To solve the equation [tex]\(\frac{x}{x-2}+\frac{x-1}{x+1}=-1\)[/tex], follow these steps:
1. Identify a Common Denominator:
The denominators are [tex]\(x-2\)[/tex] and [tex]\(x+1\)[/tex]. The common denominator will be [tex]\((x-2)(x+1)\)[/tex].
2. Rewrite the Equation with a Common Denominator:
Multiply both terms on the left-hand side by [tex]\((x+1)(x-2)\)[/tex]:
[tex]\[
\frac{x(x+1) + (x-1)(x-2)}{(x-2)(x+1)} = -1
\][/tex]
3. Simplify the Numerator:
Expand the terms in the numerator:
[tex]\[
x(x+1) + (x-1)(x-2)
\][/tex]
[tex]\[
x^2 + x + (x^2 - 2x - x + 2)
\][/tex]
Combine like terms:
[tex]\[
x^2 + x + x^2 - 3x + 2 = 2x^2 - 2x + 2
\][/tex]
4. Rewrite the Equation:
Substitute back into the equation:
[tex]\[
\frac{2x^2 - 2x + 2}{(x-2)(x+1)} = -1
\][/tex]
5. Eliminate the Denominator:
Multiply both sides by [tex]\((x-2)(x+1)\)[/tex] to get rid of the denominator:
[tex]\[
2x^2 - 2x + 2 = - (x-2)(x+1)
\][/tex]
6. Expand and Simplify:
Expand [tex]\(-(x-2)(x+1)\)[/tex]:
[tex]\[
- (x^2 - x + 2) = -x^2 + x - 2
\][/tex]
The equation becomes:
[tex]\[
2x^2 - 2x + 2 = -x^2 + x - 2
\][/tex]
7. Combine Like Terms:
Move all terms to one side of the equation:
[tex]\[
2x^2 + x^2 - 2x - x + 2 + 2 = 0
\][/tex]
This simplifies to:
[tex]\[
3x^2 - 3x + 4 = 0
\][/tex]
8. Solve the Quadratic Equation:
Use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 3\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = 4\)[/tex]:
Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = (-3)^2 - 4 \cdot 3 \cdot 4 = 9 - 48 = -39
\][/tex]
9. Analyze the Discriminant:
Since the discriminant [tex]\(\Delta\)[/tex] is negative ([tex]\(-39\)[/tex]), it indicates that there are no real solutions to the quadratic equation [tex]\(3x^2 - 3x + 4 = 0\)[/tex].
Thus, the final answer is that there are no real solutions to the given equation [tex]\(\frac{x}{x-2}+\frac{x-1}{x+1}=-1\)[/tex].
