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Ch. R - Review of Basic Concepts
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. R - Review of Basic ConceptsProblem 63a
Chapter 1, Problem 63a

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

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

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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