Adding and Subtracting Rational Expressions


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1 6.4 Adding nd Subtrcting Rtionl Epressions Essentil Question How cn you determine the domin of the sum or difference of two rtionl epressions? You cn dd nd subtrct rtionl epressions in much the sme wy tht you dd nd subtrct frctions Sum of rtionl epressions + Difference of rtionl epressions Adding nd Subtrcting Rtionl Epressions Work with prtner. Find the sum or difference of the two rtionl epressions. Then mtch the sum or difference with its domin. Eplin your resoning. Sum or Difference Domin. b. c. d. e. + A. ll rel numbers ecept + B. ll rel numbers ecept nd + C. ll rel numbers ecept + D. ll rel numbers ecept E. ll rel numbers ecept nd + + f. + F. ll rel numbers ecept 0 nd CONSTRUCTING VIABLE ARGUMENTS To be proficient in mth, you need to justify your conclusions nd communicte them to others. g. + G. ll rel numbers ecept h. + + H. ll rel numbers ecept 0 nd Writing Sum or Difference Work with prtner. Write sum or difference of rtionl epressions tht hs the given domin. Justify your nswer.. ll rel numbers ecept b. ll rel numbers ecept nd c. ll rel numbers ecept, 0, nd Communicte Your Answer. How cn you determine the domin of the sum or difference of two rtionl epressions? 4. Your friend found sum s follows. Describe nd correct the error(s) Section 6.4 Adding nd Subtrcting Rtionl Epressions
2 6.4 Lesson Wht You Will Lern Core Vocbulry comple frction, p. Previous rtionl numbers reciprocl Add or subtrct rtionl epressions. Rewrite rtionl functions. Simplify comple frctions. Adding or Subtrcting Rtionl Epressions As with numericl frctions, the procedure used to dd (or subtrct) two rtionl epressions depends upon whether the epressions hve like or unlike denomintors. To dd (or subtrct) rtionl epressions with like denomintors, simply dd (or subtrct) their numertors. Then plce the result over the common denomintor. Core Concept Adding or Subtrcting with Like Denomintors Let, b, nd c be epressions with c 0. Addition Subtrction c + b c + b c c b c b c Adding or Subtrcting with Like Denomintors b Add numertors nd simplify. Subtrct numertors. Monitoring Progress Find the sum or difference Help in English nd Spnish t BigIdesMth.com To dd (or subtrct) two rtionl epressions with unlike denomintors, find common denomintor. Rewrite ech rtionl epression using the common denomintor. Then dd (or subtrct). Core Concept Adding or Subtrcting with Unlike Denomintors Let, b, c, nd d be epressions with c 0 nd d 0. Addition Subtrction c + b d d + bc d + bc c b d d bc d bc Chpter 6 Rtionl Functions You cn lwys find common denomintor of two rtionl epressions by multiplying the denomintors, s shown bove. However, when you use the lest common denomintor (LCD), which is the lest common multiple (LCM) of the denomintors, simplifying your nswer my tke fewer steps.
3 To find the LCM of two (or more) epressions, fctor the epressions completely. The LCM is the product of the highest power of ech fctor tht ppers in ny of the epressions. Finding Lest Common Multiple (LCM) Find the lest common multiple of 4 6 nd Step Fctor ech polynomil. Write numericl fctors s products of primes ( 4) ( )( + )( ) ( 4 + 4) ()()( ) Step The LCM is the product of the highest power of ech fctor tht ppers in either polynomil. LCM ( )()( + )( ) ( + )( ) Find the sum Adding with Unlike Denomintors Method Use the definition for dding rtionl epressions with unlike denomintors ( + ) + (9 ) 9 ( + ) ( + ) ( ) 9 ( + )() ( + ) c + b d d + bc Distributive Property Fctor. Divide out common fctors. Simplify. Method Find the LCD nd then dd. To find the LCD, fctor ech denomintor nd write ech fctor to the highest power tht ppers in either denomintor. Note tht 9 nd + ( + ), so the LCD is 9 ( + ) ( + ) ( + ) ( + ) + 9 ( + ) ( + ) Fctor second denomintor. LCD is 9 ( + ). Multiply. Add numertors. Note in Emples nd tht when dding or subtrcting rtionl epressions, the result is rtionl epression. In generl, similr to rtionl numbers, rtionl epressions re closed under ddition nd subtrction. Section 6.4 Adding nd Subtrcting Rtionl Epressions
