## 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**.