WISKUNDE
GRAAD 11
NOG OEFENINGE
  
Algebraiese breuke : antwoorde.
  
MATHEMATICS
GRADE 11
MORE EXERCISES
  
Algebraic fractions : answers.
  
  
                 x2 ─ 3x                   x(x ─ 3)
       1.  ─────────   =   ─────────
              x2 ─ x ─ 6              (x ─ 3)(x + 2)
  
  
                                                x
                                       =  ─────                        [ 1 ]
                                             x + 2
  
  
                6x2 ─ 21x                 3x(2x ─ 7)
       3.  ─────────   =   ───────────
             2x2 ─ 3x ─ 14          (x + 2)(2x ─ 7)
  
  
                                                3x
                                       =  ─────                        [ 3 ]
                                             x + 2
  
  
               2x2 + 5x ─ 12             (x + 4)(2x ─ 3)
       5.  ───────────   =   ───────────
               2x2 ─ 11x + 12           (x ─ 4)(2x ─ 3)
  
  
                                                   x + 4
                                            =  ─────                        [ 5 ]
                                                   x ─ 4
  
  
               ax ─ bx + ay ─ by             x2 ─ 2xy + y2
       7.  ─────────────  X  ───────────
                       x2 ─ y2                     a2 ─ 2ab + b2
  
                    (a ─ b)(x + y)           (x ─ y)2
              =  ─────────   X   ───────
                    (x + y)(x ─ y)           (a ─ b)2
  
  
                     (x ─ y)
              =  ──────                                                 [ 7 ]
                     (a ─ b)
  
  
              2x2 + 10x             2x(x + 5)
       2.  ────────   =   ─────────
                x2 ─ 25             (x ─ 5)(x + 5)
  
  
                                                2x
                                       =  ─────                        [ 2 ]
                                            (x ─ 5)
  
  
               x2 + 5x + 6            (x + 2)(x + 3)
       4.  ─────────   =   ─────────
               x2 + x ─ 6            (x ─ 2)(x + 3)
  
  
                                             x + 2
                                       =  ─────                        [ 4 ]
                                            (x ─ 2)
  
  
               x2 ─ 2x ─ 24            (x ─ 6)(x + 4)
       6.  ──────────   =   ──────────
                  2x2 + 8x                     2x(x + 4)
  
                                               x ─ 6
                                         =  ─────                      [ 6 ]
                                                  2x
  
  
               x2 ─ 2xy ─ 3y2           x2 + 2xy ─ 3y2
       8.  ───────────  X  ───────────
                  x2 ─ 9y2                 2x2 ─ 3xy + y2
  
                     (x ─ 3y)(x + y)             (x ─ y)(x + 3y)
              =  ───────────   X   ───────────
                    (x + 3y)(x ─ 3y)            (x ─ y)(2x ─ y)
  
  
                     (x + y)
              =  ──────                                                 [ 8 ]
                     (2x ─ y)
  
  
  
               ax ─ ay + bx ─ by             ax ─ ay ─ bx + by             ax + bx + ay + by
       9.  ─────────────   X   ─────────────   X   ─────────────
                       ax + ay ─ bx ─ by             a2 ─ b2                      ax ─ ay ─ bx + by
  
  
                     (x ─ y)(a + b)               (x ─ y)(a ─ b)             (x + y)(a + b)
              =  ───────────   X   ──────────   X   ──────────
                     (x + y)(a ─ b)               (a ─ b)(a + b)             (x ─ y)(a ─ b)
  
  
                     (x ─ y)(a + b)
              =  ───────────                                                                                                                             [ 9 ]
                     (a ─ b)(a ─ b)
  
  
  
  
                    a2 + 8a + 15                  ab ─ 3a ─ 2b + 6              ab + 3a ─ 5b ─ 15
      10.  ─────────────   X   ─────────────   X   ───────────────
                ab + 3a ─ 2b ─ 6                  a2 ─ 2a ─ 15                 ab ─ 3a + 5b ─ 15
  
