Tgliche bungen - Mathematik   Klasse 11-I GK

1.) Zusammenfassen von Termen

1.1) Addition von Termen

nach Termen mit gemeinsamer Variablenkonstellation ordnen

zusammenfassen durch Ausklammern der gemeinsamen Variablenkonstellation     

  1. 3x + 7x - 14x + 35x  = x (3 + 7 14 + 35) = x 31  = 31x
  2. 33a 7b 10a + 8b 17a + 22b                                                                                  = 6a + 23b
  3. 13x + 5xy 4z 6x + 14xy + 16z 7z                                                                        = 7x + 19xy + 5z
  4. 18x + 14u 13x - 2x + 2u + 4x u                                                                          = 15u + 5x + 2x
  5. 12a + 4a + 17a - 3a - 5x                                                                                         = 21a + 9a - 5x
  6. 25ax 13x + 12x                                                                                                           = 25ax + 12x  13x
  7. 14uvw + 5uv 22w + 35uw 13uvw + 21w 35uw + 25u - 12v - 24u + 13v = u + v - w + 5uv + uvw
  1. 17y + 13x - 14ax - 7x + 20ax + 15bx 13xy + 10bx 4xy                               = 6x + 25bx + 6ax
  2. 2a - 2b - 3ab + 5b + 10a - 6a + 6ab                                                                = 6a + 3b + 3ab
  3. 0,8x + 0,2a - 1,2a 1,7x - 0,3a + 0,7a + 1,9x + a + 2a                                = -0,5a + 1,9a + x

1.2) Multiplikation von Termen

Koeffizienten und Variablen gesondert multiplizieren

bei Variablen gegebenenfalls Potenzschreibweise (und gesetze) nutzen    

  1. 47x                                                                                       = 28x
  2. 3x20y                                                                                   = 60xy
  3. 14u2v                                                                                   = 28uv
  4. 4a7c2x                                                                                = 56acx
  5. 3s4t5s                                                                                 = 60st
  6. 7s2s3t                                                                                 = 42st
  7. 2x3y4x2y  = 2342 xxyy  =  48xy
  8. 3xy5x3y                                                                             = 45xy
  9. 5x3y4xy                                                                              = 60xy
  10. 3xy7xz                                                                                  = 21xyz
  11. 3xz4y3yz                                                                             = 36xyz
  12. 2x4y2y3x                                                                          = 48xy
  13. 3a4ab2ab                                                                         = 24a4b
  14. 2a3x4ax5xz2a                                                              = 240a5x4z

1.3) Vermischte Aufgaben

erst Multiplikationsterme vereinfachen, dann zusammenfassen  

  1. 3a2b 4ab + 3x5y 2x3y                                                           = 2ab + 9xy
  2. 4x3y + 2u2v + 2u4v 3y2x                                                        = 12uv + 6xy
  3. 2y3z 2x4z + 3xyz + 2x5xz + 4z5y                                       = 3xyz + 2xz + 26yz
  4. 3x4y + 10x3a 5a2ab 2y2xy + 7ax - x2a + 15a2b      = 20ab + 35ax + 8xy
  5. 3u7 12a2b + 8x5y3z 7x4y 2y - 7u3u 15yz8x + y3y + 14y2x= y - 24ab
  1. 3x2x - x5x + 2x5x2x10x 2x3x + 10x4                        = 205x4
  2. 17xyz 5x3y + y20x 2yz3x + z5xy + 5y10x                       = 55xy + 16xyz
  3. 3a4ab + 10a2ab 6b2a - 3a2ab + ab6ab + 3ab4a  = 20ab + 24ab + 6ab - 6a4b

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2.) Rechnen mit Termen

2.1) Addition und Subtraktion

gleiche Vorzeichen:  Betrge addieren, Vorzeichen bleibt

ungleiche Vorzeichen: Betrge subtrahieren, Vorzeichen von Zahl mit grerem Betrag

Additionszeichen zwischen Termen kann wegfallen

  1. (-5) + (-4)                                                                     = -9
  2. (-3,5) + (-7) + (-5) + (-3)                                          = -18,5
  3. 7,4 + 5  =  (-7,4) + (+5)  =  -(7,4 5)  =  -2,4
  4. 3 11                                                                              = -8
  5. 5 + 3 11 10 + 3 5                                               =-25
  6. 5,2 7,4 + 3,2 4,3 5,3 + 3,4                                 = -5,2
  7. 1,5x + 0,9x 2,5x + 1,1x 3x + 5                           = -5x + 5
  8. 3y 3x + 3y + 5x 8y 10x                                       = -8x 2y
  9. 4s + 6t 6s 4t + 2t 2s                                            = -4s + 4t
  10. 3,4p 0,4x + 0,4p 5,4x + 4,6p + (-9,4p)               = -p 5,8x
  11. 12xy 12x 12y 18xy + 6x + 4y                             = -6x 6xy 8y
  12. 1,4s 0,4t + 1,4st 0,9s 0,1t 0,6st                      = 0,5s + 0,8st 0,5t
  13. 1,5t + 3,7s + (-1,7s) 1,5t + 5t 2s                        = 2t
  14. 3t 4x + 7t 10t 14t + 2x 10t + 2x                      = -24t 
  15. 1,2a 0,2b 1,8a 1,2b + 0,2a + 0,8c 0,8a          = -1,2a 1,4b + 0,8c

