x2x6x376x 2x2x3 MATH 1300 Fundamenmls anathemancs 337 CHAPTER 5 RananalExpressmns Equatums and Functmns 2x2x73373 2x2x730 2x3x710 2x3 or x7 0 2x373073 x71101 F1 2 3 2 Check Substitute 7 forx m the ongmal equation The soluuons are x g andx 338 Umversxty aHaustan Department anaLhemancs SECTION 5 5 Salvmg Ranamzl Equanans Addi nnzlExample 4 Solve and check x73 Snlutinn We rst note thatx cannot be equal 03 smee ths wouldnge a 0 m a denominator Thus m 3 men we can multiply both sides ofthe gwen equauon by 3 LCD to clear he equauon of fracuons x30 or x730 3737073 xe33eo3 F3 Check Note that 3 does not sausfythe ongmal equauon smee subsmuung 3 for x ml resultm a 0 m the denominator 3 15 an extraneous solution We must ducard 3 as a solution of me equauon smee x cannot be equal to 3 Check Subsmute e3 forx m the ongmal equauon Th svluuvn As A e 73 MATH 1300 Fundamenmls anathemancs 339 Exercise Set 55 Solving Rational Equations Solve the following Remember to identify any 3x 71 17 7 extraneous solutlons x 5 2x x 1 7772 3x 18 72 5 3 x7 2 l 19 7 1 x279 3c 20 3 iiiyzz 5 20 0 5 x274 4 1354 5 8 4 21 2 7 x77x12 5 Six 3i2 22 117x 1 6 10 39 x23x710 6 Belts 23 3 9 8 20 39 7 13 4x7 x73 7 7 3 12 7 7 24 7i3 x 5 x 5 x71 5 8 3x4x8 25 Zix8771 54 x 2 399 x1 9 x52x6 26 a27177 x1 x1 9 4 10 3x45x77 27 x77 373 x6 x6 x7 3 11 534l73 28 git2772 x x 393 173 7 5 12 iii2 6x 4x 29 2139 w1 4 12 3 13 x7574 30 Haifa 39x9 2 14 75 14 3 x7 31 1x 7 3 x4 x4 2 15 750 32 x 7 1xi3 39 x72 7 x72 16 5x 0 x2 340 University of H ouston Department of M athematics Exercise Set 55 Solving Rational Equations 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 if 78 x75 3 3x715 7 4 x2 3x6 2 3 1 41178731176 3c15 7 2010 12 3 1 7 7 x5 x73 x22x715 2 1 4 x71 x2 7 x2 x72 iiii x73 x1 x272x73 MATH 1300 Fundamentals of M athematz39 cs 49 50 51 52 53 54 55 56 57 58 4 1 71 x4 x1 2 1 6x 2x7372x73 341 CHAPTER 5 Rational Expressions Equations analFunctions SCCtiOIl 562 Rational Functions gt Working with Rational Functions Working with Rational Functions De nition of a Rational Function A rational functionf is afunction ofthe forrnfx PU where P and Q are X polynomials Here are three examples of rational functions x2 3xl 4x JCX fX fx X241 x 3 x2 5X4 Domain of a Rational Function Poo X The domain of arational fx consists of all real numbers it except those values ofx for which Qx 0 To determine the domain of a rational function exclude from the set ofreal numbers the real solutions to the equation Qx 0 Example Find the domain the following rational functions and express in interval notati on 1 o 3 mg 11 i 32cl lb 339 xi x 395 03339 fix q r 52 4 342 University of H ouston Department of M athematics SECTION 5 s RananalFunmans Snllltinn a Solve the equamh x2 74 0 x2 40 x2x720 x20 or x720 x27 x72202 x2 The domam ofthe gwen funcuon 5 the set of all real numbers except 72 and 2 The domam m mterval hocamh 15 7w 2u72 2u2o b Solve the equahohxeko x730 x73303 F3 The domam othe gwen funenon is the set of all real numbers except 3 The domhhmtemlmhoms w3v3w e Solve the equamh x2 7 5x 4 0 2 5x40 x71x740 x u or x 40 Kemeon x74404 x1 F4 The domam ofthe gwen gunmen 5 the set ofall real numbers except 1 mm The domam m mcemx munch x5 ew1u14u4w MATH 1300 Fundamenmls anathemancs 343 CHAPTER 5 RananalExpressmns Equatums and Functmns Graph of a Rational Function Example 7 rs shown below The graph of the ratrona1 funetron x e X a State the domam ofthe tunetron m rntervat notahon to hnd the xrmtercepts ot the graph andtahet the pornts on the graph where the graph erosses the xraxxs 2 End the ydntercept ofthe graph andtabel the pornt on the graph where the graph erosses theyraxxs a Label the pornt on the graph whose rst coordmate rs 1 Snlutinn a So1ve the equatronr zeo x720 xe2202 x2 