  
                     (a + 3)(a + 5)               (a ─ 2)(b ─ 3)             (a ─ 5)(b + 3)
              =  ───────────   X   ──────────   X   ──────────
                     (a ─ 2)(b + 3)               (a ─ 5)(a + 3)             (a + 5)(b ─ 3)
  
  
              =  1                                                                                                                                                    [ 10 ]
  
  
                 a2 ─ a ─ 6              a2 ─ 2a ─ 3                a2 + 3a + 2
      11.  ─────────   X   ──────────   X   ──────────
                 a2 + 3a + 2             a2 + 4a + 3                 a2 + 4a + 4
  
  
                     (a ─ 3)(a + 2)               (a ─ 3)(a + 1)             (a + 1)(a + 2)
              =  ───────────   X   ──────────   X   ───────────
                     (a + 1)(a + 2)               (a + 1)(a + 3)             (a + 2)(a + 2)
  
  
                     (a ─ 3)(a ─ 3)
              =  ───────────                                                                                                                          [ 11 ]
                     (a + 3)(a + 2)
  
  
  
  
                 p2 ─ p ─ 2                 p2 ─ 3p + 2               p2 ─ 4p + 4
      12.  ──────────   X   ──────────   ÷   ──────────
                p2 + 5p + 6                   p2 ─ 1                     p2 + 4p + 3
  
  
                     (p ─ 2)(p + 1)               (p ─ 2)(p ─ 1)             (p + 1)(p + 3)
              =  ───────────   X   ──────────   X   ───────────
                     (p + 2)(p + 3)               (p ─ 1)(p + 1)             (p ─ 2)(p ─ 2)
  
  
                     (p + 1)
              =  ───────                                                                                                                                   [ 12 ]
                     (p + 2)
  
  
  
  
                  2x2 ─ 3x + 1             2x2 ─ 7x + 3               x2 ─ 3x + 2
      13.  ──────────   ÷   ──────────   ÷   ──────────
                2x2 ─ 3x ─ 2               x2 + x ─ 6               2x2 ─ 5x ─ 3
  
                    (x ─ 1)(2x ─ 1)              (x ─ 2)(x + 3)               (x ─ 3)(2x + 1)
              =  ───────────   X   ───────────   X   ───────────
                    (x ─ 2)(2x + 1)             (x ─ 3)(2x ─ 1)               (x ─ 2)(x ─ 1)
  
  
                    (x + 3)
              =  ──────                                                                                                                                          [ 13 ]
                    (x ─ 2)
  
  
  
  
                 6a2 ─ 7ab ─ 3b2             2a2 + ab ─ 6b2               9a2 ─ 3ab ─ 2b2
      14.  ─────────────   ÷   ────────────   ÷   ─────────────
                2a2 ─ 3ab ─ 2b2             3a2 ─ 8ab + 4b2              2a2 ─ 3ab ─ 2b2
  
  
                    (2a ─ 3b)(3a + b)            (a ─ 2b)(3a ─ 2b)              (a ─ 2b)(2a + b)
              =  ────────────   X   ─────────────   X   ─────────────
                    (a ─ 2b)(2a + b)              (a + 2b)(2a ─ 3b)              (3a ─ 2b)(3a + b)
  
  
                    (a ─ 2b)
              =  ───────                                                                                                                                       [ 14 ]
                    (a + 2b)
  
  
  
  
                  2x2 + xy ─ y2                        x2 ─ y2                      3x2 ─ xy ─ 2y2
      15.  ─────────────   ÷   ────────────   X   ────────────
                4x2 + 4xy ─ 3y2                2x2 + 3xy ─ 2y2             6x2 + xy ─ 2y2
  
  
                    (x + y)(2x ─ y)                  (x + 2y)(2x ─ y)                  (x ─ y)(3x + 2y)
              =  ────────────   X   ─────────────   X   ─────────────
                    (2x ─ y)(2x + 3y)                (x + y)(x ─ y)                    (2x ─ y)(3x + 2y)
  
  
                      (x + 2y)
              =  ────────                                                                                                                                   [ 15 ]
                     (2x + 3y)
  
  
  