2.2) Multiplikation und Division

Gesamtvorzeichen Minus bei ungerader Anzahl von Minuszeichen

Punktrechnung (RO zweiter Stufe) geht vor Strichrechnung (RO erster Stufe)    

  1. 3,25                                                                                                                                                     = 16
  2. (-3,2)(+5)                                                                                                                                          = -16
  3. (-3,2)(-5)                                                                                                                                           = 16
  4. 4,2:(-1,2)                                                                                                                                             = -7/2   = -3,5
  5. (-4,2):(-2)                                                                                                                                           = 2,1
  6. 1,7(-2)                                                                                                                                                = -3,4
  7. (-4,8)(-1)                                                                                                                                           = 4,8
  8. (-3)(-4)(-2)(+4)                                                                                                                            = -96
  9. 7x(-2y)(-0,5x)                                                                                                                                 = 7xy
  10. (-5y)3y(- 0,2 x)(-2)                                                                                                            = -6xy
  11. 3x +(-4x)(5y) (6x3y) +(-5x)(-2) = -3x +(-20xy) 18xy +(+10x) = -3x +10x20xy 18xy = 7x 38xy
  1. 7s3 (-4s)(-6) + (4t)(-2) 7t                                                                                                     = -3s 15t
  2. (-3x)(2y)(-4z) + (-2z)(-x)(-12y) + (-0,5y)(24xz)(-2) (-8xyz)(-6)(-0,5)              = 0xyz = 0
  1. (-3x)(-3x)(2x)(-2y) (7x)(-6y) + (14y)(-3x) (12x4)(3y)            = 0xy - 72x4y = -72x4y
  2. (-2a)(-2a) + (0,5 a)( - 0,5 )(8a) ( b)(- b) + ( 0,5 b)(-                                     = 2a - b

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3.) Rechnen mit Klammern

3.1) Auflsen von Klammern

Plusklammer: Pluszeichen und Klammer fallen weg; alle Terme in der Klammer behalten ihr Vorzeichen bei

Minusklammer: Minuszeichen und Klammer fallen weg; von allen Terme in der Klammer ist das Entgegen- gesetzte zu bilden

Mehrfachklammern: schrittweise von innen nach auen auflsen  

  1. 7 + (+ x 12 )                                            =  7 + x 12                                                  =  x 5
  2. 2s (+ 4s 4 )                                           =  2s 4s + 4                                                =  -2s + 4
  3. +(+ 3x 2y ) ( +4x 5 5y )               =  3x 2y 4x + 5 + 5y                              =  -x +3y + 5
  4. 24k -(+40k + 16k - 4) (+24k + 6)     =  24k - 40k 16k + 4 24k 6            =  8k - 64k 2
  5. 2x[+5x -(+5x 2)]+(-3x + 2) + 5x =  2x [ 5x - 5x + 2 ] 3x +2+5x                                                                       =  2x 5x + 5x 2 3x + 2 + 5x          =  4x
  6. 4t [+3s (+2t 3s ) + 6 ] 6               =  4t [ 3s 2t + 3s + 6 ] 6                                                                       =  4t 3s + 2t 3s 6 6                          =  -6s + 6t 12
  7. (+4a3)[(+3x+x)(+4x+4x)(-3xx)] =  -4a + 3 [ 3x + x 4x 4x + 3x +x]                                                                          =  -4a +3 3x -x + 4x +4x -3x - x     =  -4a + 3
  8. (-3x+2x+5)(+4x+2x+5) +(+6x-10) = 3x 2x -5 4x -2x 5 +6x - 10    =  x - 20

3.2) Distributivgesetz

jeder Summand in der Klammer ist mit dem (vorzeichenbehafteten) Faktor auerhalb der Klammer zu multiplizieren

  1.  5( 3x 2 )                                                                                                               =  15x 10

  2.  7( 4s 3t ) - 8( 2s + 5t )                =  28s 21t 16s 40t                            =  12s 61t

  3.  4t( 2s 3 ) + 2s( 4t 3 )             =  -8st + 12t + 8st 6s                             =  12t 6s

  4.  ( 5k 4 )5 - 2( 8k 10 )                 =  25k 20 16k + 20                              =  9k

  5.   2x( 2 2x + 5y ) 2y( 5x 2 )     =  4x 4x + 10xy 10xy + 4y              =  -4x + 4x + 4y

  6.   3(4a3b3b+2a)4(-3a1,5a =  -12a +9b +9b -6a +12a +6a         =  9b + 9b

  7.  4x( x y) [ 3x( x +y) 2x( x y)] =  4x - 4xy [ 3x + 3xy 2x + 2xy]                                                               =  4x - 4xy 3x - 3xy + 2x - 2xy      =  3x - 9xy

  8.  5(-2a+b)2[(3a5b)4+(129a)a]= -10a + 5b 2[ 12a 20b + 12a 9a]                                                             =-10a +5b 4a +40b 24a +18a         =  18a - 58a + 45b

  9.             4[ 4s(ts) (s+4t)3] 7s(s+8)     =  4[ 4st 4s - 3s 12t ] 7s - 56s                                                                                =  16st 16s -12s 48t 7s -56s         =  -23s - 68s + 16st 48t

