Lesson 8: Rational Expressions Digging Deeper solutions

Transcription

1 Lesson : Raional Expessions Raional Equaions Acivi Solve. 1. Each weekda a mohe dives 5 miles o pick up he son fom socce pacice. She hen dives 5 miles o he gm o pick up he daughe. Due o heav affic, he speed going o he gm is 5 mph less han he speed going o he socce field. Afe picking he daughe up, she akes a 6- mile shocu home a he same speed ha she began wih. If he whole ip akes 34 minues, how fas does she dive on each leg of he ip Gm 6 5 Home 5 Socce D D socce 5 gm 5 5 home 6 Toal ime 34 minues 34 hou 60 Toal ime + + socce gm home 009 Duke Univesi Talen Idenificaion Pogam Page 1 of

2 socce gm home Lesson : Raional Expessions ( 5) + 60( 5) 5 60 ( ) + + ( ) ( ) ( 17 55)( 30) o o If 55, hen If 30, hen he speed fom home o socce and fom he gm o home was 30 mph. He speed fom socce o he gm was 5 mph. Check: 5 1 socce h 10 min gm h 1 min home h 1 min 30 5 Toal ime 10 min + 1 min + 1 min 34 minues 009 Duke Univesi Talen Idenificaion Pogam Page of

3 Lesson : Raional Expessions. Fill in he numeaos fo he aional expession below so ha he equaion has one inege soluion and one exaneous soluion. Hin: In ode fo an equaion o have wo soluions, i mus be a quadaic equaion. b b+ b b Thee ae man was o appoach his poblem and man coec answes. The soluion given below is one saeg fo solving his poblem. To have an exaneous soluion, we need fo one of ou soluions o make he denominao zeo. We could have eihe b 5 o b. I chose b 5 o be one soluion and b-3 o be he ohe soluion. b 5 o b 3 b 5 0 o b+ 3 0 ( b )( b ) b b 15 0 Afe mulipling boh sides of he above equaion b he LCD and simplifing, I wan he equaion o be b b I le he fis numeao be b, and he second numeao be 7. (Noe: These wee chosen andoml.) I ll empoail le K be he hid numeao. b 7 K b b+ b b Mulipling boh sides b he LCD and simplifing gives: 009 Duke Univesi Talen Idenificaion Pogam Page 3 of

4 ( ) 7( 5) b + b + 7b 35 K b b K Lesson : Raional Expessions b 7 K b 5 b+ b 3b 10 b 7 K b 5 b+ b 5 b+ b 7 K b 5 b+ + b 5 b b 5 b + + ( b 5)( b+ ) b b+ + b K We wan his equaion o be b b If we se he equaions equal o each ohe, we ge: b b K b b b K b 0 K So, one equaion ha will wok is: b 7 11b 0 b b+ b b Check: ( ) ( ) ( b )( b ) 009 Duke Univesi Talen Idenificaion Pogam Page 4 of b 7 11b 0 b 5 b+ b 3b 10 b 7 11b 0 b 5 b+ b 5 b+ b b b b b 7 11b 0 b 5 b+ + b 5 b b 5 b + + ( b 5)( b+ ) b b+ + 7 b 5 11b b b b 5 0 o b+ 3 0 b 5 o b 3

5 Lesson : Raional Expessions When we subsiue b 5, we ge zeo in he denominao. This is an exaneous soluion. When we subsiue b 3, we ge: b 7 11b 0 b 5 b+ b 3b ( ) ( ) ( ) So b 3 is an inege soluion. 3. If x + 5 and x, find x We could o solve his ssem of equaions fo x and, using subsiuion: x + 5 x x Subsiuing ino he fis equaion gives: ( 5) Duke Univesi Talen Idenificaion Pogam Page 5 of

6 Lesson : Raional Expessions Unfounael, his won faco. (In Lesson 10, ou ll lean how o solve hese using he quadaic fomula.) We can he woking backwads saeg. Sa wih x and simplif: x 1 x x + x x + x x x + x Since we know ha x + 5 and x, we can subsiue o ge: 1 1 x + 5 x x 4. You hie a moving compan o pack and move ou business. One emploee, Joseph, could do he enie job in 6 hous. Anohe emploee, Logan, could do he enie job in hous. Logan woks alone fo an hou, hen Joseph shows up, and he wok ogehe. Joseph leaves afe wo hous of wok, and Logan finishes he es of he job. a) How much longe would Logan have o wok o complee he job 009 Duke Univesi Talen Idenificaion Pogam Page 6 of

7 Lesson : Raional Expessions In one hou, Logan can do 1 of he job. In x hous, he can do x 1 of he job. In one hou, Joseph can do 6 of he job. Logan woked fo 1 hou and did 1 of he job. Joseph joined 1 1 him, and he woked fo hous and ogehe did + 6 x of he job. Afe Joseph lef, Logan woked x hous and did of he job. The complee job is epesened b: x Solving fo x gives: x x x x 1 4 ( 1) 4 3 ( x) x + 4 3x x hs 0 min Duke Univesi Talen Idenificaion Pogam Page 7 of

8 Lesson : Raional Expessions b) If he compan bills $50 pe hou each emploee woked, wha would he oal bill be Logan woked 1 h + hs + hs 0 min 5 hs 0 min. Joseph woked hs. The oal numbe of billable hous is 1 7 hs 0 min, which is 7 hous. The oal bill would be: 3 1 $50 7 $ Duke Univesi Talen Idenificaion Pogam Page of