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# Fundamentals of Math MATH 1300

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This 60 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1300 at University of Houston taught by Staff in Fall. Since its upload, it has received 63 views. For similar materials see /class/208409/math-1300-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15

Exercise Set 23 Slope and Intercepts of Lines State whether the slope of each of the following lines is Find the slope of each of the following lines If positive negative zero or undefined unde ned state Unde ned AL 139 P 5 y 23 c e y 2 q r C 3 r 24 d 2 Z 4 s 25 e 5 I Z 4 5 26 f 74 V Z 4 6 w f q S d Find the slope of the line that passes through the For each of the following following points If unde ned state Unde ned a Complete the given table b Plot the points on a coordinate plane and 7 0 0 and 3 7 graph the line c Use two points from the table to nd the slope 8 8 0 and 3 6 0f the line 9 2 5 and410 10 7 3 and5 9 27 y 4xt1 x y 11 6 4 and 2 4 0 12 51 and 5 8 2 73 13 72 3 and 677 0 14 72 76 and 75 10 1 15 73 7 8 and 73 7 4 2 16 8 77 and 71 77 17 2 78 and 0 73 1s 1 7 4 and 77 2 28 y 3 2 x y 2 mewmea 20 2 and G g 4 4 21 7 and 7 73 73 3 7 7 2239 gt 197 Math 1300 Fundamentals of Mathematics The University ofHausmn Chapter 2 Points Lines and Functions Exercise Set 23 Slope and Intercepts of Lines 29 y x 4 x y For each of the following graphs 74 a State the xintercept b State the yintercept 5 c State the coordinates of the xintercept 9 d State the coordinates of the yintercept e Find the slope of the line 78 33 y 30 y 7 x 6 x y 75 0 7 8 0 Answer the following 31 Examine the relationship in numbers 2730 between each of the equations and the corresponding slope that you found for each line Do you see any pattern Can you determine the slope of the line from simply looking at its equation W N Based on the pattern found in the previous problem state the slope of the following lines Without graphing the line or performing any calculations 2 9 33 y J 5 For each of the following equations y x 4 a Find the x and yintercepts 0f the line 3 7 tate t e coor inates 0 t e interce ts c y x 2 b s h d f h p d y x 4 c Plot the x and yintercepts on a coordinate plane d Graph the line based on the intercepts 35 y 72x 8 36 y 3x 6 37 y 4x7 5 38 y 73x7 7 39 5x 2y 20 Math 1300 Fundamentals of Mathematics The University of Houston Chapter2 Points Lines and Functions Exercise Set 23 Slope and Intercepts of Lines 40 2x3y18 41 3x5y30 42 3x244y 43 2x73y710 44 4x6y79 45 5x73y210 46 4x7y780 47 2x2y7 48 3x7l5 49 4y12 50 4x4yil5 51 76x24 52 2y7l4 For each of the following a Complete the given table b Plot the points on a coordinate plane and graph the line c Find the x and yintercepts of the line d Find the slope of the line 53 y2x78 x y 0 0 2 6 705 54 y7x3 x y 0 0 73 15 72 Math 1300 Fundamentals quatheman39cs The University ofHausion Answer the following 55 Examine the relationship in numbers 53 and 54 between each of the equations and the corresponding yintercept that you found for each line Do you see any pattern Can you determine the yintercept of the line from simply looking at its equation Based on the pattern found in the previous problem state the yintercept of the following lines Without graphing the line or performing any calculations a y 2x 9 b y 77x 5 c y i xi 2 3 d y 7 7x 4 Chapter 2 Points Lines and Funcn39ans Exercise Set 26 An Introduction to Functions For each of the examples below determine whether the mapping makes sense within the context of the given situation and then state whether or not the mapping represents a function 1quot Erik conducts a science experiment and maps the temperature outside his kitchen Window at various times during the morning A Time Temp 39F 2 Dr Kim counts the number of people in attendance at various times during his lecture this afternoon 85 87 Time of People State whether or not each of the following mappings represents a function A B 4 2 A B 5 A B 6 0 8 39 4 A B Math 1300 Fundamentals quatheman39cs The University of Houston Express each of