[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)
= 9x2 - 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