THE RATIONAL NUMBERS CHAPTER

Transcription

1 CHAPTER THE RATIONAL NUMBERS When divided by b is not n integer, the quotient is frction.the Bbylonins, who used number system bsed on 60, epressed the quotients: 0 8 s 0 60 insted of 8 s 7 60,600 0 insted of 8 Note tht this is similr to sying tht 0 hours divided by 8 is hours, 0 minutes nd tht hours divided by is hours, 7 minutes, 0 seconds. This nottion ws lso used by Leonrdo of Pis (7 0), lso known s Fiboncci. The bse-ten number system used throughout the world tody comes from both Hindu nd Arbic mthemticins. One of the erliest pplictions of the bse-ten system to frctions ws given by Simon Stevin (8 60), who introduced to 6th-century Europe method of writing deciml frctions. The deciml tht we write s.7 ws written by Stevin s 7 or s 7. John Npier (0 67) lter brought the deciml point into common usge. CHAPTER TABLE OF CONTENTS - Rtionl Numbers - Simplifying Rtionl Epressions - Multiplying nd Dividing Rtionl Epressions - Adding nd Subtrcting Rtionl Epressions - Rtio nd Proportion -6 Comple Rtionl Epressions -7 Solving Rtionl Equtions -8 Solving Rtionl Inequlities Chpter Summry Vocbulry Review Eercises Cumultive Review 9

2 0 The Rtionl Numbers - RATIONAL NUMBERS When persons trvel to nother country, one of the first things tht they lern is the monetry system. In the United Sttes, the dollr is the bsic unit, but most purchses require the use of frctionl prt of dollr. We know tht penny is $0.0 or 00 of dollr, tht nickel is $0.0 or 00 0 of 0 dollr, nd dime is $0.0 or 00 0 of dollr. Frctions re common in our everydy life s prt of dollr when we mke purchse, s prt of pound when we purchse cut of met, or s prt of cup of flour when we re bking. In our study of mthemtics, we hve worked with numbers tht re not 8 integers. For emple, minutes is 60 or of n hour, 8 inches is or of foot, 8 nd 8 ounces is 6 or of pound. These frctions re numbers in the set of rtionl numbers. DEFINITION A rtionl number is number of the form b where nd b re integers nd b 0. For every rtionl number b tht is not equl to zero, there is multiplictive b inverse or reciprocl such tht b? b. Note tht b? b b b. If the non-zero numertor of frction is equl to the denomintor, then the frction is equl to. EXAMPLE Write the multiplictive inverse of ech of the following rtionl numbers:. b. 8 c. Answers 8 8 Note tht in b, the reciprocl of negtive number is negtive number.

3 Rtionl Numbers Deciml Vlues of Rtionl Numbers The rtionl number b is equivlent to b. When frction or division such s 00 is entered into clcultor, the deciml vlue is displyed. To epress the quotient s frction, select Frc from the MATH menu. This cn be done in two wys. ENTER: 00 ENTER ENTER: 00 MATH ENTER MATH ENTER DISPLAY: /00 Ans Frc. / DISPLAY: /00 Frc / 8 When clcultor is used to evlute frction such s or 8, the deciml vlue is shown s The clcultor hs rounded the vlue to ten 8 deciml plces, the nerest ten-billionth. The true vlue of, or, is n infinitely repeting deciml tht cn be written s 0.6. The line over the 6 mens tht the digit 6 repets infinitely. Other emples of infinitely repeting decimls re: Every rtionl number is either finite deciml or n infinitely repeting deciml. Becuse finite deciml such s 0. cn be thought of s hving n infinitely repeting 0 nd cn be written s 0.0, the following sttement is true: A number is rtionl number if nd only if it cn be written s n infinitely repeting deciml.

4 The Rtionl Numbers EXAMPLE Find the common frctionl equivlent of 0.8. Solution Let How to Proceed () Multiply the vlue of by 00 to write number in which the deciml point follows the first pir of repeting digits: () Subtrct the vlue of from both sides of this eqution: () Solve the resulting eqution for nd simplify the frction: Check The solution cn be checked on clcultor c 0.888c ENTER: ENTER DISPLAY: / Answer EXAMPLE Epress 0.8 s common frction. Solution: Let How to Proceed () Multiply the vlue of by the power of 0 tht mkes the deciml point follow the first set of repeting digits. Since we wnt to move the deciml point plces, multiply by 0 0,000: () Multiply the vlue of by the power of 0 tht mkes the deciml point follow the digits tht do not repet. Since we wnt to move the deciml point plces, multiply by 0 00: () Subtrct the eqution in step from the eqution in step : () Solve for nd reduce the frction to lowest terms: Answer 0 8 0,000, ,000,8.88c 00.88c 9,900,6,6 9,

