Add or subtract, as indicated. See Example 4. (3/a - 2) - (1/2 - a)

Verified step by step guidance

1. Identify the given expression: \(\frac{3}{a - 2} - \frac{1}{2 - a}\).
2. Notice that the denominators \(a - 2\) and \$2 - a\( are related. Recall that \)2 - a = -(a - 2)$, so rewrite the second fraction as \(\frac{1}{2 - a} = \frac{1}{-(a - 2)} = -\frac{1}{a - 2}\).
3. Substitute this back into the expression to get \(\frac{3}{a - 2} - \left(-\frac{1}{a - 2}\right)\), which simplifies to \(\frac{3}{a - 2} + \frac{1}{a - 2}\).
4. Since both fractions now have the same denominator, combine the numerators: \(\frac{3 + 1}{a - 2} = \frac{4}{a - 2}\).
5. The simplified expression is \(\frac{4}{a - 2}\). This is the combined result of the original subtraction.

Key Concepts

Finding a Common Denominator

When adding or subtracting fractions, it is essential to express them with a common denominator. This allows the numerators to be combined directly. For algebraic fractions, the common denominator is typically the least common multiple of the individual denominators.

Simplifying Algebraic Expressions

After combining fractions, simplifying the resulting algebraic expression involves factoring and reducing terms. This step ensures the expression is in its simplest form, making it easier to interpret or use in further calculations.

Handling Variable Expressions in Denominators

When denominators contain variables, special care is needed to avoid division by zero and to correctly manipulate expressions. Understanding how to factor and find common denominators involving variables is crucial for accurate addition or subtraction.