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\( are closely related. Notice that \)2 - a\( can be rewritten as \)-(a - 2)\( because \)2 - a = -(a - 2)$.
2. Rewrite the expression using this relationship: \(\frac{3}{a - 2} - \frac{1}{2 - a} = \frac{3}{a - 2} - \frac{1}{-(a - 2)}\).
3. Simplify the second fraction by factoring out the negative sign in the denominator: \(\frac{3}{a - 2} + \frac{1}{a - 2}\).
4. Since both fractions now have the same denominator \(a - 2\), combine the numerators over the common denominator: \(\frac{3 + 1}{a - 2}\).
5. Add the numerators to get \(\frac{4}{a - 2}\). This is the simplified form of the original expression.

Key Concepts

Understanding Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. Simplifying or performing operations on them requires careful handling of variables and expressions to avoid undefined values, especially when denominators involve variables.

Recognizing Equivalent Expressions

Expressions like (a - 2) and (2 - a) are related but differ by a sign. Recognizing that (2 - a) = -(a - 2) helps in rewriting terms with a common denominator, which is essential for adding or subtracting rational expressions.

Adding and Subtracting Rational Expressions

To add or subtract rational expressions, they must have a common denominator. This often involves factoring or rewriting denominators to find a common base, then combining the numerators accordingly before simplifying the result.