The domam ofthe gwen funetron rs the set of all rea1 numbers exeept 2 The domam m rnterval notatron rs rw2u2w There rs no pornt on the graph whose rst coordmate rs 2 344 Umversxty afHaustan Department aMaLhemancs SECTION 5 6 RanamlFunmans b To ndthe xrmtercepts flndthe real solutions ofthe equauOhx 0 18 The X1ntcrccptls 3 e To nd theyelhleheepl ma0 3 The yrmtercept ls 3 a The pomlwhose hsleoordlhalelslls ll Flnd m x73 ltxgt X72 173 72 m E Thepmntx 12 The graph of the fuheueh ls shown below labeled wlth the lhfomlamh from pans bed MATH 1300 Fundamenmls anathemancs 345 CHAPTER 5 RananalExpressmns Equatums and Functums Vertical Asymptotes Avemcal hne r a rs eaned a vemcal asymptote othe graph ofafuncuon y x xfy rnereases wrthoutbound cry deereases wrthout bound as r gets e1ose to a from the nght ofa or from the le ofa The hne 2 rs averuea1 asymptote for the graph of the ratrona1 funeuon 7 fx X From the graph we see that funcuonal values deerease wrthout X bound as xgets elose to 2 from the nght of2 and funeuonal values mcrease wrthout bound as x gets elose to 2 from the left of 2 vemcal asvmptote 346 Umversxty afHaustan Department aMaLhemancs SECTION 5 6 RanamlFunmans Finding Vertical Asymptotes 1 Letx Q be arataona1 funetron To fmd vemcal asymptotes rst srmphfy P0 QR the rat ona1 expresssr on by dwxdmg out any eomm on factors m numerator and denomrnator Then the vemcal asymptotes are ofthe form r a where 1 rs a rea1 number for whreh the denomrnator ofthe srmphfred expressron rs equal to 0 Example Fmd the vemcal asymptotes hr any of the graphs of the toll owrng ran ona1 runetrons a x 4x b m e m Snllltinn a Rewnte the funeta on by faetonng the denomrnator and then 64de out any eomm on factors x 22 m 1 M W n2 The denomrnator x72 rs equal to 0 whenever r 2 Thus the vemcal asymptote rs x 2 for x 72 The graph rs shoWn below 72 em x 2 vemcal asymptote MATH 1300 Fundamenmls anathemancs 347 CHAPTER 5 RananalExpressmns Equatums and Functxzms b Rewnte the function by faetprrhg the dehprhrrratpr and ther dunde put any eprhrh on factors Values for whreh the dehpmmatprrs equal to 0 are x1andx 4 Thus the vemcal asymptotes are x 1 and 4 The graph 15 shown below vemcal asymptotes Horizontal Asymptotes A hprerhta1 hney 5 rs eaned a hpruphta1 asymptote for the graph of a function y x xfy gets close to b as x rhereases wrthput bound or deereases wrthput bound The honzomal hm y 3 rs ahonzomal asymptote forth graph orthe ratrohal 3x1 X fuhetrphr From the graph we see that fuhehphal values get close to 3 asx mereases wrthoutbouhd and deereases wnhout bound 348 Umversxty afHaustan Department anaLhemancs SECTION 5 6 RanamlFunmans horizontal asymptote Techmques forfrhdlhg horizontal asymptotes wlll be mtxoducedm College Algebra MATH 1310 Additinnzl Example 1 The graph of the functlon x ls shown below Fmd the xrlntercepts the yrmtercept el and 2 and label the correspondmg pomts on the graph Snlntinn To nd the xrmtercepts ndthe real solutlons of the equamhr 0 x0 MATH 1300 Fundamenmls anathemancs 349 CHAPTER 5 RananalExpressmns Equatums and Functmns The xrmterceptxs 3 To nd Lhaydntercept nd0 Thcy1ntcrccptxs 3 Fmd 71 x3 A H 3 2 The comespondmg pomt on the graph s 712 Fmd 2 x 27271717 Th roprowdwg pom 0quot mp graph N 74 350 Umversxty DfHDustan Department anaLhemancs SECTION 5 6 RanamlFunmans Additinnzl Example 2 Fmdthe domam of the followmg rational funeuons and express each domam m mteml notanon x7 2 a f 17 2x b x M1 Snlutinn a We needto nd the values of forwhxeh the denommatons equal to o Solve the equauonxz2xr 0 x22x 3 0 x3xe10 x30 or x7 70 x373 073 xrl101 x x1 The domam ofthe given funeuon is the set of all real numbers exeept 73 and 1 The domam m mterval notahon ts rm73u731u1w b We heedto nd the values ofx forwhxch