3/2x-4 - 5/3x-6 = 3/5 .. the book says the answer is 31/18 but I have NO idea what the steps are..Help please.?

....3............5............3
---------- - ----------- = ---------
...2x-4......3x-6..........5

...3............5............3
---------- - ---------- = --------- lcd 30(x-2)
.2(x-2)......3(x-2)........5

45-50=18(x-2)
-5=18x-36
-5+36=18x
31=18x
......31
x=--------- answer//
......18

(3/[2x - 4]) - (5/[3x - 6]) = 3/5
(3/[2{x - 2}]) - (5/[3{x - 2}]) = 3/5
(9 - 10)/(6[x - 2]) = 3/5
5(- 1) = 3(6x - 12)
- 5 = 18x - 36
18x = 31
x = 31/18

Answer: x = 31/18

Proof:
(3/[2{31/18} - 4]) - (5/[3{31/18} - 6]) = 3/5
(3/[31/9 - 36/9]) - (5/[31/6 - 36/6]) = 3/5
(3/[- 5/9]) - (5/[- 5/6]) = 3/5
3(- 9/5) - 5(- 6/5) = 3/5
- 27/5 + 30/5 = 3/5
3/5 = 3/5

3 (* 3)...................5 (* 2)...............3
---------------....-....-----------------... = ----
(2x - 4) (* 3)..........(3x - 6) (* 2).....5

9 - 10.........3
--------- = ----
6x - 12.......5

cross it

(9 -10) * 5 = 3 (6x - 12)

-5 = 18x - 36

36 - 5 = 18x

31 = 18x

x = 31/18 solved!

this one is usually junior high school / high school common steps to solve it.

3/(2x - 4) - 5/(3x - 6) = 3/5
3/[2(x - 2)] - 5/[3(x - 2)] = 3/5
30(x - 2){3/[2(x - 2)] - 5/[3(x - 2)]} = 30(x - 2)(3/5)
15(3) - 50 = 18(x - 2)
45 - 50 = 18(x) - 18(2)
-5 = 18x - 36
-18x = 5 - 36
x = -31/(-18)
x = 31/18

[(3(3x-6)) - (5(2x-4)] / [(3x-6)(2x-4)] = 3/5

.....
[(9x-18-10x+20)] / [(6x^2 - 24x + 24)] = 3/5

.....
cross multiply..

.....
5(2-x) = 3(6x^2 - 24x + 24)
.....
10-5x = 18x^2 - 72x + 72

....
0 = 18x^2 - 67x + 62

...
0 = (18x-31) (x-2)..

so the possible values of x are 31/18 and 2, but if you substitute 2 to x, the denominators will be 0 which will make the terms undefined, so the answer would be 31/18.

Find the LCD, then combine like terms (for the numerator):

3(3x-6) - 5(2x-4) / (2x-4)(3x-6) = 3/5

9x-18-10x+20 / (2x-4)(3x-6) = 3/5

2-x / (2x-4)(3x-6) = 3/5

2-x can also be written as: -1(x-2), so:

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

in this step you could cancel (x-2) on both the numerator and the denominator, which will leave you with:

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

Factor out 3 from (3x-6) --> on the denominator:

-1 / 2*3(x-2) = 3/5 or

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

Analyzing the whole equation, we can rearrange this as:

-1 [1 / 6(x-2) ] = 3/5

Then use the reciprocal of the equation:

-1 * [ 6(x-2) ] = 5/3

-1 (6x-12) = 5/3

-6x+12 = 5/3

-6x = 5/3 - 12

in this step, convert 12 into a fraction that has the same denominator as 5/3. in this case i converted it into 36/3.

-6x = 5/3 - 36/3

-6x = -31/3

-x = -31/18, just get rid of the - sign by dividing both sides to -1.

x = 31/18 --> final answer.