x-1/x-2+x-2/x-3=4 find x?

Answer 1

I think that you should add some brackets to this question in order to clarify the situation.
As it stands, I feel it might be a bit dodgy!!

I think it should perhaps read:-

(x - 1) / (x - 2) + (x - 2) / (x - 3) = 4

Multiply by (x -2)(x - 3):-

(x - 1)(x - 3) + (x - 2)(x - 2) = 4(x - 2)(x - 3)

x² - 4x + 3 + x² - 4x + 4 = 4(x² - 5x + 6)

2x² - 8x + 7 = 4x² - 20x + 24

0 = 2x² - 12x + 17

x = 12 ± √(144 - 136 ) / 4

x = [12 ±√8 ] / 4

x = [12 + 2√2] / 4 or x = [ 12 - 2√2] / 4

x = 3 + (1/2 )√2 or x = 3 - (1/2) √2

Answer 2

(x - 1)/(x - 2) + (x - 2)/(x - 3) = 4
(x - 1) + (x - 2)(x - 2)/(x - 3) = 4(x - 2)
(x - 1)(x - 3) + (x - 2)(x - 2) = 4(x - 2)(x - 3)
(x² - 4x + 3) + (x² - 4x + 4) = 4(x² - 5x + 6)
2x² - 8x + 7 = 4x² - 20x + 24
0 = 2x² - 12x + 17

Using quadratic formula

x = 12 ± √((-12)² - 4*2*17)/4
= 12 ± √(144 - 136)/4
= (12 ± √8)/4
= (12 ± 2√2)/4
= (6 ± √2)/2

Answer 3

(x - 1) / (x - 2) + (x - 2) / (x - 3) = 4

Multiply both sides by (x - 2)(x - 3) :
(x - 1)(x - 3) + (x - 2)^2 = 4(x - 2)(x - 3)

Expand :
x^2 - 4x + 3 + x^2 - 4x + 4 = 4x^2 - 20x + 24

Gather like terms :
2x^2 - 8x + 7 = 4x^2 - 20x + 24

Subtract the LHS from both sides :
2x^2 - 12x + 17 = 0

Solve for x by the quadratic formula :
x = {12 ± sqrt[(-12)^2 - (4)(2)(17)]} / (2*2)

x = [6 ± sqrt(2)] / 2

Answer 4

Get a common denominator on the left:

(x-1)(x-3) + (x-2)(x-2)
--------------------------
(x-2)(x-3)

Simplify the top:
x^2 - 4x + 3 + x^2 - 4x + 4
= 2x^2 - 8x + 7

Simplify the bottom:
x^2 - 5x + 6

Now, you have:

(2x^2 - 8x + 7)/(x^2 - 5x + 6) = 4
2x^2 - 8x + 7 = 4(x^2 - 5x + 6)
2x^2 - 8x + 7 = 4x^2 - 20x + 24
0 = 2x^2 -12x + 17

Using quadratic formula:
x = 3 + (1/2)sqrt 2
x = 3 - (1/2)sqrt 2