How do you solve #1=(x+3)/(-2x+2)# and find any extraneous solutions?

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To solve the equation 1=(x+3)/(-2x+2) and find any extraneous solutions, we can follow these steps:

1. Multiply both sides of the equation by (-2x+2) to eliminate the denominator: (-2x+2) * 1 = (-2x+2) * (x+3)/(-2x+2)

2. Simplify the equation: -2x + 2 = x + 3

3. Move all the x terms to one side and the constant terms to the other side: -2x - x = 3 - 2

4. Combine like terms: -3x = 1

5. Divide both sides of the equation by -3 to solve for x: x = 1/-3

6. Simplify the result: x = -1/3

To check for extraneous solutions, substitute the found value of x back into the original equation and see if it holds true. In this case, substituting x = -1/3 into the original equation gives:

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

After simplifying, we find that both sides of the equation are equal to 1. Therefore, there are no extraneous solutions in this case.