the following rules in function notation For example Subtract 3 then square would be written as fx x 32 7 a Divide by 7 then add 4 b Add 4 then divide by 7 8 a Multiply by 2 then square b Square then multiply by 2 9 a Take the square root then subtract 6 squared b Take the square root subtract 6 then square 10 a Add 4 square then subtract 2 b Subtract 2 square then add 4 Complete the table for each of the following functions 11 fx x3 75 x fx 12 gx x7 42 1 x fx Find the domain of each of the following functions Write the domain rst as an inequality and then express it in interval notation 13 fx 14 fx 7 ChaplerZ Points Lines and Functions Exercise Set 26 An Introduction to Functions H 5quot 7 5 fXx73 77 x8 H P fx x76 x4 H gt1 fx x4 x76 1s hx 19 ft 21455 2 20 mpg 21 gx 4x71 x 4 9 5x7 2239 x 3x7 x71 x279 23 gx x2 24 hx O x2725 25 fxx272x724 26 fx772x 27 gx3x5 28 hxx2716 29 ftJ7 30 2004 31 fxJTS 32 gh m 33 fxE Math 1300 Fundamentals of Mathematics The University of Houston W 5 W UI W a W l W W W G 5 UI 5 6 UI 5 WWW hxm hl3IT2 WWW fxm fx8J 2x fx2m Hx M x76 Gx 3 x x fl gx3 2x79 732 39 ha 15 73 2x79 fx7 4x77 20025 20045 gxl 3x75 gxx5 2x7 fxx gxx72 Hx2x76 fx3x5 Chapzer2 Points Lines and Functions Exercise Set 26 An Introduction to Functions 55 fxx27 56 fx57l 57 fx 58 fx Lj Evaluate the following 59 If fx5x74 21 Find f 3 b Find x when f x 3 c Find fi d Find x when fx 7 e Find f 0 1 Find x when fx 0 60 If fx 3x1 21 Find f 75 b Find x when f x 75 c Find f d Find x when f x e Find f 0 1 Find x when fx 0 2 4 61 If hxx73 21 Find h1 b Find x when hx 1 c Find M72 d Find x when hx 72 e Find h7 1 Find x when hx 7 Math 1300 Fundamentals quatheman39cs The University of Houston 62 a W a 5 6 UI a a l a W If gx I x77 21 Find g0 b Find x when gx 0 c Find g 2 d Find x when gx 2 e Find g 73 1 Find x when gx 73 If hx Jx2 find a h7 b h25 m he If hxJx2fmd a h7 b h25 c h If fx73 find a f 16 h f 12 c f 9 If fxJ 73find a f 16 h f 12 c f 9 If gxx275x6 21 Find g3 b Find g74 c Find g7 d Find g0 If hr42 72115 21 Find h0 b Find h6 c Find h75 d Find 7 Chapzer2 Points Lines and Functions Exercise Set 26 An Introduction to Functions 2x x73 21 Find f77 b Find f0 c Find f 5 d Find f3 e Find f72 69 If foo 572x x4 21 Find g2 b Find g74 c Find d Find g73 e Find g0 70 If gx Math 1300 Fundamentals quIathematics The University of Houston Chapter 2 Points Lines and Functions 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 is the ratio of two monomials 2 The rational expression 37 1 1233 2 2 x 4 a The rational expresm on 2 y is the ratio of two binomial s x 4y 2 7r 4 5 The rational expreSSion 2 is the ratio oftwo tr1nomials x X 2 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 x I 7 cannot be equal to x b expres sion 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 m boun numerator and denominator Example Simphfy each rat onal expression 3x2y 12m3 x2 74y2 2x4y a b x24x75 E 7 0 x2x72 Snllltinn 3ny 3 a 3 Pasta ooun numeratax ma denammatm39 ma 12W 112 2 1g y y Lhandwxdeautmecamanmanamalfactms 1 x 5 My 4y p Factaxthe difference onwo squares mLhe numeratm Foam autLhe ch mLhe denammatm Dwxde autLhe camman bmarmalfactax afx 2y Simphfv poem the mnamul m the numeratm Factaxthe tnnm39mahnthe denammatm Dwxde autLhe camman bmarmalfactax afx 71 Simphfv 298 Umversxty afHaustan Department anaLhemancs SECTION 5 1 11m an Razmmz Expressans Additinnzl Example 1 1mp11fy each 1111101111 expressmn a 1819 24W 27 b 7282 72 I z smug a My Emmanummmammmm 24W 11m 11m nut 11 camman 11111111111 factms 5mm 1 1 28 7 41271 Factax 11111111 GCFanhe 111111111 1nd b igyw 11mm 1111mm 11mm 11 722 x x 1 1 gammamus 11m 1 1 41271 7 7 51111an gt3 AddidnnzlExamplel 51mp11fy each quot11101111 expressxon smug 1 1 2X 7 2 Factaxautarlmthenumentm Factaxthe a 7 7 dn exence 11m 51111111511111 denammatm 0 412 1 211 M Then mm 11111111 cammanbxnamalfactm 1 1 7 s1 2x1 Pm MATH 1300 Fundamenmls anathemancs 299 CHAPTER 5 RananalExpressmns Equatums and Functmns 1527 x2 2x4 Mann difference aftwa cubes mLhe numeratax b x3 8 7 x2 7 4 x 2 M Then am nut an camman bxnm39mal mm 1 Snnn x 2 AdditinnzlExam 12 3 hmphfy each rational expresmn a x274x712 