5 Rtionl Numbers Procedure To convert n infinitely repeting deciml to common frction:. Write the eqution: deciml vlue.. Multiply both sides of the eqution in step by 0 m, where m is the number of plces to the right of the deciml point following the first set of repeting digits.. Multiply both sides of the eqution in step by 0 n, where n is the number of plces to the right of the deciml point following the non-repeting digits. (If there re no non-repeting digits, then let n 0.). Subtrct the eqution in step from the eqution in step.. Solve the resulting eqution for, nd simplify the frction completely. Eercises Writing About Mthemtics.. Why is coin tht is worth cents clled qurter? b. Why is the number of minutes in qurter of n hour different from the number of cents in qurter of dollr?. Eplin the difference between the dditive inverse nd the multiplictive inverse. Developing Skills In 7, write the reciprocl (multiplictive inverse) of ech given number In 8, write ech rtionl number s repeting deciml In, write ech deciml s common frction

6 The Rtionl Numbers - SIMPLIFYING RATIONAL EXPRESSIONS EXAMPLE A rtionl number is the quotient of two integers. A rtionl epression is the quotient of two polynomils. Ech of the following frctions is rtionl epression: b 7 Division by 0 is not defined. Therefore, ech of these rtionl epressions hs no mening when the denomintor is zero. For instnce: hs no mening when 0. b hs no mening when 0 or when b 0. y y hs no mening when y or when y. y 6 y (y )(y ) For wht vlue or vlues of is the frction b y y y 6 undefined? Solution A frction is undefined or hs no mening when fctor of the denomintor is equl to 0. How to Proceed () Fctor the denomintor: ( )( ) () Set ech fctor equl to 0: 0 0 () Solve ech eqution for : Answer The frction is undefined when nd when. If b nd c d re rtionl numbers with b 0 nd d 0, then c b? d c bd c nd bd b? d c d? b c For emple: nd. We cn write rtionl epression in simplest form by finding common fctors in the numertor nd denomintors, s shown bove.

7 Simplifying Rtionl Epressions EXAMPLE Simplify: Answers. 6?? b. c. b ( ) y y y 6? b? b b ( 0, ) y (y )(y ) y y? y? y y (y, ) Note: We must eliminte ny vlue of the vrible or vribles for which the denomintor of the given rtionl epression is zero. b The rtionl epressions,, nd y in the emple shown bove re in simplest form becuse there is no fctor of the numertor tht is lso fctor of the denomintor ecept nd. We sy tht the frctions hve been reduced to lowest terms. When the numertor or denomintor of rtionl epression is monomil, ech number or vrible is fctor of the monomil. When the numertor or denomintor of rtionl epression is polynomil with more thn one term, we must fctor the polynomil. Once both the numertor nd denomintor of the frction re fctored, we cn reduce the frction by identifying fctors in the numertor tht re lso fctors in the denomintor. In the emple given bove, we wrote: y y y 6 y (y )(y ) y y? y? y y (y, ) We cn simplify this process by cnceling the common fctor in the numertor nd denomintor. (y ) (y ) (y ) y (y, ) Note tht cnceling (y ) in the numertor nd denomintor of the frction given bove is the equivlent of dividing (y ) by (y ).When ny number or lgebric epression tht is not equl to 0 is divided by itself, the quotient is.

8 6 The Rtionl Numbers Procedure To reduce frction to lowest terms: METHOD. Fctor completely both the numertor nd the denomintor.. Determine the gretest common fctor of the numertor nd the denomintor.. Epress the given frction s the product of two frctions, one of which hs s its numertor nd its denomintor the gretest common fctor determined in step.. Write the frction whose numertor nd denomintor re the gretest common fctor s nd use the multipliction property of. METHOD. Fctor both the numertor nd the denomintor.. Divide both the numertor nd the denomintor by their gretest common fctor by cnceling the common fctor. EXAMPLE Write in lowest terms. Solution METHOD METHOD ( )?? ( ) Answer ( 0) Fctors Tht Are Opposites The binomils ( ) nd ( ) re opposites or dditive inverses. If we chnge the order of the terms in the binomil ( ), we cn write: ( ) ( ) ( )

9 Simplifying Rtionl Epressions 7 We cn use this fctored form of ( ) to reduce the rtionl epression to lowest terms. ( ) ( )( ) ( ) ( )( ) ( ) (, ) EXAMPLE 8 Simplify the epression: 8 6 Solution METHOD METHOD 8 ( ) 8 6 ( )( ) ()( ) ( )( ) 8 ( ) 8 6 ( )( ) ()( ) ( )( ) ( ) ( )? ( ) ( ) ( )( ) Answer ( ) Eercises Writing About Mthemtics. Abby sid tht cn be reduced to lowest terms by cnceling so tht the result is. Do you gree with Abby? Eplin why or why not.. Does for ll rel vlues of? Justify your nswer. Developing Skills In 0, list the vlues of the vribles for which the rtionl epression is undefined.. d. 6c. b 6. b c b 9. c 0. b 6 c 6

10 8 The Rtionl Numbers In 0, write ech rtionl epression in simplest form nd list the vlues of the vribles for which the frction is undefined. 6.. b y. 0 0 y. 9cd y y c d 6y 8. 0d 8c y d 0d. 8c 6c 9y 6 y. y y y 6. (b ) ( ) b b b 8b b 6b ( b) b 6 - MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS Multiplying Rtionl Epressions We know tht nd tht In generl, the product of two c rtionl numbers b nd d is b? d c bd c for b 0 nd d 0. This sme rule holds for the product of two rtionl epressions: The product of two rtionl epressions is frction whose numertor is the product of the given numertors nd whose denomintor is the product of the given denomintors. For emple: ( )? 6( ) ( 0) This product cn be reduced to lowest terms by dividing numertor nd denomintor by the common fctor,. ( )? 6( ) 6 9 ( ) We could hve cnceled the fctor before we multiplied, s shown below. ( ) ( ) 9( ) 9( ) Note tht is not common fctor of the numertor nd denomintor becuse it is one term of the fctor ( ), not fctor of the numertor.