  10. 10.  (-2)[(-3t)(2a b) (-4a)(10 2t)] (-6)(3a + 7t) = (-2)[-6at + 3bt + 40a 8at] + 18a +42t                                                            = 12at 6bt 80a + 16at + 18a + 42t   = -62a + 28at 6bt + 42t

3.3) Multiplikation von Summen

  jeder Summand der einen Klammer ist mit jedem Summanden der anderen Klammer zu multiplizieren

  beachten Sie die Struktur der Terme (Summe, Differenz, Produkt, Quotient, Potenz)   

  1. ( 7t + 8e )( x + 1 )                                                                                  = 7tx + 7t + 8ex + 8e  

  2. ( 5t - 8t )( 2t + 4 )                                                                               = 10t5 16t4 + 20t - 32t 

  3. ( 4x + 2,5y )( -x 2y )                                                                          = -4x - 8xy 2,5xy 5y 

  4. (2x3y2)(42x+2y)=8x 4x +4xy 12y +6xy 6y -8 +4x 4    = -4x + 12x + 10xy 16y 6y - 8y 

  5. ( x 2)( -3x 5 y) = -3x -5x xy +6x +10 +2y                            = -3x + x xy + 2y + 10

  6. (2x+y)(xy)[(x+3)(2x4y)+3(4y2x)]=2x-2xy+xyy-[2x-4xy+6x-12y+12y6x] = + 3xy y 

  7. ( a + b )( 2a 1 )b       = (2a - a + 2ab b)b                                  = 2ab ab + 2ab - b 

  8. (x 1)(x + 2x 1) (x - 2x + 1)(x + 2) = x +2x -x x -2x +1 = x + x - 3x + 1 

  9. a(3a b + 2ab 1) (a b)(3a b) = 3a -ab +2ab a [3a -3ab ab +b] =  2a2b + 3ab a b 

  10. (-3x)(2x+3y)-(2x3y)(3+4x2y)=-6x-9xy[6x+8x-4xy9y12xy+6y]= -14x - 6x + 7xy + 9y 6y

3.4) Ausklammern (Faktorisieren)

gemeinsame Faktoren aller Summanden vor die neugebildete Klammer schreiben 

die Restglieder in der Klammer erhlt man durch Division der Ausgangssummanden mit dem auszuklam- mernden Term   

  1. 4xy + 2x 6xy          = 22xy + 2x 23xy                                             =  2x ( 2y + 1 3y )  = 2x(-y +1)

  2. a + b - c         = 2a + 5b - c                                                 =  (2a + 5b c)

  3. 42abc - 30abc 28abc = 237abcc 235abbc 227aabc     =  2abc ( 21c 15b 14a)

  4. a - 2a + 4a                          =  aaa 2aa + 22a                                        =  a ( a - 2a + 4 )  

  5. a b                        =  -1a 1b                                                                         =  -1 ( a + b )

  6. a( 5x 2y ) b( 5x 2y )                                                                                    =  ( a b ) ( 5x 2y )

  7. x( 3a b ) + y(-3a + b ) + 3az bz  =  x( 3a b ) - y( 3a b ) + z( 3a b ) =  (3a b) (x y + z)  

  8. 3at 5bt + 3as 5bs  = 3a ( t + s ) 5b ( t + s )                                           =  ( 3a 5b ) ( t + s )

  9. 3bx7ax+6b-14ab12ab+28a=3bx7ax+23bb27ab-223ab+227aa=(3b-7a)(x+2b-4a)                                                                                                                                                                                                                    oder =  3b (x + 2b) +a (-7x - 26b + 28a)                                                                                                     oder =  x (3b 7a) + 2 (3b - 13ab +14a)                                                                                                     oder  =  b (3x + 6b 26a) + 7a (-x +4a) 

  10. sxytxy+3ps3pt4as+4at=sxxytxxy+3ps3pt4as+4at=xy(st)+3p(st)4a(st)                                                                                                                                  =  (s t) (xy + 3p 4a)

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4.) Binomische Formeln

benutzen Sie die Formeln aus dem Tafelwerk     

verwandeln Sie Binome in Summen  bzw. Summen zurck in Binome

  1. ( s + t )                                             =  s + 2st + t   

  2. ( 12s + 4t )                                       =  144s + 96st + 16t

  3. ( 5x + 6y )                                        =  25x + 60xy + 36y

  4. ( 9a + 6b )                                        =  81a + 108ab + 36b

  5. 4s + 12st + 9t                                 =  (2s + 3t)

  6. 9x + 4y + 12xy                               =  (3x + 2y)

  7. t + 2t + 1                                           =  (t + 1)

  8. 400n + 9m + 120mn                      =  (20n + 3m)

  9. ( 1,2x + 5y )                                     =  1,44x + 12xy + 25y

  10. ( s t )                                              =  s - 2st + t

  11. ( 8n 0,5m )                                    =  64n - 8mn + 0,25m

  12. k - 24kl + 144l                                =  (k 12l)

  13. 16s + 25t - 40st                             =  (4s - 5t)

  14. x -x + 0,25                                       =  (x - 0,5 )

  15. x - 2x + 1                                          =  (x 1)

  16. 25d + 36c - 60cd                            =  (5d 6c)

  17. 9x + 64y - 24xy                              = kein vollstndiges Quadrat  (-24xy ist kein doppeltes Produkt!)

