Question

solve for all values of x.
(1)/(x-1) -2= (x)/(x-1)

Answer 1

everything after this step is wrong:
(3 - 2x)/(x-1) = (x)/(x-1)
if x ≠1, then this is just
3-2x = x
x = 1
but x=1 was not allowed, so there are no solutions

Answer 2

To solve for all values of x, we need to first find a common denominator for the fractions on both sides of the equation.

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

Multiplying the second fraction by (x-1)/(x-1) to get a common denominator:

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

Simplifying the fractions:

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

Expanding the numerator:

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

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

Cross-multiplying to eliminate the denominators:

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

Expanding both sides:

3x - 2x^2 = x^2 - x

Rearranging terms:

2x^2 + x - 3x - 3 = 0

2x^2 - 2x - 3 = 0

Solving this quadratic equation using the quadratic formula:

x = (-B ± sqrt(B^2 - 4AC))/(2A)

x = (2 ± sqrt(2^2 - 4*2*(-3))) / (2*2)

x = (2 ± sqrt(4 + 24))/ 4

x = (2 ± sqrt(28)) / 4

x = (2 ± 2sqrt(7)) / 4

Since the equation can't be simplified further, the values of x are:

x = (2 + 2sqrt(7)) / 4
x = (2 - 2sqrt(7)) / 4

Answer 3

You are correct. My apologies for the mistake. Here is the correct solution:

(1)/(x-1) - 2 = (x)/(x-1)
Multiply both sides by (x-1) to get rid of the denominators:
1 - 2(x-1) = x
1 - 2x + 2 = x
3 - 2x = x
3 = 3x
x = 1

However, since x = 1 is not allowed in the original equation as it would make the denominator zero, there are no solutions for this equation. Thank you for pointing out the error.