### Instruction: Find all solutions to the equation \[\frac{\left(\frac{x}{x+1}\right)^2 + 11}{\left(\frac{x}{x+1}\right)^2 + 1} = 2.\]Enter all the solutions, separated by commas.### Response: This equation looks complicated, but maybe I can simplify it by making a substitution.
Let me try to let \[y = \frac{x}{x+1}\] and see what happens.
If I do that, then the equation becomes \[\frac{y^2 + 11}{y^2 + 1} = 2.\]
That looks much easier to work with.
Maybe I can clear the fractions by multiplying both sides by \[y^2 + 1\].
That gives me \[y^2 + 11 = 2y^2 + 2.\]
Now I can bring everything to one side and get a quadratic equation: \[y^2 - 9 = 0.\]
To solve for y, I can factor the equation as \[(y + 3)(y - 3) = 0.\]
Then I get two possible values for y: \[y = -3 \text{ or } y = 3.\]
But I'm not done yet, because I need to find the values of x that correspond to these values of y. I can use the original substitution to do that: \[y = \frac{x}{x+1}\] and solve for x.
For y = -3, I get \[-3 = \frac{x}{x+1}\] which means \[-3x - 3 = x\] which simplifies to \[x = -\frac{3}{4}.\]
For y = 3, I get \[3 = \frac{x}{x+1}\] which means \[3x + 3 = x\] which simplifies to \[x = -\frac{3}{2}.\]
So I have two solutions for x: \[x = -\frac{3}{4} \text{ or } x = -\frac{3}{2}.\]