  18. x4 + 100 20x                                  =  (x - 10)

  19. 4x + 0,25 y - xy                            =  (2x - 0,5 y)

  20. ( 2s + 5t )( 2s 5t )                           =  4s - 25t

  21. ( x + 12s )( x 12s )                        =  x - 144s2

  22. ( 16x 11y )( 16x + 11y )               =  256x - 121y

  23. 9y - 25                                                =  (3y + 5)(3y 5)

  24. 169 144x                                         =  (13 + 12x) (13 12x)

  25. 196a - 225b                                     =  (14a + 15b)(14a 15b)

  26. 256p - 16pq + 0,25 q                     =  (16p - 0,5 q)

  27. x - y                                       =  ( x + y) ( x - y)

  28. 36a - 4ab + b                                 =  (6a 2b)

  29.  + 2x + 100x                              =  ( + 10x)

  30. ( x - 6x )                                     =  x4 x + 36x

  31. x4 y4                                                   =  (x + y)(x - y)

  32. 4x4 25x                                            =  (2x + 5x)(2x - 5x)

  33. a -  b                                   =  ( a + b) ( a + b)

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5.) Faktorisieren von Termen

  1. 3as + 5at 6bs 10bt                                 =  a(3s + 5t) 2b(3s + 5t)                      =  (a 2b) (3s + 5t)
  2. 15a + 30ab + 15b                                     =  15 (a + 2ab + b)                               =  15 (a + b)
  3. 8g - 8h                                                        =  8 (g - h)                                             =  8 (g + h) (g h)
  4. 18x + 12ux + 2ux 27y 18uy 3uy   =  2x (9+6u+u)-3y(9+6u+u)             =  (2x 3y) (u + 3)
  5. 18s - 60st + 50st                                     =  2s (9s - 30st + 25t)                          =  2s (3s 5t)
  6. 6axz+12avx+6ayz+12avy = 6az(x+y)+12av(x+y) =(6az+12av)(x+y)=  6a(z+2v)(x+y)
  7. 5a-5b+4a+8ab+4b=5(a-b)+4(a+2ab+b)=5(a+b)(ab)+4(a+b)      =  (a+b)[5(a-b)+4(a+b)]
  8. 4x-2xy+12x+3y-12xy=2x(2xy)+3(4x-4xy+y)=2x(2xy)+3(2xy)       =  (2xy)[2x+3(2xy)]
  9. 4x5 24x4 + 36x                                         =  4x (x - 6x + 9)                                  =  4x (x 3)
  10. 3x - 48x                                                        =  3x (x - 16)                                           =  3x (x + 4) (x 4)

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6.) Bruchterme

6.1) Grundmenge und Definitionsbereich

Aufgabe: Bestimmen Sie den Definitionsbereich folgender Bruchterme bzgl. des Grundbereichs der rationalen Zahlen!

beachten Sie: ein Bruch ist nicht definiert , wenn der Nenner Null und der Zhler ungleich Null ist

sollten beide Terme fr eine Zahl Null werden, kann der Bruch gekrzt werden

  1.                                       NR: 3 x    0                     DB = { x  R |  x ≠ 3}
  2.                                       NR: a + 4    0                     DB = { a  R |  a ≠ -4}
  3.                               NR: 2x    0                          DB = { x  R |  x ≠ 0}
  4.                                  NR: 3 8a    0                    DB = { a  R |  a ≠ 3/8}
  5.                NR: a 7 ≠ 0 und a ≠ 0      DB = { a  R |  a ≠ 0 und a ≠ 7}
  6.                                    NR: b + 4    0                   DB = { b  R }
  7.            Krzen von x mglich                            DB = { x  R}
  8.    NR: 12x20 ≠ 0     DB = { x  R | x ≠ 5/3 }
  9.               NR: x 1    0                      DB = { x  R |  x ≠ 1}
  10.      NR: 3x + 5    0                   DB = { x  R |  x ≠ - 5/3}

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6.2) Erweitern von Bruchtermen

Aufgabe: Erweitern Sie den Bruchterm mit dem angegebenen Erweiterungsfaktor!

beachten Sie: Erweitern eines Bruches bedeutet , Zhler und Nenner mit dem gleichen Term multiplizieren

nutzen Sie danach ihre Fertigkeiten beim Multiplizieren von Summen

  1.            erweitern mit  2x       
  2.            erweitern mit  ( 2a 4 )        
  3.          erweitern mit  ( 3xy 2 )        
  4.          erweitern mit  ( a b )        
  5.         erweitern mit  ( a + b )        

    Aufgabe: Erweitern Sie den Bruchterm so dass der angegebene Nenner entsteht!

    ermitteln Sie zunchst den Erweiterungsterm durch Division des neuen Nenners durch den alten Nenner
    erweitern Sie danach den Bruch wie in den vorhergehenden Aufgaben
  6.         auf den Nenner:  10b 15            
  7.            auf den Nenner:  10ab - 15ab       
  8.          auf den Nenner:  12xz + 18xz       
  9.          auf den Nenner:  a - b        

    Aufgabe: Machen Sie die jeweils gegebenen Bruchterme gleichnamig!

    ermitteln Sie zunchst den Hauptnenner beider Brche als kleinstes gemeinsames Vielfache (kgV) beider Nenner

    erweitern Sie dann beide Brche auf diesen Hauptnenner

  10.            auf den Nenner:  9a + 6ab + b        
  11. ;             
  12.              ;
  13.             
  14.             
  15.             