4 COMMON ERROR When subtrcting rtionl epressions, remember to distribute the negtive sign to ll the terms in the quntity tht is being subtrcted. Find the difference Subtrcting with Unlike Denomintors ( ) ( )( ) + ( ) ( )( ) 6 ( )( ) 4 ( )( ) 6 ( 4 ) ( )( ) + 4 ( )( ) ( )( + 4) ( )( ) Fctor ech denomintor. LCD is ( )( ). Multiply. + 4, Simplify. ( ) Subtrct numertors. Simplify numertor. Fctor numertor. Divide out common fctor. Monitoring Progress Help in English nd Spnish t BigIdesMth.com. Find the lest common multiple of nd 0. Find the sum or difference Rewriting Rtionl Functions Rewriting rtionl function my revel properties of the function nd its grph. In Emple 4 of Section 6., you used long division to rewrite rtionl function. In the net emple, you will use inspection. Rewriting nd Grphing Rtionl Function Rewrite g() + in the form g() + k. Grph the function. Describe the + h grph of g s trnsformtion of the grph of f(). 4 g 4 y Rewrite by inspection: ( + ) + ( + ) The rewritten function is g() +. The grph of g is trnsltion unit + left nd units up of the grph of f(). Monitoring Progress Help in English nd Spnish t BigIdesMth.com 9. Rewrite g() 4 in the form g() + k. Grph the function. h Describe the grph of g s trnsformtion of the grph of f(). 4 Chpter 6 Rtionl Functions
5 Comple Frctions A comple frction is frction tht contins frction in its numertor or denomintor. A comple frction cn be simplified using either of the methods below. Core Concept Simplifying Comple Frctions Method If necessry, simplify the numertor nd denomintor by writing ech s single frction. Then divide by multiplying the numertor by the reciprocl of the denomintor. Method Multiply the numertor nd the denomintor by the LCD of every frction in the numertor nd denomintor. Then simplify. + 4 Simplify Method Simplifying Comple Frction ( + 4) ( + 4) ( + 4) ( + 4)( + 8) + 8, 4, 0 Add frctions in denomintor. Multiply by reciprocl. Divide out common fctors. Simplify. Method The LCD of ll the frctions in the numertor nd denomintor is ( + 4) ( + 4) ( + 4) + 4 ( + 4) + 4 ( + 4) + Divide ( + 4) Monitoring Progress + ( + 4) + 8, 4, 0 Multiply numertor nd denomintor by the LCD. out common fctors. Simplify. Simplify. Help in English nd Spnish t BigIdesMth.com Simplify the comple frction Section 6.4 Adding nd Subtrcting Rtionl Epressions
6 6.4 Eercises Dynmic Solutions vilble t BigIdesMth.com Vocbulry nd Core Concept Check. COMPLETE THE SENTENCE A frction tht contins frction in its numertor or denomintor is clled (n).. WRITING Eplin how dding nd subtrcting rtionl epressions is similr to dding nd subtrcting numericl frctions. Monitoring Progress nd Modeling with Mthemtics In Eercises 8, find the sum or difference. (See Emple.) In Eercises 9 6, find the lest common multiple of the epressions. (See Emple.) 9., ( ) 0., 4 +., ( ). 4, 8 6., , , , + 7 ERROR ANALYSIS In Eercises 7 nd 8, describe nd correct the error in finding the sum ( + ) ( + )( ) In Eercises 9 6, find the sum or difference. (See Emples nd 4.) REASONING In Eercises 7 nd 8, tell whether the sttement is lwys, sometimes, or never true. Eplin. 7. The LCD of two rtionl epressions is the product of the denomintors. 8. The LCD of two rtionl epressions will hve degree greter thn or equl to tht of the denomintor with the higher degree. 9. ANALYZING EQUATIONS How would you begin to rewrite the function g() 4 + to obtin the form + g() h + k? 4( + ) 7 A g() + 4( + ) + B g() + ( + ) + ( ) C g() + D g() ANALYZING EQUATIONS How would you begin to rewrite the function g() to obtin the form g() h + k? ( + )( ) A g() B g() + C g() + D g() 6 Chpter 6 Rtionl Functions
