(x-1)/(x²-1)+(x+2)/(x-1)=2(x-3)

To solve this equation, we first need to find a common denominator for the fractions on the left side of the equation. The common denominator for the fractions is (x-1)(x+1).

So, the equation becomes:

[(x-1)(x+2) + (x+2)(x+1)] / (x-1)(x+1) = 2(x-3)

Expanding the numerators of the fractions, we get:

[(x²+x-2) + (x²+3x+2)] / (x-1)(x+1) = 2(x-3)

Combining like terms in the numerator, we get:

(2x² + 4x) / (x-1)(x+1) = 2(x-3)

Now, we can simplify the equation by multiplying both sides by (x-1)(x+1) to get rid of the denominator:

2x² + 4x = 2(x-3)(x-1)(x+1)

Expanding the right side of the equation, we get:

2x² + 4x = 2(x² - 2x - 3)

Simplifying further, we get:

2x² + 4x = 2x² - 4x - 6

Now, we can subtract 2x² and 4x from both sides of the equation to get:

0 = -8x - 6

Dividing by -8, we get:

x = -3/4

Therefore, the solution to the equation is x = -3/4.