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6.3) Krzen von Bruchtermen

ermitteln Sie zunchst den grten gemeinsamen Teiler (ggT) von Zhler und Nenner (gegebenenfalls durch geschicktes Ausklammern)

krzen sie dann gemeinsame Faktoren in Zhler und Nenner

  1. =  
  2. =
  3. nicht krzbar, weil Zhler nicht in Produkt zerlegbar
  4. nicht krzbar, weil der Nenner nicht in Produkt zerlegbar

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6.4) Vereinfachung von Bruchtermen, Polynomdivision

Aufgabe: Vereinfachen Sie den Bruchterm durch Zerlegen von Zhler und Nenner in

                  Faktoren und anschlieendes Krzen!

Verfahren der Polynomdivision siehe Arbeitblatt  

 

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7.) Rechnen mit Bruchtermen

7.1) Addition gleichnamiger Bruchterme

Zhler addieren, Nenner beibehalten

  1. + = = x2
  2. - =
  3. - - =
  4. - - = =
  5. - - = =

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7.2) Addition ungleichnamiger Bruchterme

ermitteln Sie zunchst das kleinste gemeinsame Vielfache (kgV) der beiden Nenner (Hauptnenner)

erweitern sie dann beide Brche auf den gemeinsamen Hauptnenner

addieren sie nun die gleichnamigen Brche

  1. + =
  2. -  =
  3. - -  =
  4. - =
  5. + = 
  6. +  = 
  7. +  = 
  8. -  +  = =
  9. -  = 
  10. -  =  =
  11. -  +  =  =  = 
  12. +  =  = 
  13. -  + =  =    = = 
  14. -  +  -  =  =  =  = 

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7.3) Multiplikation von Bruchtermen  

multiplizieren Sie zunchst Zhler mit Zhler und Nenner mit Nenner

krzen sie dann gemeinsame Faktoren in Zhler und Nenner und vereinfachen Sie den Bruch  

  1. 20xy  =  = 16uy
  2. =  
  3. =  
  4. =  
  5. =  
  6. =  
  7. = apr
  8.   ( 20x 25y )  = xy  
  9. =
  10. =  
  11. =  
  12. ( a + 2 )  
  13.  

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7.4) Division von Bruchtermen 

ersetzen Sie zunchst die Division mit dem 2.Bruch (Divisor) durch eine Multiplikation mit dessen Reziproken 

gehen sie nun vor wie beim Multiplizieren zweier Brche  

  1. =  
  2. : 2x  =  
  3. 4x :   =  
  4. =  
  5. =  
  6. =
  7. =
  8. =  
  9. =  
  10. =  
  11. =  
  12. =  
  13. =
  14. =

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8.) Lineare Gleichungen und Bruchgleichungen

8.1) Grundbereich, Definitionsbereich und Lsungsmenge

 

Bestimmen Sie den Definitionsbereich und die Lsungsmenge der Gleichung jeweils fr den Grundbereich der rationalen Zahlen Q, der ganzen Zahlen Z und der natrlichen Zahlen N! 

           

  1.   3 + x  =  2                    DB = R                                         x = -1           LQ = {-1}        LZ = {-1}          LN =

  2.   7 x  =  12 2x          DB = R                                         x = 5             LQ = {5}          LZ = {5}           LN = {5}         

  3.   =  2                          DB = {xR|x0}                       x = 4             LQ = {4}          LZ = {4}           LN = {4}  

  4. =                      DB = {xR|x≠2 und x≠0,25}  x = 1            LQ = {1}          LZ = {1}           LN = {1}

  5.   =  1                      DB = {xR|x≠0,75}                x = -1           LQ = {-1}        LZ = {-1}         LN =

        home                                                            zurck                                                            top

8.2) Bestimmung der Lsungsmenge einfacher Gleichungen

 

1.      5x + 7  =  3x + 25   /-3x                                                                                                2.           2x 17  =  17 3x   /+3x

 2x + 7  =  25   /-7                                                              5x 17  =  17   /+17

        2x  =  18   /:2                                                                     5x   =  34   /:5

        x = 9                                                                                        x = 6,8 

   

3.      2x 9( 3x + 1 )  =  3( x 7 ) + 5( x + 2 )         4.   3( 5x 2 ) + 7( 4 x )  =  4(5 + x ) 11( 2x 7 ) + 4x

       2x 27x 9  =  3x 21 + 5x + 10                              15x 6 + 28 7x   =  20 + 4x 22x + 77 + 4x

              -25x 9  =  8x 11   /+25x                                                    8x + 22  =  -14x  + 97   /+14x

                        -9   =  33x 11   /+11                                                  22x + 22  =   97   /-22

                          2   =  33x   /:33                                                                    22x  =  75   /:22

                         x   =   ( 0,0606)                                                                 x  =     (3,40909)

 

5.      13( 2x 6 ) 5( 3 + x )  =  4( 23 4x ) + 2( x 7 ) + 7( x 13 )

                    26x 78 15 5x  =  92 16x + 2x 14 + 7x 91

                                     21x 93  =  -7x 13   /+7x

                                     28x 93  =  -13   /+93

                                             28x  =  80   /:28 

                                                 x  =    (2,857142)  