11 Multiplying nd Dividing Rtionl Epressions 9 Procedure To multiply frctions: METHOD. Multiply the numertors of the given frctions nd the denomintors of the given frctions.. Reduce the resulting frction, if possible, to lowest terms. METHOD. Fctor ny polynomil tht is not monomil.. Cncel ny fctors tht re common to numertor nd denomintor.. Multiply the resulting numertors nd the resulting denomintors to write the product in lowest terms. EXAMPLE b. Find the product of b? b 9 in simplest form. b 9b b. For wht vlues of the vrible re the given frctions nd the product undefined? Solution. METHOD How to Proceed () Multiply the numertors of the frctions nd the denomintors of the frctions: () Fctor the numertor nd the denomintor. Note tht the fctors of (b ) re (b ) nd the fctors of (b 9b) re b(b ). Reduce the resulting frction to lowest terms: b b? b 9 b 9b b (b 9) (b )(b 9b) b (b )(b ) b(b )(b ) b(b ) b(b ) b(b )? (b ) b(b ) (b )? b(b ) (b ) Answer

12 0 The Rtionl Numbers METHOD How to Proceed () Fctor ech binomil term completely: () Cncel ny fctors tht re common to numertor nd denomintor: () Multiply the remining fctors: b b? b 9 b 9b b (b ) b b (b ) b. The given frctions nd their product re undefined when b 0 nd when b. Answer b(b ) (b )? (b )(b ) b(b ) (b ) (b )? b(b ) Answer Dividing Rtionl Epressions We cn divide two rtionl numbers or two rtionl epressions by chnging the c division to relted multipliction. Let b nd d be two rtionl numbers or rtionl epressions with b 0, c 0, d 0. Since d is non-zero rtionl number or c d c rtionl epression, there eists multiplictive inverse such tht d? d c c. b c d b c d d b? c c d d c b? d c c d? d c We hve just derived the following procedure. Procedure b? d c b? d c To divide two rtionl numbers or rtionl epressions, multiply the dividend by the reciprocl of the divisor. For emple:? ( )( ) 0 (, 0) This product is in simplest form becuse the numertor nd denomintor hve no common fctors. Recll tht cn be written s nd therefore, if 0, the reciprocl of is. For emple: b (b ) (b )? b (b )? b (b )

13 Multiplying nd Dividing Rtionl Epressions EXAMPLE Divide nd simplify: 0 Solution How to Proceed () Use the reciprocl of the divisor to write the division s multipliction: () Fctor ech polynomil: () Cncel ny fctors tht re common to the numertor nd denomintor: () Multiply the remining fctors: ( ) Answer ( 0, ) 0 0? ( )( )? ( ) ( ) ( )? ( ) ( ) EXAMPLE Perform the indicted opertions nd write the nswer in simplest form: 6? Solution Recll tht multiplictions nd divisions re performed in the order in which they occur from left to right. 6?? ( )?? ( )? 6? (, 0) Answer

14 The Rtionl Numbers Eercises Writing About Mthemtics ( ). Joshu wnted to write this division in simplest form: 7. He begn by cnceling ( ) in the numertor nd denomintor nd wrote following: ( ) Is Joshu s nswer correct? Justify your nswer.. Gbriel wrote 0 ( 0). Is Gbriel s solution correct? Justify your nswer. Developing Skills In, multiply nd epress ech product in simplest form. In ech cse, list ny vlues of the vribles for which the frctions re not defined... y.? 8y 6. b 7.? b 8. 7y 9. 7y? y ?. 7? 0? ? 6? ? In, divide nd epress ech quotient in simplest form. In ech cse, list ny vlues of the vribles for which the frctions re not defined b c 0c b 6y y y y 9 8 c 9. 6c 9 w 0. w w w c c b.. 8 b (b ) ( ). ( 7). ( ) 7

15 In 0, perform the indicted opertions nd write the result in simplest form. In ech cse, list ny vlues of the vribles for which the frctions re not defined ? (b) b b? b b 0. Adding nd Subtrcting Rtionl Epressions? ( )?? - ADDING AND SUBTRACTING RATIONAL EXPRESSIONS We know tht nd tht. Therefore: 7 A 7 B We cn lso write: 7 A 7 B 7 7 A 7 B A 7 B ( ) A 7 B A 7 B 7 ( ) A B ( ) A B ( ) A B ( 0) In generl, the sum of ny two frctions with common denomintor is frction whose numertor is the sum of the numertors nd whose denomintor is the common denomintor. ( ) In order to dd two frctions tht do not hve common denomintor, we need to chnge one or both of the frctions to equivlent frctions tht do hve common denomintor. Tht common denomintor cn be the product of the given denomintors. For emple, to dd 7, we chnge ech frction to n equivlent frction whose denomintor is by multiplying ech frction by frction equl to

