[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