6.      1,5( x 4 ) - 0,5 ( 7 8x )  =  0,75 ( 16x 7 ) 8( 12x + 0,5 ) + 74,5x

                          1,5x 6 3,5 + 4x  =  12x 5,25 - 96x 4 + 74,5x

                                         5,5x 9,5  =  -9,5x 9,25   /+9,5x

                                         15x 9,5   =  -9,25   /+9,5

                                                    15x  =  0,25   /:15

                                                         x  =      (0,0166)  

7.              ( x 7 ) - 15( x + 7 )  =  ( x 2 )( x + 2 ) 5( 3 + 4x ) + 8x

         x - 14x + 49 15x 105  =  x - 4 15 20x + 8x  

                              x - 29x 56  =  x - 12x 19   /-x

                                   - 29x 56  =   -12x 19   /+29x + 19

                                              - 37  =  17x

                                                   x  =      (-2,176470588)  

8.               3x( x 5 ) + 2( 3 2x )  =  2x( 9 + 2x ) + ( 5 3x )( 2 5x ) 8x

           3x - 15x + 2( 9 12x + 4x ) = 18x + 4x + 10 25x 6x + 15x - 8x

                3x - 15x + 18 24x + 8x  =  18x + 4x + 10 25x 6x + 15x - 8x

                                   11x - 39x + 18  =  11x - 13x + 10   /-11x

                                             - 39x + 18  =  - 13x + 10   /+39x -10

                                                         + 8  =  26x   /:26

                                                             x  =     (0,307692)  

9.      ( x 5 )( 5 x ) 4x( 2 x )  =  ( 7x 2 )( x + 5 ) 2x( 2x + 3 )

            5x x - 25 + 5x 8x + 4x  =  7x + 35x 2x 10 4x - 6x

                                     3x + 2x 25  =  3x + 27x 10   /-3x

                                                2x 25  =  27x 10   /-2x + 10

                                                       -15  =  25x   /:25

                                                           x  =    (-0,6)  

10.                                                       5x( x 2 ) + 0,2 ( 4x 2 )( 8 5x ) + 5x  =  2( 6x + 4 )( 7 2x ) x( 7 5x )

           5x( x - 4x + 4 ) + 0,2( 32x 20x - 16 + 10x ) + 5x  =  2( 42x 12x + 28 8x ) 7x + 5x

                          5x - 20x + 20x + 6,4x 4x - 3,2 + 2x + 5x  =  84x 24x + 56 16x 7x + 5x

                                                               5x - 24x + 33,4x 3,2  =  5x - 24x + 61x + 56   /-5x

                                                                       -24x + 33,4x 3,2  =  -24x + 61x + 56   /+24x

                                                                                      33,4x 3,2  =  61x + 56   /-33,4x - 56

                                                                                                -59,2  =  27,6x   /:27,6

                                                                                                                                  x  =     (-2,14493)

    11.                     4 -   =  3 -   /20

                420 -   = 320 -

                    80 - 4( 7 3x )  =  60 - 2( 3 7x ) + 10( x + 1 )

                        80 28 + 12x  =  60 6 + 14x + 10x + 10

                                12x + 52  =  24x + 64   /-24x

                               -12x + 52  =  64   /-52

                                        -12x  =  12   /:( -12)

                                              x  =  -1                              Probe:   2  =  2  w.A.

    12.                    - 4  =    1 -    /12

               - 124  = 121 -

                 4( 4x 1 ) 48  =  12 - 2( x 4 ) + 3( 3x + 5 ) - 317

                       16x 4 48  =  12 2x + 8 + 9x + 15 51

                

                              16x 52  =  7x 16   /-7x

                                9x 52  =  -16   /+52

                                        9x  = 36   /:9

                                           x  =  4                             Probe:   1  =  1  w.A.

    13.                                    + 6  =     /30

            ( x + 3 ) + 306  = 

               10( 7x 2 ) - 24( x + 3 ) + 180  =  45( x + 2 )

                           70x 20 24x - 72 + 180  =  45x + 90

                                                         46x 88  =  45x + 90   /-45x

                                                             x + 88  =  90   /-88

                                                                      x  =  2          Probe:   6  =  6  w.A.

    14.                             / 120  

                 

          8( 2x 3 ) - 6( 4x 9 )  =  4( 8x 27 ) 5( 16x 81 ) - 39

                  16x 24 24x + 54  =  32x 108 80x + 405 27

                                      -8x + 30  =  -48x + 270   /+48x

                                      40x + 30  =  270   /-30

                                               40x  =  240   /:40

                                                   x  =  6                      Probe:      w.A.

8.3) Bruchgleichungen

        Ermitteln Sie die Lsungsmenge folgender Bruchgleichungen, in dem Sie:

-         den Definitionsbereich bestimmen

-         mit dem Hauptnenner beide Seiten multiplizieren

-         die bruchfreie Gleichung lsen

-         berprfen, ob die Lsung im Definitionsbereich liegt

             -         eine Probe in der Ausgangsgleichung machen

 

1.      =  - - 33 /4x                          DB = { x  R |  x ≠ 0} }

     5 = -28 132x /+28

   33 = - 132x

     x = -1/4 = -0,25          Probe: -5 = -5 w.A.          L = {-1/4}  

 

2.          =    /(2-x)(3-x)                     DB = { x  R |  x ≠ 2 und x ≠ 3 }

3(3 x) = 2(2 x)