16 The Rtionl Numbers To dd the frctions nd y, we need to find denomintor tht is multiple of both nd of y. One possibility is their product, y. Multiply ech frction by frction equl to so tht the denomintor of ech frction will be y:? y y y y nd y y? y 6 y y y 6 y 6 y y ( 0, y 0) The lest common denomintor (LCD) is often smller thn the product of the two denomintors. It is the lest common multiple (LCM) of the denomintors, tht is, the product of ll fctors of one or both of the denomintors. For emple, to dd, first find the fctors of ech denomintor.the lest common denomintor is the product of ll of the fctors of the first denomintor times ll fctors of the second tht re not fctors of the first.then multiply ech frction by frction equl to so tht the denomintor of ech frction will be equl to the LCD. Fctors of : ( ) Fctors of : ( ) ( ) LCD: ( ) ( ) ( )? ( )( ) nd ( )( )? ( )( ) ( )( ) ( )( ) Since this sum hs common fctor in the numertor nd denomintor, it cn be reduced to lowest terms. ( )( ) ( )( ) ( ) (, ) Any polynomil cn be written s rtionl epression with denomintor of. To dd polynomil to rtionl epression, write the polynomil s n equivlent rtionl epression. For emple, to write the sum b b s single frction, multiply b (b ) by in the form b. (b ) b b A b b B b b 6b b b b 6b b (b 0)

17 Adding nd Subtrcting Rtionl Epressions EXAMPLE Write the difference s single frction in lowest terms. Solution How to Proceed () Find the LCD of the frctions: () Write ech frction s n equivlent frction with denomintor equl to the LCD: () Subtrct: () Simplify: () Reduce to lowest terms: ( ) ( ) ( ) ( ) LCD ( ) ( ) ( ) ( )( )? ( )( )( ) ( )( )? 6 9 ( )( )( ) ( 6 9) ( )( )( ) 9 9 ( )( )( ) 9 ( )( ) Answer 9 ( )( ) (,, ) EXAMPLE Solution Simplify: STEP. Rewrite ech epression in prentheses s single frction. A B STEP. Multiply. A BA B A BA nd ( )( ) B Q R A B ( )( ) A B

18 6 The Rtionl Numbers Alterntive Solution STEP. Multiply using the distributive property. A BA B A B A ( ) B STEP. Add the frctions. The lest common denomintor is ( ). ) ( ) Q( ( ) R STEP. Simplify. ( ) ( ) ( ) A B A B ( ) ( ) ( ) ( ) ( ) Answer ( 0, ) ( ) ( )( ) ( ) Eercises Writing About Mthemtics ( )( ). Ashley sid tht ( )( ) for ll vlues of ecept. Do you gree with Ashley? Eplin why or why not.. Mtthew sid tht b d c d bc bd when b 0, d 0. Do you gree with Mtthew? Justify your nswer. Developing Skills In 0, perform the indicted dditions or subtrctions nd write the result in simplest form. In ech cse, list ny vlues of the vribles for which the frctions re not defined y y y b b b

19 Applying Skills In, the length nd width of rectngle re epressed in terms of vrible.. Epress ech perimeter in terms of the vrible. b. Epress ech re in terms of the vrible.. l nd w. l nd w. l nd w. l nd w Rtio nd Proportion 7 - RATIO AND PROPORTION We often wnt to compre two quntities tht use the sme unit. For emple, in given clss of students, there re students who re boys. We cn sy tht of the students re boys or tht the rtio of students who re boys to ll students in the clss is :. DEFINITION A rtio is the comprison of two numbers by division. The rtio of to b cn be written s b or s : b when b 0. A rtio, like frction, cn be simplified by dividing ech term by the sme non-zero number. A rtio is in simplest form when the terms of the rtio re integers tht hve no common fctor other thn. For emple, to write the rtio of inches to foot, we must first write ech mesure in terms of the sme unit nd then divide ech term of the rtio by common fctor. inches foot inches foot inches foot In lowest terms, the rtio of inches to foot is :. An equivlent rtio cn lso be written by multiplying ech term of the rtio by the sme non-zero number. For emple, : 7 () : 7() 8 :. In generl, for 0: : b : b

20 8 The Rtionl Numbers EXAMPLE The length of rectngle is yrd nd the width is feet. Wht is the rtio of length to width of this rectngle? Solution The rtio must be in terms of the sme mesure. yd ft ft yd Answer The rtio of length to width is :. EXAMPLE The rtio of the length of one of the congruent sides of n isosceles tringle to the length of the bse is :. If the perimeter of the tringle is.0 centimeters, wht is the length of ech side? Solution Let AB nd BC be the lengths of the congruent sides of isosceles ABC nd AC be the length of the bse. AB : AC : : B Therefore, AB, BC, nd AC. A C AB BC AC Perimeter. cm AB BC (.) 7. cm AC (.) 7.0 cm Check Answer AB BC AC cm The sides mesure 7., 7., nd 7.0 centimeters.