  
                3p2 + 8pq + 4q2                 4p2 ─ 5pq ─ 6q2              4p2 + 11pq + 6q2
      16.  ─────────────   X   ──────────────   ÷   ─────────────
                2p2 ─ 7pq + 6q2              10p2 ─ 11pq + 3q2             4p2 ─ 8pq + 3q2
  
  
                    (p + 2q)(3p + 2q)            (p ─ 2q)(4p + 3q)             (2p ─ q)(2p ─ 3q)
              =  ────────────   X   ─────────────   X   ─────────────
                    (2p ─ 3q)(p ─ 2q)           (2p ─ q)(5p ─ 3q)             (p + 2q)(4p + 3q)
  
  
                     (3p + 2q)
               =  ────────                                                                                                                                   [ 16 ]
                     (5p ─ 3q)
  
  
  
  
                  a ─ b           a + b
      17.  ──────  +  ──────
                2a ─ b          a ─ 2b
  
  
                         (a ─ b)(a ─ 2b)  +  (a + b)(2a ─ b)
                   =  ────────────────────────
                                         (2a ─ b)(a ─ 2b)
  
  
                         a2 ─ 3ab + 2b2  +  2a2 + ab ─ b2
                   =  ────────────────────────
                                         (2a ─ b)(a ─ 2b)
  
  
                         3a2 ─ 2ab + b2
                   =  ────────────                           [ 17 ]
                         (2a ─ b)(a ─ 2b)
  
  
  
  
                  a ─ b           a + b
      19.  ──────    ──────
                2a ─ b          a ─ 2b
  
  
                         (a ─ b)(a ─ 2b)    (a + b)(2a ─ b)
                   =  ────────────────────────
                                         (2a ─ b)(a ─ 2b)
  
  
                         a2 ─ 3ab + 2b2  ─  2a2 ─ ab + b2
                   =  ────────────────────────
                                         (2a ─ b)(a ─ 2b)
  
  
                         ─ a2 ─ 4ab + 3b2
                   =  ────────────                           [ 19 ]
                         (2a ─ b)(a ─ 2b)
  
  
  
  
                 a + b                2b              2a ─ b
      21.  ──────  +  ──────    ──────
                a ─  b             a + b            a2 ─ b2
  
  
                   a + b                2b                   2a ─ b
             = ──────  +  ──────    ──────────
                  a ─  b             a + b             (a ─ b)(a + b)
  
  
                   (a + b)(a + b) + 2b(a ─ b) ─ (2a ─ b)
             = ─────────────────────────
                                   (a ─ b)(a + b)
  
  
                   a2 + 2ab + b2 + 2ab ─ 2b2 ─ 2a + b
             = ─────────────────────────
                                   (a ─ b)(a + b)
  
  
                   a2 + 4ab ─ b2 ─ 2a + b
             = ─────────────────
                           (a ─ b)(a + b)                         [ 21 ]
  
  
  
  
               3p + 2q          p ─ 3q
      18.  ──────  +  ──────
                 p + 3q          2p + q
  
  
                     (3p + 2q)(2p + q)  +  (p ─ 3q)(p + 3q)
               =  ────────────────────────
                                         (p + 3q)(2p + q)
  
  
                     6p2 + 7pq + 2q2  +  p2 ─ 9q2
               =  ────────────────────
                                   (p + 3q)(2p + q)
  
  
                     7p2 + 7pq ─ 7q2
               =  ────────────                           [ 18 ]
                     (p + 3q)(2p + q)
  
  
  
  
               3p + 2q          p ─ 3q
      20.  ──────    ──────
                 p + 3q          2p + q
  
  
                     (3p + 2q)(2p + q)    (p ─ 3q)(p + 3q)
               =  ────────────────────────
                                         (p + 3q)(2p + q)
  
  
                     6p2 + 7pq + 2q2  ─  p2 + 9q2
               =  ────────────────────
                                   (p + 3q)(2p + q)
  
  
                     5p2 + 7pq ─ 11q2
               =  ────────────                           [ 20 ]
                     (p + 3q)(2p + q)
  
