(x+1)/(x-1)=(-3)/(x+3)+(8)/(x^2+2x-3)

2 min read Jun 16, 2024
(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.

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