2/3= 2 - (5x-3) / (x-1) Solve for x.?

I got x= -1/17. XD

2(x - 1) = 6(x - 1) - 15x + 9
2x - 2 = 6x - 6 - 15x + 9
11x = 5
x = 5/11

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

2/3 = 2 − (5x − 3)/(x − 1) // Divide 2 by 3, and 5x − 3 by x − 1. The result of the second division is 5 with a remainder of 2.
0.(6) = 2 − (5 + 2/(x − 1)) // Distribute the minus and reorganize.
0.(6) = 2/(1 − x) − 3 // Multiply both sides by 3 − 3x. Originally it was x − 1, but a bit of foresight says that we need the 3 to fix up 0.(6).
2 − 2x = 9x − 3 // Add 2x + 3 to both sides and flip the equation around the "equals" sign.
11x = 5 // Divide by 11.
x = 0.(45)

Check:
2/3 = 2 − (5x − 3)/(x − 1), x = 0.(45)
0.(6) = 2 − 0.(72)/0.(54)
0.(6) = 2 − 1.(3)
0.(6) = 0.(6)
True
Q.E.D.

The answer is 5/11
2/3= 2-(5x-3) / (x -1)
(5x-3)/ (x-1)= 2 - 2/3
(5x-3) / (x-1) = 4/3
5x - 3 = 4/3 (x-1)
5x - 3 = 4/3x - 4/3
5x - 4/3x = 3 - 4/3
11/3x = 5/3
x = (5/3) / (11/3)
x= (5/3) x (3/11)
x= 5/11

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

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

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

4x - 4 = 15x - 9

11x = 5

x = 5 / 11

2/3 = 2 - (5x - 3)/(x - 1)
(x - 1)2/3 = (x - 1)[2 - (5x - 3)/(x - 1)
(2x - 2)/3 = 2x - 2 - 5x + 3
2x - 2 = 6x - 6 - 15x + 9
2x - 6x + 15x = 2 - 6 + 9
-4x + 15x = -4 + 9
11x = 5
x = 5/11