[addition of rational expressions][section 55]

(1.)    3a     (a + 9)
       ____ + ________           here is the problem
         2        2

=    (4a + 9)/2       add fractions

(2.)   2a + 3     5a - 7
      ________ - ________       here is the problem
         6          9

=     3(2a + 3)      2(5a - 7)       multiply by 3/3
      ___________ - ___________      multiply by 2/2
         18              18

=    6a + 9 - 10a + 14      multiply thru parentheses
     __________________
            18

=   (23 - 4a)/18         combine like terms

(3.)   x - 1      x2 - x
       _____ + ___________              here is the problem
        3xy        9x2y2  

=     3xy(x - 1)       x2 - x         multiply by 3xy/3xy
     _____________ + ____________
       9x2y2              9x2y2  

=      3x2y - 3xy + x2 - x
     _________________________    add fractions
             9x2y2  

=      (3y + 1)(x2 - x)/(9x2y2)    factor the top

=    (3y + 1)(x - 1)/(9xy2)    divide thru by x

(4.)   4a       a - 1
     _______ - _______
     2a + 6     a + 3            here is the problem

=      4a          2a - 2          multiply by 2/2
    ___________ - ___________
     2a + 6         2a + 6

=       2a + 2
      ___________         subtract fractions combine like terms
        2a + 6

=   (a + 1)/(a + 3)    divide thru by 2

(5.)    a + b        a + b                   here is the problem
     ___________ - _________
       ax + ay       bx + by

=       a + b              a + b       factor
     ______________ - ______________
       a(x + y)         b(x + y)

=      a2 + ab     ab - b2       multiply by a/a
       ________ - __________   multiply by b/b
       ab(x +y)    ab(x +y)

=      a2 + b2            subtract fractions
     __________    carry the minus sign thru on
      ab(x +y)    the 2nd fractions (multiply thru)

(6.)    4x + y       3x - 4y          here is the problem
         _________ - ____________
            x           x2 

=      4x2 + xy       3x - 4y           multiply by x/x
     ___________  - ___________
         x2               x2  

=     4x2 + 3x + xy - 4y      subtract fractions, carry the
     _____________________  minus sign thru
              x2  



(7.)   a - b      a
       _____ - ________             here is the problem
       a + b    a2 - b2  

=      a - b         a
       _____ - ______________        factor
       a + b   (a + b)(a - b)

=   (a - b)(a - b)          a          multiply by (a-b)/(a-b)
    _______________ - _______________
     (a + b)(a - b)   (a + b)(a - b)

=    a2 - 2ab + b2 - a
    _____________________   multiply, subtract fractions
      (a + b)(a - b)

(8.)       5            x        3x2
       __________ + _______ - _________  here is the problem
         3x - 3      2x + 2    x2 - 1

=       5                2                3x2
    __________  +   ____________ - _______________
     3(x - 1)       2(x + 1)       (x + 1)(x - 1)   factor

=     5*2(x + 1)          2*3(x - 1)          6*3x2
    __________________ + _______________ - ____________
      6(x - 1)(x + 1)      6(x-1)(x+1)     6(x+1)(x-1)

[multiply by these forms of "1"]

=    10x + 10 + 6x - 6 - 18x2    multiply thru, add fractions
    _____________________________
          6(x - 1)(x + 1)

=    -18x2 + 16x + 4
    ____________________       combine like terms on top
        6(x - 1)(x + 1)

=    -2(9x2 - 8x - 2)
    ______________________   factor on top
        6(x - 1)(x + 1)

=    9x
2 - 8x - 2
     ________________           reduce
     -3(x - 1)(x + 1)

(9.)     x + 3     x - 5
         ______ + ________
         x2 - 4     x + 2          here is the problem

=      x + 3            x - 5
    ________________ + _______
    (x + 2)(x - 2)      x + 2      factor

=      x + 3                  (x - 5)(x - 2)    multiply by
    ___________________  +   _______________    (x-2)/(x-2)
     (x + 2)(x - 2)          (x + 2)(x - 2)

=      x + 3 + x2 - 7x + 10     foil multiply
      _________________________  combine like terms
            (x + 2)(x - 2)         add fractions

=     x2 - 6x + 13
      ______________
      (x + 2)(x - 2)     combine like terms

(10.)       4a              3b        here is the problem
         ____________ + __________
           6a - 2b       9a - 3b

=       2a               b            divide thru by 2,
     _________  +   ____________     divide thru by 3
      3a - b         3a - b

=  (2a + b)/(3a - b)           add fractions