4730 5 2x23x75 3x2 5x2 Snllltinn a an mnamuls um appear m an numeratax and an denammalm Then divide Dunne camman bmarmalfactm 2 2 5 mmnnnnnnnnnnn b m M n n Mann Tnn 3x275x2 MW am Dunne cammanbxnamalfactm 2x 5 Snnn 3x 7 2 Additinnzl Exam 12 4 xmphfy each rational expression a 4x271 3x2 3x 6 b W75x3y715 W smAy zo 300 Umversxty aHaustan Department anaLhemancs SECTION 5 1 11111an Razmmz Expressans 811111111111 0 412716 XL 31273176 327172 1 mm am the m 1n bum 1111111111111 1nd denammatax mm 1111 difference aftwa s Lures 1n 4 N 2 W7 1 mm m 11cm 1 31111111quot 1111 11111111111111 Th mm 11111111 MUM 111111111111 1 7 4 x 2 7 3x1 5quot th W75x3y715 7 xy53v5 Factaxby gaupmgmbamnummm 7 7X ma 111mm 4y75 W i x My y i 1 3 UseLhe ms bu v m e mbaLh 14 W mm 11111111 camman bmamnalfactm 51mm MATH 1300 Fundamenmls anathemancs 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 1 E 25 2 7Q 36 3 g 64 4 E 39 5 60x2y5 48x5y3 6 749114129 561171710 7 75x3xy7 10x5xy3 8 806cid2 120cid3 9 x y yix 10 67 dic 11 2aibcid 6b7a 12 712x7ywiz 6ziwx7y 13 4x8 x2 14 x73 5x715 15 x5 x2725 302 x3 x29 12er2 ab x2716 x74 49702 c279c18 x2711x10 1004c2 x272x715 x210x21 m27m720 m2m730 31 x25x6 x2x712 x27x12 x277x730 x278x12 x2713x42 x277x10 x27x10 x2 736 x212x36 x278x16 x2716 9x36 x2 4x 7x2 714x x72 10x2 730x 5x2 le University of H ouston Department of M athemattcs Exercise Set 51 Simplifying Rational Expressions 6x2 7 8x x 5 32 7 48 9x3 42x2 x3 125 x27x6 x327 8x28x x273x9 2 7 4x 720x 50 2x 1 x2 4x 5 x x1 6x224x18 4x278x760 5x210x740 10x2730x20 4x217x15 5x213x76 w 4x278x721 8x2724x714 6x275x74 10x29x2 15x24x74 5x2722x8 8x230x7 16x24 9x245 6x2713x5 m3 m2 m1 m3 mm2nn axiaybxiby axiay2x72y xyi3x2y76 yzi3275y15 ab75a2b710 a2b4b75a2 720 x72 MATH 1300 Fundamentals of M athematics 303 CHAPTER 5 Rational Expressions Equations analFunctions Section 522 Multiplying and Dividing Rational Expressions gt Multiplication and Division Multiplication and Division Multiplication of Rational Expressions Recall the rule for multiplieati on nffi39aetions To multiply two fractions place the product of the numerators over the product of the denominators If 153 C and d are real numbers and 6 0 and d t 0 then a E a 32 C d Cd Ware apply the mle for multiplication of fractions to find the product of rational express in 39 quot Example Find the follmvii39ig prtjidutzt 1 2x E x l x3 1 2 4 Solution 2 2 2 8 1 x 2x 8 xl X x 7 J Write the product ofthe numerators over the 39 product ofthe denominators x2 1 x4 x2 1x4 Factor the trinomial in the numerator and 1 1 Wx factor the difference oftwo squares inthe r X I M denominator Then divide out the common W binomial factors 1 1 X 2 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 fractitxns To nd 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 2 U andd i 0 then a ad 9 Cd of Two numbers are reciprocals of each otheriftheir product is 1 For example 5 8 and g are reciprocals since El Tore Epplj the rule for division of fractions to find the quotient of rational exgn esssmns Example Find the following quotient Solution 3 18 9 x3x2 25 X5 x2 25 X5 x2 9 x3x2 25 paw 9 5 3x531557 LV5TLMlX 3 x5 x 3 MATH 1300 Fundamentals of M athemati cs Multiply the rst rational expression by the reciprocal of the second rational expression Write the product of the numerators over the product of the denominators Factor the Merence 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 5 a x2 0 3x6 2x710 b Snllltinn a 2 5 2 5 szntethe rstexpxesslanln acuan 3x6 l 3x6 farm wmc unc pmduct onnc numerath overunc pmduct onnc denamlnatms Facial ounlnc ch mule denamlnatax Tncn dlvlde ounlnc common blnamlal facial Slmphfy sznte unc seconu expreslen rnrmcuon farm Muluply me rst rouonol expreslen by me xeclpmcal onnc seconurouonol expreslen wmc unc pmduct onnc numerath overunc pmduct onnc denamlnat Facial ounlnc ch mule numeratm Tncn dlvlde ounlnc common blnamlal facial Simphfv Additinnzl Example 2 Perform the followlng operaoons All results should be ln slmpllfled form 2 a x In x 73x728 6x2 18 b rhzrelo 306 Umverslty aHaustan Department anaLhemancs SECTION 5 2 Multiplying and wadmg Razmmz 5111111111111 