12.1 Introduction to Rational Expressions
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1 . Introduction to Rtionl Epressions A rtionl epression is rtio of polynomils; tht is, frction tht hs polynomil s numertor nd/or denomintor. Smple rtionl epressions: 0 EVALUATING RATIONAL EXPRESSIONS To evlute rtionl epression for given vlue of, simply substitute the vlue for nd simplify. Smple Problem: Evlute ) 6 ) Solution: ) 6 if ) Student Prctice: Evlute the following rtionl epressions if... DETERMINING WHEN A RATIONAL EXPRESSION IS UNDEFINED A rtionl epression is undefined when the vlue of it s denomintor is 0 becuse dividing by 0 is not llowed. To determine for wht vlue of rtionl epression is undefined, simply set the denomintor equl to 0 nd solve for. 7 Smple Problem: Determine wht vlue of mkes undefined. 0 Solution: We need to determine wht mkes the denomintor 0, so we strt by: 0 0 Check: 0
2 Student Prctice:Determine wht vlue of mkes ech rtionl epression undefined SIMPLIFYING RATIONAL EXPRESSIONS Simplifying rtionl epressions is much like simplifying frctions. First fctor ech polynomil, then cncel like fctors. Use cution when cnceling) Tip: You cn NOT cncel ny individul term tht is being ADDED OR SUBTRACTED. You cn ONLY cncel fctors, or terms tht re being MULTIPLIED! Smple Problem: Simplify ) ) ) ) Solution: 6 9) ) 9) ) 9 Student Prctice: Simplify ech rtionl epression. 6y y 6
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4 . MULTIPLYING RATIONAL EXPRESSIONS Smple Problem: Multiply. 9 Solution: 9 ) ) ) ) 7) ) ) ) ) ) 7) ) 7) ) Student Prctice: Multiply the following rtionl epressions b b 8 9 To multiply rtionl epressions,. First fctor ll polynomils tht cn be fctored.. Cncel like fctors.. Multiply the fctors tht remin cross.
5 DIVIDING RATIONAL EXPRESSIONS Smple Problem: Divide. Solution: ) ) ) ) ) ) Student Prctice: Divide the following rtionl epressions n m n m 7. 6 b b To divide rtionl epressions, multiply the first by the reciprocl of the second.
6 . nd. Adding/Subtrcting Rtionl Epressions Adding or subtrcting rtionl epressions is much like dding nd subtrcting regulr frctions. Unlike multipliction or division, it cn only be done if the two frctions hve common denomintor. To dd or subtrct rtionl epressions,. If the denomintors re the sme, go to step.. If the denomintors re different, fctor ech denomintor nd multiply the numertor nd denomintor of ech frction by the fctor tht it s missing from the other denomintor. rrrgghhh???). Add/Subtrct the numertors. Write this over the common denomintor.. Fctor numertor nd simplify. Student Prctice: Add or subtrct s indicted
7 Smple Problem: Add. 8 7 Solution: 8 7 ) 8) ) ) 8) ) ) ) 8) 8) ) ) 0 8) ) ) 6 7
8 Student Prctice: Add or subtrct s indicted Different denomintors). 7. y y
9 multiply by -) 8
10 . Solving Rtionl Equtions Solving rtionl equtions is much like solving liner equtions with frctions. We must first multiply both sides ech term) by the LCD to cncel out ll denomintors. To solve rtionl equtions,. Rewrite the eqution nd fctor ll denomintors.. Multiply ech side of the eqution by the LCD so s to get the fctors in the denomintor to cncel.. Solve the resulting eqution.. Ensure tht your nswer mkes no rtionl epression undefined in the eqution. Smple Problem: Solve: 9 Solution:. Fctor: ) ). The only two distinct fctors here re ) nd -), so the LCD is ) ). ) ) ) ). ) ). Note tht ½ does not mke either denomintor 0 in the originl problem. Student Prctice: Solve ech rtionl eqution
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12 9.
13 .6 Applictions Involving Rtionl Equtions nd Proportions There re vriety of pplictions tht require the use of rtionl equtions to find the solution. Smple Problem WORK: It tkes Omr hr to mow the lwn lone. It tkes Mndy hr to mow the lwn lone. How long would it tke them, working together, to mow the lwn? Solution: A common error is to think tht together it will tke them hrs, since the verge of hr nd hr is hr. Tht doesn t mke sense becuse if they work together, they should obviously finish fster thn hr. A very prcticl wy to solve this problem is to look t the mount of work ech person ccomplishes in hr nd put the informtion in tbulr form: Time to finish the job Omr Mndy Together Prt of job done in hr Now we consider tht if we dd the mount of work done by ech person in hr, it should totl the mount of work done together in hr. So, Now solve hr. hr 9 9 So together, it will tke them hr which is bout hrs nd minutes) to finish. 9
14 Smple Problem MOTION: A cr trvels 80 miles in the sme time tht truck trvels 0 miles. If the cr s speed is 0 miles per hour fster thn the truck s, find the cr s speed nd the truck s speed. Solution: Once gin, it my help to put the informtion in tbulr formt listing the distnce, rte, nd time for ech vehicle. We ll let represent the speed of the truck. We lso wnt to consider populr formul D R T Distnce Rte Time Cr 80 0 Truck 0 No specific time ws given; however, we cn solve for time in the formul D R T to D get T. R Distnce Rte Time Cr Truck 0 0 Since we know they trveled the sme mount of time, we must hve tht To solve, we multiply by LCD or get cross products) ) ) The truck ws trveling t 0 mph, nd the cr ws trveling t 60 mph.
15 Smple Problem PROPORTION: Use the fct tht corresponding lengths of sides re proportionl in similr figures to solve for the missing side of figure EFGH Solution: There re vriety of proportions tht cn be set up to solve this problem. The min thing is tht your consistent while setting up the proportion. AB EF BC FG Cross products must be equl when solving proportions. Side FG is units long. Student Prctice: Solve ech ppliction problem.. Bill nd Ted work for plyground instlltion compny. On verge, it tkes Bill 8 hours to instll plyground set for children. It tkes Ted 0 hours to instll the sme set rookie worker). Working together, how long would it tke them to complete the job?. A drin cn empty swimming pool in dys. A second drin cn empty the sme pool in dys. How long will it tke to empty the pool if both drins re used?
16 . A pipe cn fill n empty pool in hours. A dringe pipe cn empty full pool in bout 0 hours. How long would it tke to fill this empty pool if the dringe ws left uncovered?. A bot cn trvel 9 miles upstrem in the sme mount of time it tkes to trvel miles downstrem. If the current of the river is miles per hour, find the speed of the bot in still wter.. A cyclist rides 6 miles per hour on level ground on still dy. He finds tht he rides 8 miles with the wind behind him in the sme mount of time tht he rides 6 miles into the wind. Find the rte of the wind.
17 6. If Δ ABC ~ Δ DEF, solve for the missing side. 7. A 6 ft mn csts shdow ft long t the sme time tht tree ft tll csts shdow. How long is the shdow of the tree? 8. To estimte the popultion of lrgemouth bss t Doolittle prk, scientists ctch nd tg 8 lrgemouth bss, then relese them bck into the pond. Lter, scientists return nd ctch 60 lrgemouth bss nd find of them to be tgged. Approimtely how mny lrgemouth bss re in the lke?
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