Add and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions.

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1 TEKS 8. A..A, A.0.F Add nd Subtrct Rtionl Epressions Before Now You multiplied nd divided rtionl epressions. You will dd nd subtrct rtionl epressions. Why? So you cn determine monthly cr lon pyments, s in E. 4. Key Vocbulry comple frction As with numericl frctions, the procedure used to dd (or subtrct) two rtionl epressions depends upon whether the epressions hve like or unlike denomintors. KEY CONCEPT For Your Notebook Adding or Subtrcting with Like Denomintors To dd (or subtrct) rtionl epressions with like denomintors, simply dd (or subtrct) their numertors. Then plce the result over the common denomintor. Let, b, nd c be epressions with c Þ 0. Properties Addition Subtrction c b c b c c b c b c Emples 9 9 E XAMPLE Add or subtrct with like denomintors Perform the indicted opertion b b. 0 Add numertors nd simplify result Subtrct numertors GUIDED PRACTICE for Emple Perform the indicted opertion nd simplify Chpter 8 Rtionl Functions

2 KEY CONCEPT For Your Notebook Adding or Subtrcting with Unlike Denomintors 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). Let, b, c, nd d be epressions with c Þ 0 nd d Þ 0. Addition Subtrction b d bc d bc c b d bc d bc c d cd cd cd d cd cd cd You cn lwys find common denomintor of two rtionl epressions by multiplying their denomintors, s shown bove. However, if you use the lest common denomintor (LCD), which is the lest common multiple (LCM) of the denomintors, you my hve less simplifying to do. E XAMPLE Find 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 Form the LCM by writing ech fctor to the highest power it occurs in either polynomil. LCM ( )()( )( ) ( )( ) E XAMPLE Add with unlike denomintors Add: 9 REVIEW LCDS For help with finding lest common denomintors, see p To find the LCD, fctor ech denomintor nd write ech fctor to the highest power it occurs. Note tht 9 nd ( ), so the LCD is ( ) 9 ( ). 9 Fctor second denomintor. 9 ( ) p p LCD is 9 ( ). 9 ( ) 9 ( ) 9 ( ) Multiply. 9 ( ) Add numertors. 8. Add nd Subtrct Rtionl Epressions 8

3 E XAMPLE 4 Subtrct with unlike denomintors Subtrct: 4 4 ( ) ( )( ) Fctor denomintors. ( ) p ( )( ) p LCD is ( )( ). 6 4 Multiply. ( )( ) ( )( ) AVOID ERRORS After you simplify the numertor, check to see if the numertor hs fctor in common with the denomintor. If so, the epression cn be simplified further. 6 (4 ) ( )( ) Subtrct numertors. 4 ( )( ) Simplify numertor. ( )( 4) ( )( ) Fctor numertor. Divide out common fctor. 4 ( ) Simplify. GUIDED PRACTICE for Emples,, nd 4 Find the lest common multiple of the polynomils.. nd nd Perform the indicted opertion nd simplify KEY CONCEPT For Your Notebook Simplifying 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. Method : If necessry, simplify the numertor nd denomintor by writing ech s single frction. Then divide the numertor by the denomintor. Method : Multiply the numertor nd the denomintor by the lest common denomintor (LCD) of every frction in the numertor nd denomintor. Then simplify. 84 Chpter 8 Rtionl Functions

4 E XAMPLE Simplify comple frction (Method ) PHYSICS Let f be the focl length of thin cmer lens, p be the distnce between n object being photogrphed nd the lens, nd q be the distnce between the lens nd the film. For the photogrph to be in focus, the vribles should stisfy the lens eqution below. Simplify the comple frction. Lens eqution: f p q f p qpq q p Write denomintor s single frction. q p pq pq pq q p Divide numertor by denomintor. E XAMPLE 6 Simplify comple frction (Method ) 4 Simplify: 4 The LCD of ll the frctions in the numertor nd denomintor is ( 4). 4 4 ( 4) Multiply numertor nd p ( 4) denomintor by the LCD. 4 4 ( 4) Simplify. 8 Simplify. GUIDED PRACTICE for Emples nd 6 Simplify the comple frction Add nd Subtrct Rtionl Epressions 8

