Solve each equation in Exercises 83–108 by the method of your choice. 2x/(x - 3) + 6/(x + 3) = - 28/(x² - 9)

Verified step by step guidance

1. Rewrite the equation and recognize that the denominator \(x^2 - 9\) is a difference of squares, which can be factored as \((x - 3)(x + 3)\). This will help simplify the equation.
2. Multiply through by the least common denominator (LCD), which is \((x - 3)(x + 3)\), to eliminate the fractions. Be careful to distribute the LCD to each term in the equation.
3. Simplify each term after multiplying by the LCD. For example, \(\frac{2x}{x - 3} \cdot (x - 3)(x + 3)\) simplifies to \(2x(x + 3)\), and so on for the other terms.
4. Combine like terms and simplify the resulting equation. This will likely result in a quadratic equation.
5. Solve the quadratic equation using factoring, the quadratic formula, or completing the square. Be sure to check for any restrictions on the variable (e.g., \(x \neq 3\) and \(x \neq -3\)) to ensure the solution is valid.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including finding common denominators and simplifying, is crucial for solving equations involving them. In this problem, the presence of rational expressions necessitates careful handling to combine and solve the equation effectively.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components (factors) that, when multiplied together, yield the original polynomial. In this equation, recognizing that the denominator x^2 - 9 can be factored into (x - 3)(x + 3) is essential for simplifying the equation and eliminating the fractions.

Finding Common Denominators

Finding a common denominator is a key step in adding or subtracting rational expressions. It allows for the combination of fractions into a single expression. In this equation, identifying the least common denominator (LCD) of the fractions involved will facilitate the elimination of the denominators, making it easier to solve for x.