21 Rtio nd Proportion 9 Proportion An eqution tht sttes tht two rtios re equl is clled proportion. For emple, : : is proportion. This proportion cn lso be written s. In generl, if b 0 nd d 0, then : b c : d or b d c re proportions.the first nd lst terms, nd d, re clled the etremes of the proportion nd the second nd third terms, b nd c, re the mens of the proportion. If we multiply both sides of the proportion by the product of the second nd lst terms, we cn prove useful reltionship mong the terms of proportion. b d b b c d bd A b B bd A c d B bd d c d bc In ny proportion, the product of the mens is equl to the product of the etremes. EXAMPLE In the junior clss, there re more girls thn boys. The rtio of girls to boys is :. How mny girls nd how mny boys re there in the junior clss? Solution How to Proceed () Use the fct tht the number of girls is more thn the number of boys to represent the number of girls nd of boys in terms of : () Write proportion. Set the rtio of the number of boys to the number of girls, in terms of, equl to the given rtio: () Use the fct tht the product of the mens is equl to the product of the etremes. Solve the eqution: Let the number of boys, the number of girls. ( )

22 60 The Rtionl Numbers Alterntive Solution () Use the given rtio to represent the number of boys nd the number of girls in terms of : () Use the fct tht the number of girls is more thn the number of boys to write n eqution. Solve the eqution for : () Use the vlue of to find the number of girls nd the number of boys: Let the number of girls the number of boys () 0 girls () 96 boys Answer There re 0 girls nd 96 boys in the junior clss. Eercises Writing About Mthemtics. If b d c, then is c d b? Justify your nswer. b. If b d c, then is b c d d? Justify your nswer. Developing Skills In 0, write ech rtio in simplest form.. : 8. :. : 8 6. : :9, , 0 In 9, solve ech proportion for the vrible y. y Applying Skills 0. The rtio of the length to the width of rectngle is :. The perimeter of the rectngle is 7 inches. Wht re the dimensions of the rectngle?. The rtio of the length to the width of rectngle is 7 :. The re of the rectngle is 6 squre centimeters. Wht re the dimensions of the rectngle?. The bsketbll tem hs plyed gmes. The rtio of wins to losses is :. How mny gmes hs the tem won?

23 Comple Rtionl Epressions 6. In the chess club, the rtio of boys to girls is 6 :. There re more boys thn girls in the club. How mny members re in the club?. Every yer, Jvier mkes totl contribution of $ to two locl chrities. The two dontions re in the rtio of :. Wht contribution does Jvier mke to ech chrity?. A cookie recipe uses flour nd sugr in the rtio of 9 :. If Nichols uses cup of sugr, how much flour should he use? 6. The directions on bottle of clening solution suggest tht the solution be diluted with wter. The rtio of solution to wter is : 7. How mny cups of solution nd how mny cups of wter should Christopher use to mke gllons ( cups) of the miture? -6 COMPLEX RATIONAL EXPRESSIONS A comple frction is frction whose numertor, denomintor, or both contin frctions. Some emples of comple frctions re: A comple frction cn be simplified by multiplying by frction equl to ; tht is, by frction whose non-zero numertor nd denomintor re equl. The numertor nd denomintor of this frction should be common multiple of the denomintors of the frctionl terms. For emple: 6 A comple frction cn lso be simplified by dividing the numertor by the denomintor A comple rtionl epression hs rtionl epression in the numertor, the denomintor, or both. For emple, the following re comple rtionl epressions b b b

24 6 The Rtionl Numbers Like comple frctions, comple rtionl epressions cn be simplified by multiplying numertor nd denomintor by common multiple of the denomintors in the frctionl terms.? ( 0)? ( 0) b b b b b b b b b b? b b (b )(b ) b(b ) (b )(b ) b(b ) b b (b 0, ) Alterntively, comple rtionl epression cn lso be simplified by dividing the numertor by the denomintor. Choose the method tht is esier for ech given epression. Procedure To simplify comple frction: METHOD m Multiply by m, where m is the lest common multiple of the denomintors of the frctionl terms. METHOD Multiply the numertor by the reciprocl of the denomintor of the frction. EXAMPLE Epress in simplest form. Solution METHOD Multiply numertor nd denomintor of the frction by the lest common multiple of the denomintors of the frctionl terms. The lest common multiple of nd is.? 8

25 Comple Rtionl Epressions 6 METHOD Write the frction s the numertor divided by the denomintor. Chnge the division to multipliction by the reciprocl of the divisor. 8 Answer ( 0)?? 8 EXAMPLE Solution Simplify: b 0 b METHOD METHOD The lest common multiple of 0,, nd is 0. b 0 b b 0 b? 0 0 b b 0 (b ) (b )(b ) Answer (b ) (b, ) (b ) b 0 b b 0 0 b b 0 b b 0 b b 0? (b ) (b ) (b ) Eercises Writing About Mthemtics. For wht vlues of is A B A B undefined? Eplin your nswer. d. Bebe sid tht since ech of the denomintors in the comple frction is non-zero d constnt, the frction is defined for ll vlues of d. Do you gree with Bebe? Eplin why or why not.

26 6 The Rtionl Numbers Developing Skills In 0, simplify ech comple rtionl epression. In ech cse, list ny vlues of the vribles for which the frctions re not defined d b b d b y y y... y y b b b b b y 8. y y y In, simplify ech epression. In ech cse, list ny vlues of the vribles for which the frctions re not defined... A B A 0 6 B A 6 b B A b b B b b -7 SOLVING RATIONAL EQUATIONS An eqution in which there is frction in one or more terms cn be solved in different wys. However, in ech cse, the end result must be n equivlent eqution in which the vrible is equl to constnt.

