Question

solve the equation

\\\frac{x + 1}{x - 1} = \frac{-2}{x + 3} + \frac{8}{x^2 + 2x - 3}\\

list all valid solutions, separated by commas.

\\(x =\\)

Explanation:

Factor the quadratic denominator

Using the Factoring Quadratics knowledge point

$$ x^2 + 2x - 3 = (x - 1)(x + 3) $$

Multiply by the least common denominator

Using the Solving Quadratic Equations knowledge point

$$
LATEXBLOCK0
$$

Expand and simplify the equation

$$
LATEXBLOCK1
$$

Solve the quadratic equation

Using the Factoring Quadratics and Solving Quadratic Equations knowledge points

$$
LATEXBLOCK2
$$

Check for extraneous solutions

Since the original equation is undefined at \(x = 1\) (as it makes the denominators \(x - 1\) and \(x^2 + 2x - 3\) equal to zero), \(x = 1\) is an extraneous solution.

Thus, the only valid solution is \(x = -7\).

Answer:

Solve the equation

$$\frac{x + 1}{x - 1} = \frac{-2}{x + 3} + \frac{8}{x^2 + 2x - 3}$$

List all valid solutions, separated by commas.

\(x =\) <blank>-7</blank>