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

Verified step by step guidance

1. First, rewrite the equation clearly to avoid confusion with the denominators: \(\frac{x}{x+2} + \frac{1}{x+3} = \frac{2}{x^{2} + 2x}\). Notice that the denominator on the right side can be factored.
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 in the equation by this LCD to eliminate the denominators.
4. After multiplying, simplify each term by canceling the common factors in the denominators and numerators, resulting in a polynomial equation without fractions.
5. Collect like terms and rearrange the equation to one side to set it equal to zero, then solve the resulting polynomial equation for \(x\). Remember to check for any restrictions where denominators would be zero.

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 a rational equation can divide into evenly. Finding the LCD allows you to combine or clear fractions by multiplying both sides of the equation, simplifying the solving process.

Solving Rational Equations

Solving rational equations involves eliminating denominators by multiplying through 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.