27 Solving Rtionl Equtions 6 For emple, solve:. This is n eqution with rtionl coefficients nd could be written s. There re three possible meth- ods of solving this eqution. METHOD Work with the frctions s given. Combine terms contining the vrible on one side of the eqution nd constnts on the other. Find common denomintors to dd or subtrct frctions. () 0 0 METHOD Rewrite the eqution without frctions. Multiply both sides of the eqution by the lest common denomintor. In this cse, the LCD is. A B A B METHOD Rewrite ech side of the eqution s rtio. Use the fct tht the product of the mens is equl to the product of the etremes. 8 8 ( ) ( 8) These sme procedures cn lso be used for rtionl eqution, n eqution in which the vrible ppers in one or more denomintors.

28 66 The Rtionl Numbers EXAMPLE Solve the eqution. Solution Use Method. How to Proceed () Write the eqution: () Multiply both sides of the eqution by the lest common denomintor, 6: () Simplify: () Solve for : 6 A B 6 A B Check? 8 Q 8 R A 8 B? A B 8? 9? 8 Answer EXAMPLE 8 When solving rtionl eqution, it is importnt to check tht the frctionl terms re defined for ech root, or solution, of the eqution. Any solution for which the eqution is undefined is clled n etrneous root. Solve for : Solution Since this eqution is proportion, we cn use tht the product of the mens is equl to the product of the etremes. Check the roots. ( ) ( ) 6 0 ( 6) Check: 0 0 0? 0 (0) Ech side of the eqution is undefined for 0. Therefore, 0 is not root. Check: 6 6 6? 6 (6) 6? 8 Answer 6

29 Solving Rtionl Equtions 67 EXAMPLE Solve: ( ) Solution Use Method to rewrite the eqution without frctions. The lest common denomintor is ( ). ( ) A ( ) 0 0 ( )( ) 0 ( ) 0 0 B ( ) A B ( ) A b ( ) 6 b ( ) ( ) B ( ) b Check the roots: Check: Check: ( )? ( ) 7? 7 7? Since leds to true sttement, is root of the eqution. Answer is the only root of ( ). ( )? ( ) 0? (0) Since leds to sttement tht is undefined, is not root of the eqution. It is importnt to check ech root obtined by multiplying both members of the given eqution by n epression tht contins the vrible. The derived eqution my not be equivlent to the given eqution.

30 68 The Rtionl Numbers EXAMPLE Solve nd check: 0.y 0.0y Solution The coefficients 0. nd 0.0 re deciml frctions ( tenths nd hundredths ). The denomintor of 0. is 0 nd the denomintor of 0.0 is 00. Therefore, the lest common denomintor is y 0.0y 00(0.y ) 00( 0.0y) 0y y y 00 y Check 0.y 0.0y 0.()? 0.0() 0.8? Answer y EXAMPLE On his wy home from college, Dniel trveled miles on locl rods nd 90 miles on the highwy. On the highwy, he trveled 0 miles per hour fster thn on locl rods. If the trip took hours, wht ws Dniel s rte of speed on ech prt of the trip? distnce Solution Use rte time to represent the time for ech prt of the trip. Let rte on locl rods. Then time on locl rods. 90 Let 0 rte on the highwy. Then 0 time on the highwy. The totl time is hours. ( 0) A B ( 0) A B ( 0) ( 0) Recll tht trinomil cn be fctored by writing the middle term s the sum of two terms whose product is the product of the first nd lst term. (0) 900 Find the fctors of this product whose sum is the middle term. (60)

31 Solving Rtionl Equtions 69 Write the trinomil with four terms, using this pir of terms in plce of, nd fctor the trinomil ( ) 0( ) 0 ( )( 0) Reject the negtive root since rte cnnot be negtive number. Use 0. On locl rods, Dniel s rte is 0 miles per hour nd his time is 0 hour. On the highwy, Dniel s rte is 0 0 or 60 miles per hour. His time is hours. His totl time is hours. Answer Dniel drove 0 mph on locl rods nd 60 mph on the highwy. Eercises Writing About Mthemtics. Smnth sid tht the eqution in Emple could be solved by multiplying both sides of the eqution by. Would Smnth s solution be the sme s the solution obtined in Emple? Eplin why or why not.. Brinn sid tht is rtionl eqution but is not. Do you gree with Brinn? Eplin why or why not. Developing Skills In 0, solve ech eqution nd check b 7 0.0b b 0 ( )( ) y b b 7 y(y ) 9b