  
  
  
               p + 2q            2p + q             3p ─ 2q
      22.  ──────  ─  ───────  +  ───────
               p ─ 2q           p2 ─ 4q2            p + 2q
  
  
                  p + 2q               2p + q               3p ─ 2q
             = ────── + ───────────  ──────
                  p ─ 2q        (p ─ 2q)(p + 2q)        p + 2q
  
  
                  (p + 2q)(p + 2q) + 2p + q + (3p ─ 2q)(p ─ 2q)
             = ──────────────────────────────
                                          (p ─ 2q)(p + 2q)
  
  
                  p2 + 4pq + 4q2 + 2p + q + 3p2 ─ 8pq + 4q2
             = ──────────────────────────────
                                          (p ─ 2q)(p + 2q)
  
  
                  4p2 ─ 4pq + 8q2 + 2p + q
             = ───────────────────
                              (p ─ 2q)(p + 2q)                 [ 22 ]
  
  
  
  
                      x + 3                    x ─ 2                     x ─ 3                       x + 3                          x ─ 2                         x ─ 3
      23.  ─────────   +   ───────      ─────────   =   ──────────   +   ──────────      ──────────
                x2 + 3x + 2               x2 ─ 1                x2 + x ─ 2             (x + 1)(x + 2)             (x ─ 1)(x + 1)             (x ─ 1)(x + 2)
  
  
                                                                                                        (x + 3)(x ─ 1)  +  (x ─ 2)(x + 2)  ─   (x ─ 3)(x + 1)
                                                                                                  =  ────────────────────────────────
                                                                                                                                (x + 1)(x ─ 1)(x + 2)
  
  
                                                                                                        x2 + 2x ─ 3 + x2 ─ 4  ─ x2 + 2x + 3
                                                                                                  =  ────────────────────────
                                                                                                                      (x + 1)(x ─ 1)(x + 2)
  
  
                                                                                                               x2 + 4x ─ 4
                                                                                                  =  ───────────────                                                                 [ 23 ]
                                                                                                         (x + 1)(x ─ 1)(x + 2)
  
  
  
  
                      x + 4                    x ─ 3                     x + 1                                  x + 4                         x ─ 3                           x + 1
      24.  ─────────   +   ───────      ─────────────   =   ──────────   +   ──────────      ──────────
                x2 + x ─ 2               x2 ─ 4                x2 ─ 3x + 2                       (x ─ 1)(x + 2)            (x ─ 2)(x + 2)             (x ─ 1)(x ─ 2)
  
  
                                                                                                                  (x + 4)(x ─ 2)  +  (x ─ 3)(x ─ 1)  ─   (x + 1)(x + 2)
                                                                                                            =  ────────────────────────────────
                                                                                                                                (x ─ 1)(x ─ 2)(x + 2)
  
  
                                                                                                                  x2 + 2x ─ 8 + x2 ─ 4x + 3 ─  x2 ─ 3x ─ 2
                                                                                                            =  ───────────────────────────
                                                                                                                                (x ─ 1)(x ─ 2)(x + 2)
  
  
                                                                                                                            x2 ─ 5x ─ 7
                                                                                                            =  ───────────────                                                                 [ 24 ]
                                                                                                                   (x ─ 1)(x ─ 2)(x + 2)
  
  
  
  
                      y ─ 1                     y ─ 6                    2y ─ 3                     y ─ 1                          y ─ 6                        2y ─ 3
      25.  ─────────      ─────────   +   ───────   =   ──────────      ──────────   +   ──────────
                y2 + 5y + 6              y2 + y ─ 6                y2 ─ 4               (y + 2)(y + 3)             (y ─ 2)(y + 3)             (y ─ 2)(y + 2)
  
  
                                                                                                                  (y ─ 1)(y ─ 2)  ─  (y ─ 6)(y + 2)  +   (2y ─ 3)(y + 3)
                                                                                                            =  ────────────────────────────────
                                                                                                                                (y ─ 2)(y + 2)(y + 3)
  
