# How do you solve #(x-1)/(x+1)= (-3x)/(2x+2) + x/(6x+6)#?

Need to get denominator same for the two additions

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To solve the equation (x-1)/(x+1) = (-3x)/(2x+2) + x/(6x+6), we can start by simplifying the equation. First, we can find a common denominator for the fractions on the right side of the equation, which is (2x+2)(6x+6).

Next, we can multiply each term by the common denominator to eliminate the fractions. This gives us (x-1)(2x+2)(6x+6) = (-3x)(6x+6) + x(2x+2).

Expanding and simplifying both sides of the equation, we get 12x^3 + 12x^2 - 12x - 12 = -18x^2 - 18x + 12x + 12 + 2x^2 + 2x.

Combining like terms, we have 12x^3 + 12x^2 - 12x - 12 = -16x^2 - 4x + 12x + 12.

Further simplifying, we get 12x^3 + 12x^2 - 12x - 12 = -16x^2 + 8x + 12.

Moving all terms to one side of the equation, we have 12x^3 + 12x^2 - 12x - 12 + 16x^2 - 8x - 12 = 0.

Combining like terms again, we get 12x^3 + 28x^2 - 20x - 24 = 0.

Now, we can try to factor the equation. By using synthetic division or factoring by grouping, we find that (x-1)(2x+3)(3x+4) = 0.

Setting each factor equal to zero, we have x-1 = 0, 2x+3 = 0, and 3x+4 = 0.

Solving these equations, we find x = 1, x = -3/2, and x = -4/3.

Therefore, the solutions to the equation (x-1)/(x+1) = (-3x)/(2x+2) + x/(6x+6) are x = 1, x = -3/2, and x = -4/3.

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