Solve each equation. x/x+2 + 1/x+3 = 2/(x²+2x)

Verified step by step guidance

1. Rewrite the equation clearly to avoid ambiguity: \(\frac{x}{x+2} + \frac{1}{x+3} = \frac{2}{x^{2} + 2x}\).
2. Factor the denominator on the right side: \(x^{2} + 2x = x(x+2)\), so the equation becomes \(\frac{x}{x+2} + \frac{1}{x+3} = \frac{2}{x(x+2)}\).
3. Identify the least common denominator (LCD) for all terms, which is \(x(x+2)(x+3)\), and multiply every term by this LCD to eliminate the denominators.
4. After multiplying, simplify each term by canceling common factors, resulting in a polynomial equation without fractions.
5. Collect like terms and solve the resulting polynomial equation for \(x\), then check for any restrictions from the original denominators to exclude invalid solutions.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. Understanding how to manipulate these expressions, including simplifying and finding common denominators, is essential for solving equations involving rational terms.

Least Common Denominator (LCD)

The least common denominator is the smallest expression that all denominators in an equation can divide into without remainder. Finding the LCD allows you to combine or clear fractions by multiplying through, simplifying the process of solving rational equations.

Solving Rational Equations

Solving rational equations involves eliminating denominators by multiplying both sides by the LCD, then solving the resulting polynomial equation. It's important to check for extraneous solutions that make any denominator zero, as these are not valid.