Snlutinn a x smo 3128 5X2 6X2 X2 7amp728 p111111111r1111 1111111111111 1 1 1 z 5 F1 1111111111u111111111111 111111111111 7111 1111111 1111111 11111111111111111 1111111 111111 1111111111111111111111 3111th 7 51mm Mu uply 1111 5111 11111111 1111111111 by 1111 1111111111 11111 1111111 11111111 1111111111 1 x 4 1111111111 2XX23xiw p111111111r1111 1111111111111 3 3 2 F1c111bamnumu11ax1nd 11211111111111 2115M 1111 111111 51mm Adam11111311111111113 11f 111 1111111111 1111111111 All 1151111 11111111111111 sxmplx edform a 1 717311 1 7176 1241714110 1271314142 612123x17 312 21 1 b 4124141735 21271710 MATH 1300 Fundamenmls anathemancs CHAPTER 5 RananalExpressmns Equatums and Functmns Snllltinn 2 a 04 x 7x730 s x27x10 x2713x42 X x27x730x27x76 x27x10x2713x42 WWMUJ MWV7 m 3x27 6x223x72x27x710 4x2 4x7353x2 72x71 wwww www x x71 Wm the mam mm numeratms avexthe mam mm denaxmnatms mm a mnamuls m numeratm39 and denammatax Then amt am the camman bmamul factms 5mm Mu uply the my annual expressmn bythe xecxpmcal mm secand annual expressmn Wm the mam mm numeratms avexthe mam mm denaxmnatms mm a mnamuls m numeratm39 and denammatax Then amt am the camman bmamul factms 5mm Umversxty aHaustan Department anaLhemancs Exercise Set 52 Multiplying and Dividing Rational Expressions Multiply the following rational expressions and 3 simplify No answers should contain negative 1639 x 7 239 2 7 x exponents 1 9E 17 ma35i2 39 718 x 2 g 18 aimLEI 39 9 32 x 3 2 19 x74 7 3 4039 5x720 3 20 4x28 2 4 12 er7 all 7M3 21 away 5 csds asbg 3x12 6 x5y6 wz3 22 3x73ii MSZS x10y9 23 6x7124x12 7 7m8n2 I 721416 I 7 p612 x3 3x76 p315 5 m3n7 24 x7 6x724 x3y4 asbz x2y3a4 2x78 5x35 8 all2 I 7x7 7 I 7 b5 y y 6x710 3 25 3 5x 1579x 9 72x2 5 x 2x 4x76 26 7 5 6369 x2x 10 6x4 7 x 27 x2x76 I x276x5 H5 x3 39 x23x74 x272x715 11 73 710 x x 28 x27x72 x27x712 H6 5 39 x28x15 x279xl4 12 x71 x6 2 2 29 x 3x7102x 74x x75 39 6x2724x x5 13 x72 x72 2 2 30 6x 730xx 4x72l 14 x1x73 39 x27x76 40x78x2 xl 2 31 x4I x2 79 15 77x75x 37x x 16 x77 MATH 1300 Fundamentals of M athematz39 cs Exercise Set 52 Multiplying and Dividing Rational Expressions Divide the following rational expressions and simplify x2725x212x36 x6 x5 2x2 9x10x2 7x12 x25x6 2x23x75 x2 2x78 3x2714x75 3x216x5 x27x720 axibxayiby Iwc7x2al4 wc7x73a72 akibx2a72b a072adibc2bdi 027d2 acadibcibd 3aci3adbcibd No answers should contain negative exponents 37 310 5L15 8395 x 71 x71 x6 393x18 x 5x x24 x2 x274 27x x5 39257x2 x29i x3 x271 I x272xl 4x279 i2x73 x2710x25 x75 x273x710i x27x76 x273x728 39 x2x712 x24x4 ix278x720 x276x716 x279x8 6x2xil i3x22xil 6x25xl 39 3x24xl 10x2717x6L6x25x74 5x24x712 3x272x78 amianbmibni amian73bm3bn amanibmibn I aman73bmi3bn 0x72dxcy72dyi cxcy5dx5dy x2x73xyi3y cx5d E5d University of H ouston Department of M athematz39cs SECTION 53 Adding and Subtracting Rational Expressions Section 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 additiazln and subtraction of fractions with like denominators To add or subtract two fractions whose denominators are the same add or subtract the numerators and keep the common denominator If ad and C are real numbers and c at 0 then a E 3 c c c and aba b c c c Tale apply the rule for addition and subtraction of fractions with like denominators to find the sum and difference of rational expresmons With like denominators Example Perform the following operations All results should be in simpli ed form 2 quot o x 4 Jr 4 fa rib 7 4 x L 1 MATH 1300 Fundamentals ofMathematics 311 CHAPTER 5 RananalExpressmns Equations and Functions Snllltinn x 0 Wm lhe sum onhe humemms aver lhe common denammatax meme seem moo sums h 7 he swam when some nu he 2 he somehohomuhsm i Simplify wme lhe dn emce onhe humemms aver lhe common denammatax Factaxthe difference onwo squares m 7 lhe denammatax mu than uwlue out 3 he somehohomuhslo 5mm Addition and Subtraction of Rational Expressions with Unlike Denominators To add or subuael two haeuohs whose denominators are notth rewnte each fraction so that they have a eommo e same we must h denominator The mallest suah denominator is callethe least common denominator LCD The method offmdmg the LCM of the