    9 3x = 4 2x /+3x   /-4

            5 = x                    Probe: -1 = -1 w.A.          L = {5}

 

3.       +   =  /(2-3x)(x+3)   DB = { x  R |  x ≠ 2/3 und x ≠ -3 }

x + 3 + 2(2 3x) = 3(2 3x)

                     7 5x = 6 9x /+9x 7

                           4x = -1

                             x = -1/4 = -0,25       Probe: 12/11 = 12/11 w.A.         L = {-1/4}

 

4.        =   + 3 /2(2-x)                    DB = { x  R |  x ≠ 2}

             7 = 10 + 3(4 2x)

             7 = 22 6x /+6x 7

           6x = 15

             x = 5/2 = 2,5                        Probe: -7 = -7 w.A.          L = {5/2}

 

5.                       +     =  2 /(2+x)(3-x)               DB = { x  R |  x ≠ -2 und x ≠ 3 }   

2x(3 x) + 3(2 + x) = 2(2 + x)(3 x)

        6x 2x + 6 + 3x = 2(6 + x x)

               -2x + 9x + 6 = -2x + 2x + 12 /+2x

                           9x + 6 = 2x + 12 /-2x 6

                                  7x = 6

                                    x = 6/7           Probe: 2 = 2 w.A.       L = {6/7}

 

6.            =   /6(3x+2)                    DB = { x  R |  x ≠ -2/3 }

3(7x 4) = 2(5x 6)

   21x 12 = 10x 12 /-10x + 12

            11x = 0

                x = 0                  Probe: -1 = -1 w.A.          L = {0}

 

7.                           -   =   - /2(3+x)(2+x)       DB = { x  R |  x ≠ -3 und x ≠ -2 }

8(2 + x) 14(3 + x) = 3(3 + x) 5(2 + x)

      16 + 8x 42 14x = 9 + 3x 10 5x

                        -6x 26 = -2x 1 /+6x + 1

                                 -25 = 4x

                                     x = -25/4 = -6,25        Probe: 92/221 = 92/221 w.A.         L = {-25/4}

 

 

8.                       =  +  /(x+7)(x-7)           DB = { x  R |  x ≠ 7 und x ≠ -7 }

(3 + x)(x + 7) = x + 9(x 7)

 x + 10x + 21 = x + 9x 63 /-x

          10x + 21 = 9x 63 /-9x 21

                        x = -84          Probe: 81/91 = 81/91 w.A.          L = {-84}

 

 

9.                 +   =  /6(5+x)(5-x)       DB = { x  R |  x ≠ 5 und x ≠ -5 }

3x(5 + x) 2(5 x)  =  6(0,5x + x)

   15x + 3x - 10 + 2x  =  3x + 6x

           3x + 17x 10  =  3x + 6x /-3x

                      17x 10  =  6x /-6x + 10

                                11x = 10  

                                    x = 10/11         Probe: 32/585 = 32/585 w.A.         L = {10/11}

 

 

10.                                   -   =   -  /2(1-x)(1+x)          DB = { x  R |  x ≠ 1 und x ≠ -1 }

10(1 + x) 6x(1 x)  =  6x - 3(1 x)

     10 + 10x 6x + 6x  =  6x - 3 + 3x

                6x + 4x + 10  =  6x + 3x 3 /-6x

                           4x + 10  =  3x 3 /-3x 10

                                       x = -13                        Probe: -81/28 = -81/28 w.A.              L = {-13}

 

 

11.              -   =    /2(5+x)(2-x)  DB = { x  R |  x ≠ 2 und x ≠ -5 }

8x(2 x) 8x(5 + x)  =  16x(2 x)

  16x 8x - 40x 8x  = 32x 16x

                    -16x - 24x  =  -16x + 32x /-8x

                                -24x  =  32x /+24x

                                       0 = 56x /:56

                                       x = 0              Probe: 0 = 0 w.A.       L = {0}

 

12.              +   =   +  /3x(x-5)(x+2)     DB = { x  R |  x ≠ 5 und x ≠ -2 und x ≠ 0 }

9x(x + 2) + 6x(x 5) = 12x(x 5) + 3(x - 3x 10)

 9x + 18x + 6x - 30x  =  12x - 60x + 3x - 9x - 30

                      15x -12x  =  15x - 69x 30 /-15x

                                -12x  =  -69x 30 /+69x

                                 57x  =  -30

                                      x  =  -10/19       Probe: 57/70 = 57/70 w.A.     L = {-10/19}

13.     +   =   + /2x(x-3)       DB = { x  R |  x ≠ 0 und x ≠ 3 }

2x + 6(x 3)  =  6(x 3) + x

   2x + 6x 18  =  6x 18 + x

            8x 18  =  7x 18 /-7x + 18

                        x = 0           gehrt nicht zum DB      Probe: nicht mglich       L =

  

14.   -   =  /7(3+2x)   DB = { x  R |  x ≠ -3/2 }

              7(3 + 2x) 7x  =  3,5(3 + 2x)

                 21 + 14x 7x  =  10,5 + 7x

                           21 + 7x  = 10, 5 + 7x /-7x 10,5

                                 10,5  =  0  Widerspruch       Probe: nicht mglich              L =

    

15.                  +   =   - /2x(3+x)(3-x)    DB = { x  R |  x ≠ 3 und x ≠ -3 und x ≠ 0 }

4x(3 x) + 2x(3 + x)  =  -4x 2(9 x)

   12x 4x + 6x + 2x  =  -4x 18 + 2x

                     -2x + 18x  =  -2x - 18 /+2x

                                   18x = -18

                                        x = -1             Probe: 5/4 = 5/4 w.A.         L = {-1}

        home                                                            zurck                                                            top

9.) Ungleichungen und Bruchungleichungen

9.1) Bestimmung der Lsungsmenge einfacher Ungleichungen  

-          beachten Sie die Unterschiede bei den quivalenzumformungen zwischen Gleichungen und Ungleichungen und geben Sie die Lsungsmenge im Bereich der reellen Zahlen an!

