%2F(x-1)%3D(-3)%2F(x%2B3)%2B(8)%2F(x%5E2%2B2x-3).jpeg "(x+1)/(x-1)=(-3)/(x+3)+(8)/(x^2+2x-3)")
Solving the Rational Equation: (x+1)/(x-1)=(-3)/(x+3)+(8)/(x^2+2x-3)
This article will guide you through the steps to solve the rational equation:
(x+1)/(x-1)=(-3)/(x+3)+(8)/(x^2+2x-3)
1. Factor the denominator:
The denominator of the rightmost term can be factored:
- x² + 2x - 3 = (x + 3)(x - 1)
Now the equation becomes:
(x+1)/(x-1)=(-3)/(x+3)+(8)/((x+3)(x-1))
2. Find the Least Common Multiple (LCM):
The LCM of the denominators (x-1) and (x+3) is (x-1)(x+3).
3. Multiply each term by the LCM:
Multiply both sides of the equation by (x-1)(x+3):
- (x+1)(x+3) = -3(x-1) + 8
4. Simplify and solve the equation:
- x² + 4x + 3 = -3x + 3 + 8
- x² + 7x - 8 = 0
- (x+8)(x-1) = 0
Therefore, the solutions are:
- x = -8
- x = 1
5. Check for extraneous solutions:
It's crucial to check if any of the solutions make the original denominators equal to zero, as this would make the equation undefined.
- x = 1 makes the denominators (x-1) and (x²-2x-3) equal to zero.
- x = -8 does not make any denominator equal to zero.
Therefore, x = 1 is an extraneous solution.
Final Solution:
The only valid solution to the equation is x = -8.