  
                                                                                                                   y2 ─ 3y + 2 ─ y2 + 4y + 12 +  2y2 + 3y ─ 9
                                                                                                            =  ─────────────────────────────
                                                                                                                                (y ─ 2)(y + 2)(y + 3)
  
  
                                                                                                                        2y2 + 4y + 5
                                                                                                            =  ───────────────                                                                 [ 25 ]
                                                                                                                   (y ─ 2)(y + 2)(y + 3)
  
  
  
  
                      2a + 1                         a ─ 5                       3a + 2                    2a + 1                               a ─ 5                               3a + 2
       26.  ──────────   +   ───────────      ───────   =   ────────────   +   ────────────      ─────────────
                6a2 + 5a ─ 6              6a2 ─ 13a + 6              4a2 ─ 9            (3a ─ 2)(2a + 3)             (3a ─ 2)(2a ─ 3)              (2a ─ 3)(2a + 3)
  
  
                                                                                                                   (2a + 1)(2a ─ 3) + (a ─ 5)(2a + 3) ─  (3a + 2)(3a ─ 2)
                                                                                                           =   ───────────────────────────────────
                                                                                                                                        (3a ─ 2)(2a ─ 3)(2a + 3)
  
  
                                                                                                                   4a2 ─ 4a ─ 3 + 2a2 ─ 7a ─ 15 ─  9a2 + 4
                                                                                                           =   ─────────────────────────────
                                                                                                                               (3a ─ 2)(2a ─ 3)(2a + 3)
  
  
                                                                                                                    ─ (3a2 + 11a + 14)
                                                                                                           =   ─────────────────                                                [ 26 ]
                                                                                                                  (3a ─ 2)(2a ─ 3)(2a + 3)
  
  
  
  
                        a ─ 3b                            a + b                           2a ─ 3b                           a ─ 3b                            a + b                           2a ─ 3b
       27.  ────────────      ───────────   +   ────────────   =   ────────────      ───────────   +   ───────────
                2a2 ─ 3ab ─ 2b2           a2 ─ 3ab + 2b2              2a2 ─ ab ─ b2               (a ─ 2b)(2a + b)            (a ─ 2b)(a ─ b)              (a ─ b)(2a + b)
  
  
                                                                                                                                     (a ─ 3b)(a ─ b) ─ (a + b)(2a + b) +  (2a ─ 3b)(a ─ 2b)
                                                                                                                             =   ─────────────────────────────────────
                                                                                                                                                          (a ─ b)(a ─ 2b)(2a + b)
  
  
                                                                                                                                    a2 ─ 4ab + 3b2 ─ 2a2 ─ 3ab ─ b2 +  2a2 ─ 7ab + 6b2
                                                                                                                             =   ─────────────────────────────────────
                                                                                                                                                          (a ─ b)(a ─ 2b)(2a + b)
  
  
                                                                                                                                          a2 ─ 14ab + 8b2
                                                                                                                             =   ─────────────────                                  [ 27 ]
                                                                                                                                     (a ─ b)(a ─ 2b)(2a + b)
  
  
  
  
                        6                             1
      28.  ─────────  +  ─────────
               x2 ─ 2x ─ 8           x2 + 5x + 6
  
  
                              6                            1
               =  ─────────  +  ─────────
                    (x ─ 4)(x + 2)         (x + 2)(x + 3)
  
  
                      6(x + 3)  +  (x ─ 4)
               =  ──────────────
                     (x ─ 4)(x + 2)(x + 3)
  
  
                      6x + 18  +  x ─ 4
               =  ──────────────
                     (x ─ 4)(x + 2)(x + 3)
  
  
                               7x + 14
               =  ──────────────
                     (x ─ 4)(x + 2)(x + 3)
  
  
                               7(x + 2)
               =  ──────────────
                     (x ─ 4)(x + 2)(x + 3)
  
  
                               7
               =  ──────────                              [ 28 ]
                     (x ─ 4)(x + 3)
  
  
  
  
                  1                  1                3a
      30.  ─────  +  ─────  +  ──────
               1 + a           a ─ 1          1 ─ a2              [ 30 ]
  
