Without parentheses and proper math symbols, it is difficult to tell exactly what your problem statement is. I interpret it as follows.
x-3 = x/(x-3) -3/2
(x-3)(2)(x-3) = (x)(2) -3(x-3) (multiply both sides of the equation by 2(x-3))
2x2 -12x +18 = 2x -3x +9 (use the distributive property to eliminate parentheses)
2x2 -11x +18 = 9 (add x to both sides)
2x2 -11x +9 = 0 (subtract 9 from both sides)
(2x-9)(x-1) = 0 (factor the equation)
x = 9/2 or x = 1 (values of x that make the factors equal to zero)
Check
for x = 9/2
(9/2) -3 = (9/2)/((9/2) - 3) -3/2
9/2 -6/2 = 9/2/(9/2 -6/2) -3/2
3/2 = (9/2)/(3/2) - 3/2
3/2 = 9/3 - 3/2
3/2 = 3 - 3/2
3/2 = 6/2 - 3/2 (yes)
for x = 1
(1) - 3 = (1)/((1)-3) -3/2
-2 = 1/-2 -3/2
-2 = -1/2 -3/2 (yes)
It looks to me like the solutions found work in the equation as I interpret it from your question. The only "solution" that would not work is x=3, because that would result in an indeterminate expression on the right side of the equation. Fortunately, x=3 is not one of the solutions we came up with.
x-3 = x/(x-3) -3/2
(x-3)(2)(x-3) = (x)(2) -3(x-3) (multiply both sides of the equation by 2(x-3))
2x2 -12x +18 = 2x -3x +9 (use the distributive property to eliminate parentheses)
2x2 -11x +18 = 9 (add x to both sides)
2x2 -11x +9 = 0 (subtract 9 from both sides)
(2x-9)(x-1) = 0 (factor the equation)
x = 9/2 or x = 1 (values of x that make the factors equal to zero)
Check
for x = 9/2
(9/2) -3 = (9/2)/((9/2) - 3) -3/2
9/2 -6/2 = 9/2/(9/2 -6/2) -3/2
3/2 = (9/2)/(3/2) - 3/2
3/2 = 9/3 - 3/2
3/2 = 3 - 3/2
3/2 = 6/2 - 3/2 (yes)
for x = 1
(1) - 3 = (1)/((1)-3) -3/2
-2 = 1/-2 -3/2
-2 = -1/2 -3/2 (yes)
It looks to me like the solutions found work in the equation as I interpret it from your question. The only "solution" that would not work is x=3, because that would result in an indeterminate expression on the right side of the equation. Fortunately, x=3 is not one of the solutions we came up with.