32 70 The Rtionl Numbers Applying Skills. Lst week, the rtio of the number of hours tht Joseph worked to the number of hours tht Nicole worked ws :. This week Joseph worked hours more thn lst week nd Nicole worked twice s mny hours s lst week. This week the rtio of the hours Joseph worked to the number of hours Nicole worked is :. How mny hours did ech person work ech week?. Anthony rode his bicycle to his friend s house, distnce of mile. Then his friend s mother drove them to school, distnce of miles. His friend s mother drove t rte tht is miles per hour fster thn Anthony rides his bike. If it took Anthony of n hour to get to distnce school, t wht verge rte does he ride his bicycle? (Use rte time for ech prt of the trip to school.). Amnd drove 0 miles. Then she incresed her rte of speed by 0 miles per hour nd drove nother 0 miles to rech her destintion. If the trip took hours, t wht rte did Amnd drive?. Lst week, Emily pid $8. for pounds of pples. This week she pid $9.0 for ( ) pounds of pples. The price per pound ws the sme ech week. How mny pounds of pples did Emily buy ech week nd wht ws the price per pound? (Use totl cost number of pounds cost per pound for ech week.) -8 SOLVING RATIONAL INEQUALITIES Inequlities re usully solved with the sme procedures tht re used to solve equtions. For emple, we cn solve this eqution nd this inequlity by using the sme steps. 8 8 A B A 8 B A B A 8 B , 8 A B A 8 B, A B A 8 B 6, 8 9, 6. All steps leding to the solution of this eqution nd this inequlity re the sme, but specil cre must be used when multiplying or dividing n inequlity. Note tht when we multiplied the inequlity by, positive number, the order of the inequlity remined unchnged. In the lst step, when we divided the inequlity by, negtive number, the order of the inequlity ws reversed. When we solve n inequlity tht hs vrible epression in the denomintor by multiplying both sides of the inequlity by the vrible epression, we must consider two possibilities: the vrible represents positive number or the

33 vrible represents negtive number. (The epression cnnot equl zero s tht would mke the frction undefined.) For emple, to solve. 9, we multiply both sides of the inequlity by. When is positive, the order of the inequlity remins unchnged. When is negtive, the order of the inequlity is reversed. If 0, then. If 0, then ( ). ( )() 9., 8, Solving Rtionl Inequlities ( ), ( )() 9,. 8. Therefore, nd or There re no vlues of such. tht nd. The solution set of the eqution. 9 is { : }.. An lterntive method of solving this inequlity is to use the corresponding eqution. STEP. Solve the corresponding eqution. 9 9 ( )() ( ) ( )() ( ) 9 8 STEP. Find ny vlues of for which the eqution is undefined. 9 Terms nd re undefined when. STEP. To find the solutions of the corresponding inequlity, divide the number line into three intervls using the solution to the equlity,, nd the vlue of for which the eqution is undefined,. 0 6

34 7 The Rtionl Numbers Choose vlue from ech section nd substitute tht vlue into the inequlity. 0 6 Let : Let 0: Let :. 9.? ? ? 9 ().? (9) (0).? (9) 6.? The inequlity is true for vlues in the intervl nd flse for ll other vlues. EXAMPLE Solve for :. Solution METHOD Multiply both sides of the eqution by. The sense of the inequlity will remin unchnged when 0 nd will be reversed when 0. Let 0: Let 0: 0 nd 0 nd 0 The solution set is { : 0 or }. METHOD A B. A B.. 8. Solve the corresponding eqution for. A B A B 8 A B, A B,, 8,

35 Solving Rtionl Inequlities 7 Prtition the number line using the solution to the eqution,, nd the vlue of for which the eqution is undefined, 0. Check in the inequlity representtive vlue of from ech intervl of the grph. Answer { : 0 or } 0 6 Let : Let : Let :....?.?.?.?.? 0.? Eercises Writing About Mthemtics. When the eqution b b is solved for b, the solutions re nd. Eplin why the number line must be seprted into five segments by the numbers,, 0, nd in order to check the solution set of the inequlity b. b.. Wht is the solution set of, 0? Justify your nswer. Developing Skills In, solve nd check ech inequlity. y b.., y. 8, b. 6 0 d d y 7, y 0.,.,.,

36 7 The Rtionl Numbers CHAPTER SUMMARY A rtionl number is n element of the set of numbers of the form b when nd b re integers nd b 0. For every rtionl number b tht is not equl to b zero there is multiplictive inverse or reciprocl such tht b? b. A number is rtionl number if nd only if it cn be written s n infinitely repeting deciml. A rtionl epression is the quotient of two polynomils. A rtionl epression hs no mening when the denomintor is zero. A rtionl epression is in simplest form or reduced to lowest terms when there is no fctor of the numertor tht is lso fctor of the denomintor ecept nd. c If b nd d re two rtionl numbers with b 0 nd d 0: b? d c bd c b d c b? d c d bc (c 0) b d c b? d d d c? b b bd d cb d cb bd bd A rtio is the comprison of two numbers by division.the rtio of to b cn be written s b or s : b when b 0. An eqution tht sttes tht two rtios re equl is clled proportion. In the proportion : b c : d or b d c, the first nd lst terms, nd d, re clled the etremes, nd the second nd third terms, b nd c, re the mens. In ny proportion, the product of the mens is equl to the product of the etremes. A comple rtionl epression hs rtionl epression in the numertor, the denomintor or both. Comple rtionl epressions cn be simplified by multiplying numertor nd denomintor by common multiple of the denomintors in the numertor nd denomintor. A rtionl eqution, n eqution in which the vrible ppers in one or more denomintors, cn be simplified by multiplying both members by common multiple of the denomintors. When rtionl inequlity is multiplied by the lest common multiple of the denomintors, two cses must be considered: when the lest common multiple is positive nd when the lest common multiple is negtive. VOCABULARY - Rtionl number Multiplictive inverse Reciprocl - Rtionl epression Simplest form Lowest terms - Lest common denomintor (LCD) Lest common multiple (LCM) - Rtio Rtio in simplest form Proportion Etremes Mens -6 Comple frction Comple rtionl epression -7 Rtionl eqution Etrneous root