  
  
  
                    x ─ 3                       x ─ 1
      29.  ─────────    ──────────
               x2 ─ 3x ─ 4            x2 ─ x ─ 2
  
  
                           x ─ 3                     x ─ 1
               =  ─────────  ─  ─────────
                    (x ─ 4)(x + 1)        (x ─ 2)(x + 1)
  
  
                       (x ─ 3)(x ─ 2)  ─  (x ─ 1)(x ─ 4)
               =  ──────────────────────
                               (x ─ 4)(x ─ 2)(x + 1)
  
  
                       x2 ─ 5x + 6  ─  x2 + 5x ─ 4
               =  ────────────────────
                               (x ─ 4)(x ─ 2)(x + 1)
  
  
                                    2
               =  ───────────────                              [ 29 ]
                     (x ─ 4)(x ─ 2)(x + 1)
  
  
  
  
  
  
  
  
  
  
  
  
  
  
                  1                  1                3a
      30.  ─────  +  ─────  +  ──────
               1 + a           a ─ 1          1 ─ a2              [ 30 ]
  
  
            Die faktore van 1 ─ a2  is (1 ─ a)(1 + a)
            sodat die kleinste gemene noemer
            (1 ─ a)(1 + a) is. Dit sal dus gerieflik wees
            om die tweede noemer te verander na 1 ─ a
            Dit word gedoen deur AL die tekens van
            die noemer te verander, maar dan sal die
            teken van die hele breuk (die teken voor
            die breuk) ook verander.
            Die volgende "reël" help baie : As die teken
            van slegs die noemer verander word,
            verander die teken van die hele breuk ook.
  
  
            The factors of 1 ─ a2  are (1 ─ a)(1 + a)  
            so that the lowest common denominator is
            (1 ─ a)(1 + a). It will therefore be convenient
            if the second denominator can be changed
            to 1 ─ a. Change the signs of the denominator,
            but then the sign of the fraction will also change.
            The following "rule" is convenient : If the
            sign of the denominator alone be changed,
            the sign of the whole fraction will be changed.
  
  
  
  
  
  
                  1                  1                3a
      30.  ─────  +  ─────  +  ──────
               1 + a           a ─ 1          1 ─ a2
  
  
              Verander AL die tekens in die tweede noemer
  
              en ook die teken voor die breuk.
  
              Change ALL the signs in the second
  
              denominator and also the sign in front of
  
              the fraction.
  
  
                  1                  1                3a
             ─────  +  ─────  +  ──────
               1 + a           a ─ 1          1 ─ a2
  
  
                       1                  1                        3a
             =  ─────    ──────  +  ──────────
                   1 + a            a + 1          (1 ─ a)(1 + a)
  
  
                   1(1 ─ a)  1(1 + a) + 3a
             =  ─────────────────
                            (1 ─ a)(1 + a)
  
  
                   1 ─ a  1  a + 3a
             =  ─────────────
                       (1 ─ a)(1 + a)
  
  
                             a
             =  ──────────                                [ 30 ]
                   (1 ─ a)(1 + a)
  
  
  
  
                  2y                  1                1
      32.  ─────    ─────    ──────
               4 ─ y2           y ─ 2          2 + y
  
  
                            2y                    1                  1
              =   ───────    ─────    ──────
                        4 + y2            y ─ 2            y + 2
  
  
                       2y ─ 1(y + 2)  ─ 1(y ─ 2)
              =  ───────────────────
                              (y ─ 2)(y + 2)
  
  
                       2y ─ y ─ 2  ─ y + 2
              =  ────────────────
                              (y ─ 2)(y + 2)
  
  
                           4y
              =  ───────────                               [ 32 ]
                    (y ─ 2)(y + 2)
  
  
  
  
                     1               3                1
      34.  ──────  +  ───  +  ───────
               a(a ─ b)         ab          b(b ─ a)
  
  
                         1               3                1
              =  ──────  +  ───    ───────
                   a(a ─ b)         ab          b(a ─ b)
  
  
                    1b + 3(a ─ b)  1a
              =  ──────────────
                            ab(a ─ b)
  
  
                    1b + 3a ─ 3b  1a
              =  ──────────────
                            ab(a ─ b)
  
  
                      2(a ─ b)
              =  ──────────────
                    ab(a ─ b)
  
  
                      2
              =  ────                               [ 34 ]
                    ab
  
  
  
  
  
  
  
  
  
                  4                  3                7x
      31.  ─────  +  ─────  +  ──────
               1 + x           1 ─ x          x2 ─ 1
  
  
              Verander AL die tekens in die derde noemer
  
              en ook die teken voor die breuk.
  