denominators will produce the L See Seeuoh l 3 for amethod of ndmgthe LCM least eommoh muluple Example Fmd the followme sum The resull should be m slmphhed form 2 5 University afHaustan Department anaLhemancs SECTION 53 Adding and Subtracting Rational Expressions Solution 39J 39 I must tewnte the ldtlQll39cil explessmns S they hath L1 Li39LI IILIIUJIl deli1nlnt1tol l 2 in 2 Find the least cormnon denonm lator t the denominators X y and ny 3x2y3xxvy and 9xy233xyy The least comm on denominator is 3 x y a 3 y 9x2yz v 2 2 Express each rational express1on as an equivalent one With a denominator of 9x y 2 For the rst rational expresston 3 2 we need to multiply the denominator X y 3x2 by 3y since 3xzy3y 9x2y2 We also needto multiply the numerator 2 by By 5 For the second ranonal expresswn 9 2 we need to multiply the denominator 0 9012 by it since 91312x 9x2y2 We also need to multiply the numerator 5 by x 2 5 23 5K 2 2 2 2 3x y 92y 3x y3y 9x32 x 6 5X Peforrn the multiplications in the 2 2 2 2 numerators and the denominators 9 y 9x y 6y 57 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 2x2 x l5 MATH 1300 Fundamentals ofMathematics 313 CHAPTER 5 RananalExpressmns Equatums and Functmns Snlutinn a x 2 2x2 Wm 11e sum 111 mums we x23x2 x 3x2 x 3x2 391 Dmmd mmm 1 2 Pasta bath the numeratax anddenaxmnatax 51mpth b 51 7 15 51715 wmeme difference afthenumentms 72 X715 72 X715 72 X715 Mme cammandenaxmnatm 1 51537 Factmbaththenumentm 7W W anddenaxmnatax 1 7 Sxmphfy 215 AdditjnnzlExample Pe 01m e addmon vae the result 111 slmpll edform L l 181 6y Snlntin We mustrewnte the rational expeessmns so they have a common denommator Fmd the least eommon denommator ofthe denominators 18x and By and Theleast common denommator 1s 2 3 3 x yaw Express each mum nxpmsslon as an eqmvalem one mm a denominator oleXy Umversxty afHaustan Department anaLhemancs SECTION 5 3 Addmg and Subvamng Rat sz Expressions Forthe firstrauonal expression we needle multiply me denominator x 18x byy smce mm 18W We also needle mulllplylne numerator 1 by y 7 Forthe secondranonal expression 67 we needle multiply me denominator y 6y by 3x smce 6y 3x18xy We also needle multiply me numerator 7 by 3x 7 1 7 Er 7 77 18x 6y 18x y 6y 3x 21x perennunennunneeuensnune lm 18W nnnenlen anddenammatms 7 y21x Wnlelne sum enne nnnenlen 7 71 8V avexthe camman denammalm39 AddidnnzlExample Perform lne subtraction Give lne resultm sxmph edform xy7x7 5x 15y Snllltinn e mustrewnte lne nanenal expnemens so lney have a eennnnen denominator Fmd the least common denommator ofthe denommators 5x and 15y Theleast eemmen denominator 5 x 3 yam Express each rational expression as an eqmyalenl ene wmn a denominator ofley x Fenlhe firstrauonal expression y 5 we needle multiply me denominator x 5x by 3y smce 5x3y 15xy We alsoneedtomuluply lne numerator xy by 3y MATH 1300 Fundamenmls anathemancs CHAPTER 5 RanamzlExpressmns Equatums and Fuhczmhs x i y it y 15y byx smee 15yx15xy We also needto mulhply the numerator heybyx Pefarm the mmuphehuohs m the numerath ma denammatms e E y3 239239 In the numeratal me the rst lynamulta the heghtwe af and poxyhomm go the see 2 7 2 mm mmth me M memes memsmnm 15W tems tagethex Amer 3y2 7 m the numeratat cambxne 15 he tems Additinnzl Example Perform the subtraetxon vae the result m sxmph ed form x X13x2 x x Snlutinn Ne must rewnte the rahohal expressions so they have a common denommator Fmd the least eommoh denommator of the denommators x7 3x 2 and x2 x x1x2 and x2x x23x x1 Theleastcommon denommatoms x1 may xlx2 Express eabh mom expression as an equtvaleht ohe wuh a denommator of 316 Umversxty afHaustan Department anaLhemancs SECTION 5 3 Addmg and Subvamng Ramsz Expressions x 7 x x23x2 x1x239 the denominator xlx2 byx We alsoneedto nnu1ep1y the numerator xbyx F m m n n n 1 expressxon 1 7 1 2xx1 denommator xxl by x2 We also needle nnu1ep1y the numerator 1 by x2 Fothe secondranonal expression we needle nnu1ep1y me x x23x2 xhx x1 Perfarm une muhtphcauans In numeralms and denammatms Wm une difference athe numeralms evenune camman denammatm 1n une numexawl nee me rst palynamul in me negeuve athe secand ene Remeve une