[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
_____________________
x = -6 add

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
_______________
t = 8 add

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
___________________
x = 13 add

result: x = 13