In Exercises 69–82, simplify each complex rational expression. (3/(x−2) − 4/(x+2))/(7/x²−4)

Verified step by step guidance

1. Step 1: Recognize that the given expression is a complex rational expression, meaning it has a fraction in the numerator and a fraction in the denominator. The numerator is (3/(x−2) − 4/(x+2)), and the denominator is (7/(x^2−4)).
2. Step 2: Simplify the numerator. To combine the terms 3/(x−2) and −4/(x+2), find a common denominator. The least common denominator (LCD) of (x−2) and (x+2) is (x−2)(x+2). Rewrite each fraction with this common denominator.
3. Step 3: Simplify the denominator. Notice that x^2−4 is a difference of squares, which can be factored as (x−2)(x+2). Rewrite the denominator as 7/((x−2)(x+2)).
4. Step 4: Rewrite the entire complex fraction. The numerator will now be a single fraction with the common denominator (x−2)(x+2), and the denominator will be 7/((x−2)(x+2)).
5. Step 5: Simplify the complex fraction by dividing the numerator by the denominator. Dividing by a fraction is equivalent to multiplying by its reciprocal. Multiply the numerator by the reciprocal of the denominator, and simplify further if possible.

Key Concepts

Complex Rational Expressions

A complex rational expression is a fraction where the numerator, the denominator, or both contain rational expressions. Simplifying these expressions often involves finding a common denominator, combining fractions, and reducing the expression to its simplest form. Understanding how to manipulate these fractions is crucial for solving problems involving complex rational expressions.

Finding a Common Denominator

To simplify complex rational expressions, it is essential to find a common denominator for the fractions involved. This process allows for the combination of multiple fractions into a single fraction, making it easier to simplify. The least common denominator (LCD) is typically used to ensure all fractions are expressed with the same base, facilitating addition or subtraction.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing them to their simplest form by canceling common factors in the numerator and denominator. This process may include factoring polynomials, identifying and eliminating common terms, and ensuring that the final expression is in its lowest terms. Mastery of this concept is vital for effectively handling complex rational expressions.