Given Equation:
[tex]\[
\frac{x}{x-2} + \frac{x-1}{x+1} = -1
\][/tex]
Step 1: Combine the fractions on the left-hand side.
To combine the fractions, we need a common denominator, which is [tex]\((x-2)(x+1)\)[/tex].
[tex]\[
\frac{x(x+1) + (x-1)(x-2)}{(x-2)(x+1)} = \frac{x(x+1) + (x-1)(x-2)}{(x-2)(x+1)}
\][/tex]
Step 2: Simplify the numerators.
Expand the numerators of both fractions:
[tex]\[
x(x+1) = x^2 + x \quad \text{and} \quad (x-1)(x-2) = x^2 - 3x + 2
\][/tex]
Now combine these:
[tex]\[
x^2 + x + x^2 - 3x + 2 = 2x^2 - 2x + 2
\][/tex]
So the fraction becomes:
[tex]\[
\frac{2x^2 - 2x + 2}{(x-2)(x+1)} = -1
\][/tex]
Step 3: Solve the equation by clearing the fraction.
Multiply both sides of the equation by [tex]\((x-2)(x+1)\)[/tex] to clear the denominator:
[tex]\[
2x^2 - 2x + 2 = -1 \cdot (x-2)(x+1)
\][/tex]
Simplify the right-hand side:
[tex]\[
-1 \cdot (x-2)(x+1) = -(x^2 - x - 2)
\][/tex]
[tex]\[
-(x^2 - x - 2) = -x^2 + x + 2
\][/tex]
So now we have:
[tex]\[
2x^2 - 2x + 2 = -x^2 + x + 2
\][/tex]
Step 4: Combine like terms to form a polynomial equation.
Move all terms to one side to set the equation to zero:
[tex]\[
2x^2 - 2x + 2 + x^2 - x - 2 = 0
\][/tex]
Combine the like terms:
[tex]\[
3x^2 - 3x = 0
\][/tex]
Step 5: Factor the resulting equation.
[tex]\[
3x(x - 1) = 0
\][/tex]
The solutions to this equation are found by setting each factor to zero:
[tex]\[
3x = 0 \quad \text{or} \quad x - 1 = 0
\][/tex]
Therefore,
[tex]\[
x = 0 \quad \text{or} \quad x = 1
\][/tex]
Thus, the solutions to the equation are:
[tex]\[
x = 0 \quad \text{or} \quad x = 1
\][/tex]