nnenmeses Factax une numeratm39 and men emee em une camman bmamul factm 5mm Additinnzl Example 5 Perform the followmg epenaeens Gwe all results 1n sxmplexed form 3 Key yen a MATH 1300 Fundamenmls aMazhemanes CHAPTER 5 RananalExpressmns Equatums and Functmns Snlutinn a The denommators are negatives ofeach other We can muluplythe numerator d denommator ofthe secondrauonal expresmn by 1to obtam a common denommator y 3 2 Wntethe sum armenumemms ivy avexthe camman denammatm39 1 Snapth X y 4 1 47 b x3 1 x3 7 4x3 1x3 x3 7 4x12 7 1 x3 x3 7 124 wmeme difference afthenumentms X 3 aver me camman denammatm39 4 11 7 5mm 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 31 5 7 4a 2b 5 3 4a 9b 114127 115124 x8x7 x5 x5 3x2 7 2x6 5x720 5x720 2x3 10x79 MATH 1300 Fundamentals of M athematz39 cs x727x2 x 4 x3 x75 x71 1 39 xl 2xl 2x73 6 x xl x73 xl x2 x74 319 Exercise Set 53 Adding and Subtracting Rational Expressions 3 8 7777 x x71 x72 x71 x1 x73 x57x72 x4 x73 77 8x12 6x76 5 2 39 12x76710x40 6 3x2 5x 772 x273x74 x 7 2 5 x22x78 x272x x24x x23x718 x26x x273x x 4 2 2 2 2 x 10x24 x 12x32 x 14x48 x 2 3 2 7 2 2 x 77x12 x 74x3 x 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 2xx 3x A i and i 2 4yi 6y E 5 10 Simplifying We simplify a complex fraction by eliminating the fractions that appear in the numerator andf 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 and Functions Example Simplify the fol owi 11g complex f39r anti 2311 2x x T U Solution Method 1 rI39Jbtem a singl e fraction in the 1mm er ator Note that the denm ninatcur 15 already a single fraeti 011 To obtain a sihgl e fraction in the 1111111 erator add the fi39fthiQ S 1 V fandj t The least common denmmnater 13 in I 3 quota 2224 322 University of H ouston Department of M athematics SECTION 5 4 Complex Fractions Method 2 numerator and denominator These denominators are 3 6 and 12 The least eommon denominator is 12 MATH 1300 Fundamenmls aMazhemanes CHAPTER 5 RananalExpressmns Equatums and Functmns Addidnnzl Example 1 51mp11 y 1112 011 owmg complex mm on 25 5 6x2y 25 Snlntinn l 5 6x2y 25 25 5 6x2y 7 219 25 5 6x2y 1 1 1 1 Z X y y 5 5 5 Z 3 X x 1 1 1 5y2 115111131 Exampl Add 22 1mp11ry1111 following complex fraction 5 X Y 2 4 2xy 8 Snlutinn 11 L2 z 2 4 22 4 2xy 2xy 8 8 21 4 4 2xy 8 Umversxty aHaustan Department anaLhemancs SECTION 5 4 Complex Fractions 775W 2 Additinnal Example 3 Simplify the 011 owmg complex frame 1 73pm MATH 1300 Fundamenmls anathemancs 325 CHAPTER 5 RananalExpressmns Equatums and Functmns Additinnzl Example 4 Stmphfy the followmg complex fraction 326 Umversxty aHaustan Department anaLhemancs SECTION 5 4 Complex Fractions AdditjnnzlExample 5 Re ne the given expressxon so that 1f contams posmve exponents rather than neganve exponents and men snnphfy x 2 x 3 Snlntinn 1 xquot 2 X H3 13 x 1 zjx x lgjx x 72 x 53x X X n w MATH 1300 Fundamenmls anathemancs 327 328 Exercise Set 54 Complex Fractions Simplify the following No answers should contain 5 1 negative exponents 10 672 39 1 l 5 3 12 1 7 E 5 8 E 11 13 9 2 3 L 239 2 l 3 17 12 17 3 y 3 4 x 73 27 5y 13 Lz 12115 2 b 4 402 3 b7 14 47 g 211122 3 5 03d x7 2 3 8a 0 15 5 SM E 10 3x4 8 x7 6 MW 16 81 427 x7 12 3 7 L H 8 17 b y b7 a 11123 5b3c 2d xzy lObd 18 xy 30 33 9 2 x2 7x12 a 19 8x x2 7x7 20 6x4 University of H ouston Department of M athematz39cs Exercise Set 54 Complex Fractions 9x5 6x 2 2 7 x 76x716 x x 2 2039 18x3 30 4 x2711x24 x2 x71 1 3 x t x 7574 x x7 21 3 2 31 3 2 x x 77 E E x3 xl 2 1 71 2 x x7 22 4 5 32 3 2 11 7 4 x72 x x2 71 x2 33 x 23 1 1 x77i 75 if x9E 3x 3 34 24 257x2 x677 x 2 25 a b For each of the following expressions 39 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 3 4 71 i 7 x x 35 y x 1 1 6 x 7 37x 1 x71 36 27 9 x71 x 71 71 x y xii 37 x71 iyil 28 x 10 38 0 1 d 1 x 7 c 2 d 2 2 3 2 2 77 x 29 x75 x5 39 71 yil 2 x y x2 7 25 MATH 1300 Fundamentals of M athematz39 cs 329 Exercise Set 54 Complex Fractions 330 1 1 71f1 If2 7 at 2 xiziy 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 called rational equations Here are three examples of rational equations 3 l 5 x l x 2 4x x 6 2x x3 4 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 