5 8. EXERCISES SKILL PRACTICE HOMEWORK KEY WORKED-OUT SOLUTIONS on p. WS for Es.,, nd 4 TAKS PRACTICE AND REASONING Es., 6,, 44, 46, nd 4. VOCABULARY Copy nd complete: A frction tht contins frction in its numertor or denomintor is clled (n)?.. WRITING Eplin how to dd rtionl epressions with unlike denomintors. EXAMPLE on p. 8 for Es. 8 LIKE DENOMINATORS Perform the indicted opertion nd simplify EXAMPLE on p. 8 for Es. 9 FINDING LCMS Find the lest common multiple of the polynomils. 9. nd ( ) 0. nd 4. nd ( ). 4 nd 8 6.,, nd nd 8. TAKS REASONING Wht is the lest common multiple of the polynomils 9 nd 6? A ( ) B 6 C 6( ) D 6 ( ) EXAMPLES nd 4 on pp for Es. 6 6 UNLIKE DENOMINATORS Perform the indicted opertion nd simplify ( 4) ERROR ANALYSIS Describe nd correct the error in dding the rtionl epressions. 4 4 ( )( ) 6. Which epression is equivlent to 4 TAKS REASONING 4 6? A 4 B ( )( ) ( 4)( 4) C 8 4 ( 4)( 4) D 8 4 ( 4)( 4) UNLIKE DENOMINATORS Perform the indicted opertion(s) nd simplify Chpter 8 Rtionl Functions

6 EXAMPLES nd 6 on p. 8 for Es. 6 SIMPLIFYING COMPLEX FRACTIONS Simplify the comple frction TAKS REASONING Write two different comple frctions tht ech simplify to. 4 CHALLENGE Simplify the comple frction ( ) 6 PROBLEM SOLVING EXAMPLE on p. 8 for E JET STREAM The totl time T (in hours) needed to fly from New York to Los Angeles nd bck (ignoring lyovers) cn be modeled by the eqution in the digrm, where d is the distnce ech wy (in miles), is the verge irplne speed (in miles per hour), nd j is the verge speed of the jet strem (in miles per hour). T d j d j NY NY LA LA j j j + j Rewrite the eqution so tht the right side is simplified. Then find the totl time if d 468 miles, 0 mi/h, nd j mi/h. t clsszone.com EXAMPLES nd 6 on p. 8 for Es ELECTRONICS If two resistors in prllel circuit hve resistnces R nd R (both in ohms), then the totl resistnce R t (in ohms) is given by the eqution shown. Simplify the comple frction. Then find the totl resistnce if R 000 ohms nd R 600 ohms. R t R t R R R R 8. Add nd Subtrct Rtionl Epressions 8

7 4. CAR LOANS If you borrow P dollrs to buy cr nd gree to repy the lon over t yers t n nnul interest rte of i (epressed s deciml), then your monthly pyment M is given by either formul below. t Formul : M Pi Formul : M Pi( i) i t ( i) t. Show tht the formuls re equivlent by simplifying the first formul. b. Find your monthly pyment if you borrow $,00 t n nnul interest rte of 6% nd repy the lon over 4 yers. 44. TAKS REASONING The mount A (in milligrms) of spirin in person s bloodstrem cn be modeled by A 9t t t where t is the time (in hours) fter one dose is tken. A first dose second dose A combined effect. Grph the eqution using grphing clcultor. t b. A second dose of the drug is tken hour fter the first dose. Write n eqution to model the mount of the second dose in the bloodstrem. c. Write nd grph model for the totl mount of spirin in the bloodstrem fter the second dose is tken. d. About how long fter the second dose hs been tken is the gretest mount of spirin in the bloodstrem? t 4. CHALLENGE Find the net two epressions in the pttern shown. Then simplify ll five epressions. Wht vlue do the epressions pproch?,,,... MIXED REVIEW FOR TAKS TAKS PRACTICE t clsszone.com REVIEW TAKS Preprtion p. 66; TAKS Workbook REVIEW Lesson.; TAKS Workbook 46. TAKS PRACTICE One leg of right tringle is 4 centimeters longer thn the other leg. The hypotenuse is 0 centimeters. About how long is the shorter leg? TAKS Obj. 0 A 0.4 cm B.0 cm C.6 cm D 6.0 cm 4. TAKS PRACTICE Which of the following is the solution of this system of liner equtions? TAKS Obj. 4 4y 8 y 4 F (, ) G (, ) H (, ) J (, ) 88 EXTRA PRACTICE for Lesson 8., p. 0 ONLINE QUIZ t clsszone.com

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