              Change ALL the signs in the third
  
              denominator and also the sign in front of
  
              the fraction.
  
  
                  4                  3                7x
             ─────  +  ─────  +  ──────
               1 + x           1 ─ x          x2 ─ 1
  
  
                        4                 3                   7x
               =  ─────  +  ─────    ───────
                     1 + x           1 ─ x            x2 + 1
  
  
                      4(1 ─ x) + 3(1 + x)  7x
               =  ─────────────────
                              (1 ─ x)(1 + x)
  
  
                      4 ─ 4x + 3 + 3x  7x
               =  ─────────────────
                              (1 ─ x)(1 + x)
  
  
                            7 ─ 8x
               =  ───────────                        [ 31 ]
                      (1 ─ x)(1 + x)
  
  
  
  
                  5                  2             6(1 ─ p)
      33.  ─────    ─────  +  ──────
               3 + p           3 ─ p           p2 ─ 9
  
  
                         5                    2                  6(1 ─ p)
                =  ─────  +  ──────  +  ───────────
                      p + 3           3 + p            (p ─ 3)(p + 3)
  
  
                         5(p ─ 3) + 2(p + 3) + 6 ─ 6p
                =  ──────────────────────
                      (p ─ 3)(p + 3)
  
  
                       5p ─ 3 + 2p + 6 + 6 ─ 6p
                =  ───────────────────
                              (p ─ 3)(p + 3)
  
  
                              p + 9
                =  ───────────                        [ 33 ]
                       (p ─ 3)(p + 3)
  
  
  
  
                  1                  3                1
      35.  ──────  +  ───  +  ───────
               p(p ─ 2)        2p          2(2 ─ p)
  
  
                          1                 3                1
                =  ──────  +  ───    ───────
                     p(p ─ 2)         2p          (+ 2p)
  
  
                          1                3                 1
                =  ──────  +  ───    ───────
                     p(p ─ 2)         2p          2(p ─ 2)
  
  
                       1 × 2 + 3(p ─ 2)  1p
                =  ────────────────
                             2p(p ─ 2)
  
  
                       2 + 3p ─ 6  1p
                =  ────────────
                           2p(p ─ 2)
  
  
                        2(p  2)
                =  ────────
                       2p(p ─ 2)
  
  
                      1
                =  ───                        [ 35 ]
                      p
  
  
  
  
                      5                    1                      x ─ 2                      5                      1                       x ─ 2
      36.  ───────    ───────    ────────   =  ───────  +  ─────────    ───────
               18x + 54         54 ─ 18x           3x2 ─ 27            18x + 54          (54 + 18x)          3(x2 ─ 9)
  
  
                                                                                                   5                       1                            x ─ 2
                                                                                      =  ───────  +  ─────────    ───────────
                                                                                            18(x + 3)           18(x ─ 3)             3(x ─ 3)(x + 3)
  
  
                                                                                             5(x ─ 3) + 1(x + 3)  6(x  2)
                                                                                      =  ─────────────────────
                                                                                                    18(x ─ 3)(x + 3)
  
  
                                                                                             5x ─ 15 + x + 3  6x + 12
                                                                                      =  ─────────────────
                                                                                                    18(x ─ 3)(x + 3)
  
  
                                                                                                           0
                                                                                      =  ────────────      =  0                                    [ 36 ]
                                                                                            18(x ─ 3)(x + 3)
  
  
  
  
                2x ─ 1             2x + 1             4x + 2              2x ─ 1             2x + 1                4x + 2
      37.  ──────   +   ──────   +   ──────   =   ──────   +   ──────      ────────
                x2 + x              x2 ─ x             x ─ x3              x(x + 1)           x(x ─ 1)           x( 1 + x2)
  