1.  3 + 2x    -7 3x /+3x 3                        2.  4 + 7x  >  3x 16 /-3x 4

             5x  -10 /:5                                                        4x > -20 /:4

               x  -2          L = {x R| x  -2 }                     x > -5            L = {x R| x > -5 }

 

 

3.  13x 5  10x + 23 /-10x + 5                   4.  25x 12  <  20x + 28 /-20x + 12

               3x 28 /:3                                                           5x < 30 /:5

                 x         L = {x R| x 28/3 }                     x < 6           L = {x R| x < 6 }

 

 

5.  3x + 5  <  5x 7 /-5x - 5                             6.  x 3  <  2x + 5 /-2x + 3

            -2x < -12 /:(-2)                                                 -x < 8 /(-1)

                x > 6             L = {x R| x > 6 }                     x > -8                L = {x R| x > -8 }

 

 

7.  2x 0,2  <  4x + 3 /-4x + 0,2                      8.  2x + 2,5   6x 1,5 /-2x + 1,5

              -2x < 3,2 /:(-2)                                                       4  4x /:4

                  x > -1,6      L = {x R| x > -1,6 }                      x          L = {x R| x  1 }

 

 

9.  0,5x 7  -2x 8 /+2x + 7                      10.  1,5( x 3 ) + x   2x -

              x  -1 /                                               -1,5x + 4,5 + x  2x -

                 x   -       L = {x R| x -2/5 }                            -0,5x  2x -  /-2x

                                                                                     

    - x - /(- )

      -4      -3      -2      -1       0       1        2       3                     x               L = {x R| x 3/20 }

Die Lsungsmenge einer Ungleichung ist in der Regel eine unendlich groe Zahlenmenge, die durch einen bestimmten Bereich auf der Zahlengerade veranschaulicht werden kann.

 

9.2) Bruchungleichungen  

Bestimmen Sie die Lsungsmenge der Bruchungleichungen, in dem Sie:

-          den Definitionsbereich der Ungleichung ermitteln

-          die Ungleichung mit dem Hauptnenner durchmultiplizieren

-          dabei eine Fallunterscheidung vornehmen fr den Fall, dass der Hauptnenner grer(gleich) Null bzw. dass der Hauptnenner kleiner als Null ist

-          die entstandenen Teilungleichungen lsen und mit der Fallbedingung vergleichen

-          die Gesamtlsungsmenge aus den Lsungsmengen der beiden Teilungleichungen zusammensetzen und graphisch veranschaulichen

-          mit geeigneten ganzen Zahlen aus der Lsungsmenge eine Probe machen

1.                                                >  0 /x         DB = { x R| x ≠ 0}

                                  x < 0                               x > 0

                       x  + 2  <  0 /-2                           x  +  2  >  0 /-2

                               x  <  -2                                         x  >  -2

                       L1 = { x R| x < -2}                 L2 = {x R| x > 0}             L = {x R| x < -2 und x > 0}  

               

                        -3             -2            -1             0              1              2              3

  2.                                                >  0 /x      DB = { x R| x ≠ 0}

                                  x < 0                           x > 0

                       3  -  x  <  0 /+x                          3  -  x  >  0 /+x

                              x  >  3                                         x  <  3

            L1 =                                 L2 = {x R| 0 < x < 3}        L = {x R| 0 < x < 3}

  3.                                              0 /x          DB = { x R| x ≠ 0}

                                  x < 0                           x > 0

                       x  -  5  0 /+5                     x  -  5  0 /+5

                               x  5                                    x   5

            L1 =                           L2 = {x R| 0 < x  5}           L = {x R| 0 < x  5}

 

4.                                              0 /3x       DB = { x R| x ≠ 0}

                 x < 0                           x > 0

                       1  - 2x    0 /+2x             1  - 2x  0 /+2x

                              2x    1 /:2                       2x  1

                                x   0,5                              x  0,5

               L1 =                         L2 = {x R| 0 < x  0,5}        L = {x R| 0 < x 0,5}

 

5.                                                >  0           DB = { x R| x ≠ 1}

                                                     >  0

                                                             3  >  0          L = {x R}  

6.                                              0 /(x+2)           DB = { x R| x ≠ -2}

                                  x < -2                          x > -2

                       x  -  1  0 /+1                    x  -  1  0 /+1

                               x  1                                   x  1

              L1 =                        L2 = {x R| -2 < x 1}                L = {x R| -2 < x 1}

7.                                              0 /(x-4)          DB = { x R| x ≠ 4}

                                  x < 4                           x > 4

                       2x  +  5  0 /-5                       2x + 5  0 /-5

                               2x  -5 /:2      &nbs