393 l 5 Solve and check 2 R 2 Solution We first note that it cannot be equal to CI since this would give a U in the denominator ofboth 3 and Thus ifx 0 then we can multiply both sides ofthe given it x equation by the 6x LCD to clear the equation of fractions MATH 1300 Fundamentals ofMathematics 331 CHAPTER 5 RananalExpressmns Equatums and Functmns 18x71815718 x Check Subsmute 7 forx m the ongmal equauou The soluuon x5 x Example Shhreandchenk L Snllltinn Rewme the equauou by faemnug the binomial 2x6 Thus xfx 7 men we can muluply both sudes of the ngen equauou by 4x3 LCD to clearthe equauou offracuons 332 Umversxty afHaustan Department anaLhemancs SECTION 5 5 Salvmg Ranamzl Equanans x x3 2amp2 4W3 4x3 4x3 MATH 1300 Fundamenmls anathemancs 333 CHAPTER 5 RananalExpressmns Equatums and Functmns Extraneous Solutions and obtamed anesulting equation whose solutton satis ed the orgmal nattonal equation Note that tn both eases the LCD eontatnedthe vanable i However tn some eases 4 but we quatt 0 Lnthtsease solution not sattsfy the ongmal equation and thus eannotbe a solution to the equation Example Solve and eheek Snlutinn Rewnte the equation by factoring the btnomial X2 71 4x x1 We rst note thatx eannot be equa1to4on15tneethts would gtve a 0 tn a denominator Thus if x z 1 then we Ban multiply both sides of the glVEn equation by m x41 LCD to elearthe equation offnaotions 7 An 3271 7 4i i71i1 x1xrl gt4 gt4 toto in iet A pawl X1i 47 W in MM MW 2x714x 2x724x N gt4 334 Umversxty nHnueznn Department DMaLhEmaDL S SECTION 5 5 Sulvmg Ranamzl Equanuns 2x7274x4x74x 72x720 72x72202 72x2 Check Note that 71 does not sausfy me ongmal equauon smce subsmuung 71 71 15 an extraneous soluuon Wemust for xwxll resultm a 0 m the denominator ard 71 as a soluuon ofthe equauon smcex cannoth equal to e The guzen equauon has no soluuon Additinnzl Example 1 Solve andcheck 371 x 3 3x Snllltinn ofboth 3 and 31 Thus xfx 0 men we can nnuluply both sudes ofthe guzen x x equauon by the 3x LCD to clearthe equauon offracuons MATH 1300 Fundamenmls anathemancs CHAPTER 5 Rational Expressions Equations andFunctions Check Substitute 4 for X in the original equation 52 E3 52g ZE E w337 T39EE 77 5 5 The solution is X 4 Additional Example 2 iolve and check L a L x5 3 2xlO Solution Rewrite the equation by factoring the binomial 2xlU 2 25 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 XE39 m x 2 x x5 2xm 39 L x V 2 v L x 0Jm 6l39 0LJ 2x5 2 6x 4x5 3x 6x 4x 20 3x 2x 20 3x 2x 2U 3x 3x3x x 20 0 336 University of H ouston Department of M athematics SECTION 5 5 Sulvmg Ranamzl Equanuns exe2o20 020 0 x720 Check Substitute e 20 for x m the ongmal equauou The soluuon x5 x 720 Additinnzl Example 3 olve and check 1 37 x 2x1 Snlntinn We rst note thatx cannot be equal to 0 01quot 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 6 x 2x1 x2x11x2x137x2x1 6 x 2x1 x2x112x1 Eli 21P x2x132x176x 2x2x6x376x 2x2x3 MATH 1300 Fundamenmls anathemancs CHAPTER 5 RananalExpressmns Equatums and Functmns 2x2x73373 The soluuons are x g andx 338 Umversxty aHaustan Department anaLhemancs SECTION 5 5 Salvmg Ranamzl Equanans Additinnzl Example 4 Solve and check x73 Snllltinn Thus he 3 men we can multiply both sides ofthe gwen equamn by 3 LCD to clear he equauon of fracuons 0 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 3 ducard 3 as a solution of me equauon smee x cannot be equ to 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 extraneous solutions 17 x 5 1 752 18 3x 5 3 x7 2 l 19 7 1 x279 3 3267252722 20 5 0 x274 4 1354 21 HS 8 4 x277xl2 5 5i3i2 22 117x 1 6 10 39 x23x710 7x 3x 6 iii5 5 9 23 772 8 20 7 13 4x7 x73 7 7 3 12 7 7 24 773 x 5 x 5 x71 5 8 3x4x8 25 Zix8771 54 x 2 399 xl 9 x52x6 26 a27l771 xl xl 9 4 10 3x45x77 27 x77 73 x6 x6 x7 3 11 52 4l73 28 27t2772 x x 3 173 7 5 12 if 7 7 6x 4x 29 L1 3E w1 4 12 3 13 774 x72 30 x4l9 x9 2 14 75 14 x7 31 1x 7 3 x4 x4 2 15 70 2 x75 32 x 7 x 3 x72 7 x72 16 5x x2 340 University of H ouston Department of M athematz39cs Exercise Set 55 Solving Rational Equations 33 4 fl 8 49 ii1 x75 3 3x715 x4 x1 34 7 4 50 5 2 x2 3x6 3 x74 x2 3 2 1 35 7777 51 77 ii 41178 31176 3 x5 