the denommatons equal to 0 Solve the equahon x1 0 The domain ofthe glVBn function is the set of all real numbers except 1 The domam m mterval notaeonte rwrlurlw AdditinnzlExample 3 Fmd the vemeal asymptotes or any of the graphs of the followmg rah anal funeh ons 5971 x71x24 2 x EDth a 1 00 MATH 1300 Fundamenmls anathemancs 351 CHAPTER 5 Rational Expressions Equations andFunctions Solution a Divide out common factors in numerator and denominator rm 5 5W 5 x x 1x24 Mx24 x24 Find the real values ofx for which the denominator ofthe simpli ed expression for x 1 is equal to 0 There are no real numbers it for which x2 4 is equal to 0 Therefore there are no vertical asymptotes The graph is shown below b The numerator and denominator share no common factors other than 1 Find the real values of X for which the denominator of the simplified expression is equal to 0 The value ofx for which the denominator x5 is equal to 0 is x 5 Thus the vertical asymptote is X 5 The graph is shown below 3 52 University of H ouston Department of M athematics SECTION 5 6 RanamlFunmans MATH 1300 Fundamenmls anathemancs 353 Exercise Set 56 Rational Functions Find the indicated function values If unde ned state The graph of each of the following functions has a Undefined horizontal asymptote at y 1 You will learn how to find horizontal asymptotes in a later mathematics 7 x course For each function 139 If fx 7 x 3 nd a Find the domain of the function and express it 7 1 as an inequality 3 f 0 h f 1 c f 3 b Write the equation of the vertical asymptotes of the function 5 c Find the x and yintercepts of the function nd if they exist If an intercept does not exist state None 2 f 0 h f 5 c f G d Find f1 and f 1 e Based on the features from ad match the 2 function with its corresponding graph using 3 If f x 7 a nd the choices Graphs IIV below a f0 b f3 c f 2 If fx x5 3x7 x Graph 1 Graph 11 4 If fx2x7find J y y L x76 4 4 a f0 b f4 f C 5 If fxz6find x 7x7 a f 2 h f 0 c f 5 Graph III Graph IV 6 Iffx2xfjnd 8y W x 2xl a f4 b f0 c N x 4 Y 7 7x I 73 4 74 4 8 7 1ffxx27121fmd 74 74 a f3 b f0 c f12 8 8 1 8 If 2 f1nd x 75x714 11 fxx4 a f0 b fH c f7 x73 x73 7x6 9 If fxmf1nd 12 fx7x2 a f3 b f4 c f0 13 fxx6 10 If fx2x5find x 3 x 7x712 a f0 b fez c f5 14 fxx4 x2 3 54 University of H ouston Department of M athematz39cs Exercise Set 56 Rational Functions The graph of each of the following functions has a horizo ntal asymptote at y 0 You will learn how to nd horizontal asymptotes in a later mathematics course For each function a Find the domain of the function and express it as an inequality b Write the equation of the vertical asymptotes of the function c Find the x and yintercepts of the function if they exist If an intercept does not exist state None d Find f1 and f 1 e Based on the features from ad match the 15 16 17 18 function with its corresponding graph using the choices Graphs IIV below Graph 1 Graph 11 Graph III Graph IV if 4 fx7 H 1003 x 1005 x 7 8 1003672 MATH 1300 Fundamentals of M athematz39 cs For each of the following functions a b C 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Find the domain of the function and express it as an inequality Then write the domain of the function in interval notation Write the equation of the vertical asymptotes of the function Find the x and y intercepts of the function If an intercept does not exist state None fx 7 3 fx fx Li 3 fx 3 fx 1 x 7 x2 712 x x2 2 20 f x2211 fltxgtx2i f4 x 7 x2 Sx 2 355 Exercise Set 56 Rational Functions 356 33 34 35 36 37 38 39 40 x f 57x x277x718 f x 5x 2x fx 25 x2 x1 fxx2716 x25x714 x fo 5x7 9x271 3x72 fx fx 25x 736 x275x4 2 x 7x6 x fo x275x724 x210x25 University of H ouston Department of M athematz39cs

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