To solve the equation
[tex]\[
\frac{x}{x-2} + \frac{x-1}{x+1} = -1
\][/tex]
we will follow these steps:
### Step 1: Simplify the equation by combining the fractions
First, we take the least common denominator (LCD) of the fractions [tex]\(\frac{x}{x-2}\)[/tex] and [tex]\(\frac{x-1}{x+1}\)[/tex], which is [tex]\((x-2)(x+1)\)[/tex]. Rewriting the given equation, we get:
[tex]\[
\frac{x(x+1) + (x-1)(x-2)}{(x-2)(x+1)} = -1
\][/tex]
### Step 2: Expand the numerators
Next, let's expand and simplify the numerator:
[tex]\[
x(x+1) = x^2 + x
\][/tex]
[tex]\[
(x-1)(x-2) = x^2 - 3x + 2
\][/tex]
Adding these together:
[tex]\[
x^2 + x + x^2 - 3x + 2 = 2x^2 - 2x + 2
\][/tex]
So our equation now looks like:
[tex]\[
\frac{2x^2 - 2x + 2}{(x-2)(x+1)} = -1
\][/tex]
### Step 3: Clear the denominator by multiplying both sides by [tex]\((x-2)(x+1)\)[/tex]
Multiplying through by the denominator, we get:
[tex]\[
2x^2 - 2x + 2 = - (x-2)(x+1)
\][/tex]
### Step 4: Expand and simplify the right-hand side
Expanding [tex]\((x-2)(x+1)\)[/tex]:
[tex]\[
(x-2)(x+1) = x^2 - x - 2
\][/tex]
So our equation becomes:
[tex]\[
2x^2 - 2x + 2 = - (x^2 - x - 2)
\][/tex]
Distribute the negative sign on the right-hand side:
[tex]\[
2x^2 - 2x + 2 = -x^2 + x + 2
\][/tex]
### Step 5: Bring all terms to one side and combine like terms
Bringing all terms to the left-hand side:
[tex]\[
2x^2 - 2x + 2 + x^2 - x - 2 = 0
\][/tex]
Combining like terms:
[tex]\[
(2x^2 + x^2) + (-2x - x) + (2 - 2) = 0
\][/tex]
[tex]\[
3x^2 - 3x = 0
\][/tex]
### Step 6: Factor out the common term
Factor out [tex]\(3x\)[/tex]:
[tex]\[
3x(x - 1) = 0
\][/tex]
### Step 7: Solve for [tex]\(x\)[/tex]
Set each factor to zero:
1. [tex]\(3x = 0\)[/tex]
[tex]\[
x = 0
\][/tex]
2. [tex]\(x - 1 = 0\)[/tex]
[tex]\[
x = 1
\][/tex]
Therefore, the solutions to the equation are:
[tex]\[
x = 0 \quad \text{and} \quad x = 1
\][/tex]
