Simplify each complex rational expression. [3-1/(x+3)]/[3+1/(x+3)]

Verified step by step guidance

1. Rewrite the numerator and denominator of the complex fraction separately. The numerator is [3 - 1/(x+3)] and the denominator is [3 + 1/(x+3)].
2. Find a common denominator for the terms in the numerator and denominator. The common denominator for both is (x+3).
3. Simplify the numerator: Rewrite 3 as (3(x+3)/(x+3)) to have the same denominator. Combine the terms to get [(3(x+3) - 1)/(x+3)].
4. Simplify the denominator: Rewrite 3 as (3(x+3)/(x+3)) to have the same denominator. Combine the terms to get [(3(x+3) + 1)/(x+3)].
5. Divide the simplified numerator by the simplified denominator. This is equivalent to multiplying the numerator by the reciprocal of the denominator. Simplify the resulting expression.

Key Concepts

Complex Rational Expressions

A complex rational expression is a fraction where the numerator, the denominator, or both contain rational expressions. To simplify these expressions, one must often find a common denominator for the fractions involved, allowing for easier manipulation and simplification of the overall expression.

Finding a Common Denominator

Finding a common denominator is essential when adding, subtracting, or simplifying fractions. In the context of complex rational expressions, it involves identifying a denominator that can be used to combine the fractions in the numerator and denominator, facilitating the simplification process.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms by dividing the numerator and denominator by their greatest common factor (GCF). In the case of complex rational expressions, this step is crucial after combining terms, as it leads to a more concise and manageable form of the expression.