7 In Eercises 8, rewrite the function in the form g() + k. Grph the function. Describe the h grph of g s trnsformtion of the grph of f(). (See Emple.) 7. g() g() + 6. g(). g() two resistors in prllel circuit with resistnces R nd R (in ohms) is given by the eqution shown. Simplify the comple frction. Then find the totl resistnce when R 000 ohms nd R 600 ohms. Rt + R R 8 +. g() R 4 6. g() + Rt g() 8. g() R In Eercises 9 44, simplify the comple frction. (See Emple 6.) REWRITING A FORMULA The totl resistnce Rt of PROBLEM SOLVING You pln trip tht involves 40mile bus ride nd trin ride. The entire trip is 40 miles. The time (in hours) the bus trvels is 40 y, where is the verge speed (in miles per hour) of the bus. The time (in hours) the trin trvels 00 is y. Write nd simplify model tht shows + 0 the totl time y of the trip. 48. PROBLEM SOLVING You prticipte in sprint trithlon tht involves swimming, bicycling, nd running. The tble shows the distnces (in miles) nd your verge speed for ech portion of the rce PROBLEM SOLVING The totl time T (in hours) needed to fly from New York to Los Angeles nd bck cn be modeled by the eqution below, where d is the distnce (in miles) ech wy, is the verge irplne speed (in miles per hour), nd j is the verge speed (in miles per hour) of the jet strem. Simplify the eqution. Then find the totl time it tkes to fly 468 miles when 0 miles per hour nd j miles per hour. d d T+ j +j Distnce (miles) Speed (miles per hour) Swimming 0. r Bicycling r Running 6 r+. Write model in simplified form for the totl time (in hours) it tkes to complete the rce. b. How long does it tke to complete the rce if you cn swim t n verge speed of miles per hour? Justify your nswer. 49. MAKING AN ARGUMENT Your friend clims tht NY LA NY LA A j j j +j Section 6.4 Int_Mth_PE_0604.indd 7 the lest common multiple of two numbers is lwys greter thn ech of the numbers. Is your friend correct? Justify your nswer. Adding nd Subtrcting Rtionl Epressions 7 /0/ 4: PM
8 0. HOW DO YOU SEE IT? Use the grph of the function f() h + k to determine the vlues of h nd k.. REWRITING A FORMULA You borrow P dollrs to buy cr nd gree to repy the lon over t yers t monthly interest rte of i (epressed s deciml). Your monthly pyment M is given by either formul below. Pi M t ( + i) Pi( + i) or M t ( + i) t. Show tht the formuls re equivlent by simplifying the first formul. b. Find your monthly pyment when you borrow $,00 t monthly interest rte of 0.% nd repy the lon over 4 yers.. THOUGHT PROVOKING Is it possible to write two rtionl functions whose sum is qudrtic function? Justify your nswer y f. PROBLEM SOLVING You re hired to wsh the new crs t cr delership with two other employees. You tke n verge of 40 minutes to wsh cr (R /40 cr per minute). The second employee wshes cr in minutes. The third employee wshes cr in + 0 minutes.. Write single epression R for the combined rte of crs wshed per minute by the group. b. Evlute your epression in prt () when the second employee wshes cr in minutes. How mny crs per hour does this represent? Eplin your resoning. 6. USING TOOLS The epression w + 40 models the w perimeter of the corrl in Section. Emple. Find the sum of the terms. Then use grph to justify the vlue of w found in the emple. How is the grph different from previous grphs of rtionl functions? 7. MODELING WITH MATHEMATICS The mount A (in milligrms) of spirin in person s bloodstrem cn be modeled by 9t A t t + where t is the time (in hours) fter one dose is tken. A first dose second dose A combined effect. USING TOOLS Use technology to rewrite the (97.6)(0.04) + (0.00) function g() in the. + form g() + k. Describe the grph of g s h trnsformtion of the grph of f(). 4. MATHEMATICAL CONNECTIONS Find n epression for the surfce re of the bo Mintining Mthemticl Proficiency Solve f() g() by grphing nd lgebric methods. (Section.). A second dose is tken hour fter the first dose. Write n eqution to model the mount of the second dose in the bloodstrem. b. Write model for the totl mount of spirin in the bloodstrem fter the second dose is tken. 8. FINDING A PATTERN Find the net two epressions in the pttern shown. Then simplify ll five epressions. Wht vlue do the epressions pproch? + +, +, +, Reviewing wht you lerned in previous grdes nd lessons 9. f() f() + 6. f() 4 + g() g() g() + 8 Chpter 6 Rtionl Functions