[rational equations][section 57]

(1.)  (a/3) - (a/4) = 2        here is the problem

4a - 3a = 24    multiply thru by 12, cancel as you go

a = 24           combine like terms

result:  a = 24

(2.)  (3t/4) + (5/3) = (7t/6)     here is the problem

9t + 20 = 14t          multiply thru by 12, cancel

14t = 9t + 20        rearrange like this

-9t  -9t              subtract 9t from each side
___________________
5t =      20          subtract
___      ____
5          5    divide each side by 5

t = 4          divide and cancel

result:  t = 4

(3.)   3/2 = x/(x + 2)         here is the problem

2x = 3x + 6       cross multiply

-2x = -3x - 6       multiply thru by -1

+   3x   +3x         add 3x to each side
_____________________

result:  x = -6

(4.)    4        -2     3
___ - 3 = ___ - ____
2x        5x     5             here is the problem

4*5 - 3*10x = -2*2 - 3*2x  multiply thru by 10x

20 - 30x = -4 - 6x         multiply

-20 + 30x = 4 + 6x     multiply thru by -1

+20     + 20          add 20 to each side
__________________________
30x = 24 + 6x            add

-6x        -6x       subtract 6x from each side
_____________________________
24x = 24                     subtract
_____  _____
24    24              divide each side by 24

x = 1           divide and cancel

result:  x = 1

(5.)  (b - 2)/(b + 3) = 3/8          here is the problem

8b - 16 = 3b + 9   cross multiply

+16     + 16   add 16 to each side
_________________________
8b      = 3b + 25        add

-3b      -3b         subtract 3b from each side
_____________________________
5b =       25       subtract
___      _____
5          5        divide each side by 5

b = 5          divide and cancel

result: b = 5

(6.)    t + 1        2        t
_______ =  ________ + ___
t - 1      t - 1      8            here is the problem

8(t + 1) = 2*8 + t(t - 1)  multiply thru by 8(t - 1)

8t + 8 = 16 + t2 - t        multiply

t2 - t + 16 = 8t + 8    rearrange like this

-8       - 8   subtract 8 from each side
__________________________
t2 - t + 8 = 8t         subtract

- 8t     - 8t       subtract 8t from each side
______________________
t2 - 9t + 8 = 0            subtract

(t - 1)(t - 8) = 0               factor

t - 8 = 0      set this factor equal to 0

+ 8  + 8      add 8 to each side
_______________

result:  t = 8

(7.)   2x - 1    x + 2     6x - 5
______ - _______ = ________
2       2x + 5       6            here is the problem

3(2x + 5)(2x - 1) - 6(x + 2) = (2x + 5)(6x - 5)

[multiply thru by 6(2x + 5) & cancel as you go thru]

3(4x2 + 8x - 5) - 6(x + 2) = 12x2 + 20x - 25

[foil multiply and combine like terms]

12x2 + 24x - 15 - 6x - 12 = 12x2 + 20x - 25  multiply thru

12x2 + 18x - 27 = 12x2 + 20x - 25   combine like terms

-12x2             -12x2        subtract 12x2 fr ea side
_______________________________________
18x - 27 =        20x - 25      subtract

-18x + 27 = -20x + 25   multiply thru by -1

+ 20x      +20x          add 20x to each side
________________________________
2x + 27 =        25       add

-27         -27     subtract 27 fr ea side
______________________
2x          =   -2        subtract
___             ____
2                2   divide each side by 2

x = -1            divide and cancel

result:  x = -1

(8.)    5x          x
_______  -  _______ = 4            here is the problem
x + 1       x + 6

5x(x + 6) - x(x + 1) = 4(x + 1)(x + 6)

[multiply thru by (x + 6)(x + 1) and cancel as you go thru]

5x2 + 30x - x2 - x = 4x2 + 28x + 24   multiply

4x2 + 29x = 4x2 + 28x + 24    combine like terms

-4x2        -4x2            subtract 4x2 from each side
____________________________
29x  =      28x + 24        subtract

-28x        -28x          subtract 28x from each side
_______________________________
x =          24          subtract

result:  x = 24

(9.)    x + 2    x + 6
_____ = _______
x + 4    x - 8          here is the problem

x2 + 10x + 24 = x2 - 6x - 16    cross multiply

-x2            - x2         subtract x2 from each side
_________________________________
10x + 24 =     -6x - 16        subtract

+  6x           +6x       add 6x to each side
__________________________________
16x + 24 =       -16  add

-24          -24   subtract 24 fr ea side
_________________________
16x         =   -40        subtract
____            ____
16               16         divide each side by 16

x = -2.5            divide and cancel

result: x = -2.5

(10.)    6        x + 2
_______ = _________
4x       x2 - 3x           here is the problem

6x2 - 18x = 4x2 + 8x    cross multiply

-4x2       -4x2       subtract 4x2 from each side
___________________________
2x2 - 18x =      8x       subtract

- 8x       - 8x      subtract 8x from each side
__________________________
2x2 - 26x = 0                subtract
___   ___ ___
2     2    2        divide thru by 2

x2 - 13x = 0          divide and cancel

x(x - 13) = 0          factor

x - 13 = 0   set this factor equal to 0

+ 13  +13     add 13 to each side
___________________