  
                                                                                        2x ─ 1             2x + 1                    4x + 2
                                                                                =   ──────   +   ──────      ───────────
                                                                                       x(x + 1)           x(x ─ 1)            x(x ─ 1)(x + 1)
  
  
                                                                                        (2x ─ 1)(x ─ 1) +  (2x + 1)(x + 1) ─ (4x + 2)
                                                                                =   ─────────────────────────────
                                                                                                               x(x ─ 1)(x + 1)
  
  
                                                                                        2x2 ─ 3x + 1 +  2x2 + 3x + 1 ─ 4x ─ 2
                                                                                =   ──────────────────────────
                                                                                                             x(x ─ 1)(x + 1)
  
  
                                                                                             4x2 ─ 4x
                                                                                =   ────────────
                                                                                        x(x ─ 1)(x + 1)
  
  
                                                                                             4x(x ─ 1)
                                                                                =   ────────────
                                                                                        x(x ─ 1)(x + 1)
  
  
                                                                                             4
                                                                                =   ───────                                                                 [ 37 ]
                                                                                         (x + 1)
  
  
  
  
                1 ─ 2a             1 + 2a            1 ─ 20a2           1 ─ 2a                1 + 2a                1 ─ 20a2
      38.  ──────   +   ──────      ───────   =   ──────      ────────      ───────
                1 + 2a             1 ─ 2a            4a2 ─ 1              2a + 1              (+ 2a)             4a2 ─ 1
  
  
                                                                                          1 ─ 2a               1 + 2a                      1 ─ 20a2
                                                                                   =   ──────      ────────      ───────────
                                                                                          2a + 1              (2a  1)               (2a  1)(2a + 1)
  
  
                                                                                          (1 ─ 2a)(2a  1)   (2a + 1)(2a + 1)   (1 ─ 20a2)
                                                                                   =   ─────────────────────────────────
                                                                                                                         (2a  1)(2a + 1)
  
  
                                                                                          2a ─ 1  4a2 + 2a   4a2  4a  1   1 + 20a2
                                                                                   =   ───────────────────────────────
                                                                                                                     (2a  1)(2a + 1)
  
  
                                                                                               12a2  3
                                                                                   =   ────────────
                                                                                           (2a  1)(2a + 1)
  
  
                                                                                            3(2a  1)(2a + 1)
                                                                                   =   ─────────────
                                                                                             (2a  1)(2a + 1)
  
  
                                                                                   =  3                                                                               [ 38 ]
  
  
  
               8 ─ 3a2               5a + 7                  2              ( 8 + 3a2)              5a + 7                   2
      39.  ──────      ────────   +   ─────   =   ────────      ────────   +   ─────
               1 ─ a3              a2 + a + 1            a ─ 1            ( 1 + a3)              a2 + a + 1            a ─ 1
  
  
                                                                                                 (3a2  8)                       5a + 7                  2
                                                                                  =   ─────────────      ────────   +   ─────
                                                                                         (a ─ 1)(a2 + a + 1)             a2 + a + 1            a ─ 1
  
  
                                                                                         (3a2  8)  (5a + 7)(a ─ 1) +  2(a2 + a + 1)
                                                                                  =   ────────────────────────────
                                                                                                                (a ─ 1)(a2 + a + 1)
  
  
                                                                                         3a2  8  5a2  2a + 7 +  2a2 + 2a + 2
                                                                                  =   ────────────────────────────
                                                                                                           (a ─ 1)(a2 + a + 1)
  
  
                                                                                                          1 
                                                                                  =   ──────────────                                            [ 39 ]
                                                                                          (a ─ 1)(a2 + a + 1)
  
  
            Let op dat in hierdie geval die tekens van beide die teller en noemer verander word en dus
  
            verander die teken van die hele breuk nie.
  
  
            Note that in this case the signs of both the numerator and denominator are changed and
  
            therefore the sign of the whole fraction remains unchanged.