x8 5 1 7 36 7777 52 is 776 1 3c15 2010 12 x7 x9 1 7 37 i72 53 47 x5 x73 x2x715 x5 x10 1 4 3s 77727 54 x727 7 x71 x2 x x72 x7 x3 39 iiziig 55 1 Jriii x x73 x1 x 72x73 2x75 3x 2x75 7 2 10 40 72 56 2 17 6x x74 x75 x79x20 39 3x1 x 3x1 3 4 8 41 777 57 74 i3ix x2 x72 x74 3x2 x71 x71 6 24 42 iii 2 58 L73 x x74 x4 x 716 2x73 x 2x73 1 6 43 1777 x x2 44 gl1 x 45 27i2 x 46 iz 3 x x 47 6 1 x4 x Z 4 7 x x5 MATH 1300 Fundamentals ofMathematics 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 formfx XI where P and Q are polynomials Here are three examples of rational functions x2 3xl 4X fCXX24 fX x3 m Domain of a Rational Function The domain of arational fx S consists of all real numbers it except those x values of 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 of the following rational functions and express in interval notati on 3 2 Aquot 4 3 r l ibi in x 395 03339 fix q r e 5 4 342 University of H ouston Department of M athematics SECTION 5 6 RanamlFunmans s lutinn a Solve the equamh x2 74 o x2 40 x2x720 or x720 x72202 x2 The domam ofthe gwen functxon s the set of all real numbers except 72 and 2 The domam m mterval notahon ls 7w 2u72 2M 01 b Solve the equauohxeseo The domam ofthe gwen function xs the set of all real numbers except 3 The domhh m mterval notatxon ts 003 gush e Solve the equamh x2 7 5x 4 0 5x40 xrlx740 or x AU x7444 04 x4 The domam ofthe gweh ruhemh 5 the set of all real numbers except 1 m4 The domam m mtervsl notation x5 em u14u4w MATH 1300 Fundamenmls anathemancs CHAPTER 5 RananalExpressmns Equatums and Functmns Graph of a Rational Function ample The graph of the ratrona1 funetron fx quot3 x72 rs shown below 393 x x72 a State the domam otthe tunetron m rntervat notataon b hnd the xrmtercepts ot the graph andtabet the pornts on the graph where the graph erosses the xraxxs E erosses theyraxxs a Label the pornt on the graph whose rst coordmate rs 1 Snlutinn a 5o1ve the equatronr zeo x720 xe2202 x2 The domam ofthe grven funetron rs the set of all rea1 numbers exeept 2 The domam m rntervat notatron rs rw2u2w There rs no pornt on the graph whose rst coordmate rs 2 344 Umversxty afHaustan Department anaLhemancs SECTION 5 6 RanamlFunmans b To ndthe xehceheepts fmdthe hea1 solutions ofthe equauOhx 0 The X1ntcrccptxs 3 e To nd theyemceheepc ma0 They lntErEE Ms 3 p 9 a The pomtWhose rstcoordAnatexs115 um Fmd m for f The 3 X gt4 1 2 pm MATH 1300 Fundamenmls anathemancs CHAPTER 5 RananalExpressmns Equatums and Functums A 12 032 39 72 U 3 o 4 e 74 x 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 h ofa elose to a from the ng tofu or from the le The hne x73 X bound as xgets elose to 2 from the nght of2 and funeuonal values mcrease 2 rs averuea1 asymptote for the graph of the ratrona1 funeuon x From the graph we see that funetrona1 values deerease wrthout wrthout bound as x gets elose to 2 from the left of 2 e t 2 2 D A E A 5 x73 mo 2 x 2 vemcalasvmptote 346 Umversxty afHaustan Department aMaLhemancs SECTION 5 6 Ramona Functions Finding Vertical Asymptotes P Letx be mama funcuon To fmd vemcal asymptotes rst simplify PO QR and denominator Then the vemcal asymptotes are 0mg om x a where a s a m1 number for whxch the denominator 0mg sxmphfxed expression 5 equal to 0 Example 0 x 4 b m 7 m n a Rewme the um on by factoring the denominator and men was out any comm on factors x2 m 1 x 7 onga x274 Whiz x 2 w A Thus xsx2 The graph 5 shown below x 2 vemcal asymptote MATH 1300 Fundamenmls anathemancs CHAPTER 5 RananalExpressmns Equatums and Functmns b Rewnte the meta on by faetorrhg the dehommator and ther dunde out any 0 h factors 4 x 4x X7x esr4 xemxeA Values for whreh the denommatorrs equal to 0 are x1andx 4 Thus the vemcal asymptotes are x 1 and x The graph rs shown below x 1 x 4 vemcal asymptotes Horizontal Asymptotes y fx xfy gets close to b as x rhereases wrthout bound or deereases wrthout ouhd The honzomal hm y 3 rs ahonzontal asymptote forthe graph ofthe rhtrohal r 7 3 From the graph x73 to 3 asx rhereases wrthoutbouhd and deereases wrthout bound 348 Umversxty afHaustan Department anaLhemancs

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