37 Review Eercises 7 REVIEW EXERCISES 7. Wht is the multiplictive inverse of? (b ). For wht vlues of b is the rtionl epression b(b undefined? ). Write s n infinitely repeting deciml. In, perform the indicted opertions, epress the nswer in simplest form, nd list the vlues of the vribles for which the frctions re undefined.. b? 0 b 9. 6.? d 0. d 8 d d A B? A B. 7 0 y y 8 y 9 b 0 b? 6 b A B? A B ( ) 6 In 6 9, simplify ech comple rtionl epression nd list the vlues of the vribles (if ny) for which the frctions re undefined. b 6. b b 0 6 In 0 7, solve nd check ech eqution or inequlity y y 6. d.. d 6 ( ) d # 8 8. The rtio of boys to girls in the school chorus ws :. After three more boys joined the chorus, the rtio of boys to girls is 9 : 0. How mny boys nd how mny girls re there now in the chorus? 9. Lst week Stephnie spent $0.0 for cns of sod. This week, t the sme cost per cn, she bought three fewer cns nd spent $8.0. How mny cns of sod did she buy ech week nd wht ws the cost per cn?

38 76 The Rtionl Numbers 0. On recent trip, Srh trveled 80 miles t constnt rte of speed. Then she encountered rod work nd hd to reduce her speed by miles per hour for the lst 0 miles of the trip. The trip took hours. Wht ws Srh s rte of speed for ech prt of the trip?. The res of two rectngles re 08 squre feet nd 8 squre feet. The length of the lrger is feet more thn the length of the smller. If the rectngles hve equl widths, find the dimensions of ech rectngle. re A Use length width. B Eplortion The erly Egyptins wrote frctions s the sum of unit frctions (the reciprocls of the counting numbers) with no reciprocl repeted. For emple: nd Of course the Egyptins used hieroglyphs insted of the numerls fmilir to us tody. Whole numbers were represented s follows ,000 0,000 00,000,000,000 0,000,000 or To write unit frctions, the symbol hieroglyph. For emple: ws drwn over the whole number 0 nd. Show tht for n 0, n n n(n ).. Write nd using Egyptin frctions.. Write ech of the frctions s the sum of unit frctions. (Hint: Use the results of Eercise.) 7 7. b. 0 c. d. e. 8

39 Cumultive Review 77 CUMULATIVE REVIEW CHAPTERS Prt I Answer ll questions in this prt. Ech correct nswer will receive credits. No prtil credit will be llowed.. Which of the following is not true in the set of integers? () Addition is commuttive. () Every integer hs n dditive inverse. () Multipliction is distributive over ddition. () Every non-zero integer hs multiplictive inverse.. is equl to () 0 () () 6 (). The sum of nd 7 is () () () () 6. In simplest form, ( ) (7 ) is equl to () () () () 7. The fctors of 6 re () ( )( ) () ( )( ) () ( )( ) () ( )( ) 6. The roots of the eqution re () nd () 7 nd 0 () nd () nd 7. For 0,,, the frction is equl to () () () () b 8. In simplest form, the quotient b 8b b equls b () () (b ) 7 () () (b ) 9. For wht vlues of is the frction undefined? (),, 7 (),, 7 (), 7 (), 7 0. The solution set of the eqution 7 is () {, } () () {} () {}

40 78 The Rtionl Numbers Prt II Answer ll questions in this prt. Ech correct nswer will receive credits. Clerly indicte the necessry steps, including pproprite formul substitutions, digrms, grphs, chrts, etc. For ll questions in this prt, correct numericl nswer with no work shown will receive only credit.. Solve nd check: 7. Perform the indicted opertions nd write the nswer in simplest form: Prt III 9 Answer ll questions in this prt. Ech correct nswer will receive credits. Clerly indicte the necessry steps, including pproprite formul substitutions, digrms, grphs, chrts, etc. For ll questions in this prt, correct numericl nswer with no work shown will receive only credit.. Write the following product in simplest form nd list the vlues of for which it is undefined:?. The length of rectngle is meters longer thn hlf the width. The re of the rectngle is 90 squre meters. Find the dimensions of the rectngle. Prt IV Answer ll questions in this prt. Ech correct nswer will receive 6 credits. Clerly indicte the necessry steps, including pproprite formul substitutions, digrms, grphs, chrts, etc. For ll questions in this prt, correct numericl nswer with no work shown will receive only credit.. Find the solution set nd check: 7 6. Diego hd trveled 0 miles t uniform rte of speed when he encountered construction nd hd to reduce his speed to one-third of his originl rte. He continued t this slower rte for 0 miles. If the totl time for these two prts of the trip ws one hour, how fst did he trvel t ech rte?