Add or subtract, as indicated. 3/(a - 2) - 1/(2 - a)

Verified step by step guidance

1. Recognize that the denominators \(a - 2\) and \(2 - a\) are negatives of each other. Specifically, \(2 - a = -(a - 2)\).
2. Rewrite the second fraction to have the same denominator as the first fraction: \(-\frac{1}{a - 2}\).
3. Now, the expression becomes \(\frac{3}{a - 2} - \left(-\frac{1}{a - 2}\right)\).
4. Simplify the expression by combining the fractions: \(\frac{3}{a - 2} + \frac{1}{a - 2}\).
5. Combine the numerators over the common denominator: \(\frac{3 + 1}{a - 2}\).

Key Concepts

Understanding Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. To manipulate these expressions, one must understand how to combine them through addition or subtraction, which often involves finding a common denominator. This is crucial for simplifying expressions and solving equations involving rational functions.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. When adding or subtracting rational expressions, it is essential to convert each fraction to have this common denominator to perform the operation correctly. In the given expression, recognizing that (2 - a) is equivalent to -(a - 2) helps in finding the common denominator.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which can include combining like terms and factoring. In the context of rational expressions, this may also involve canceling common factors in the numerator and denominator after performing operations. Mastery of simplification is key to solving algebraic problems efficiently.