In Exercises 33–68, add or subtract as indicated. 3/(2x+4) + 2/(3x+6)

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1. Factor the denominators of both fractions to simplify them. For the first fraction, the denominator \( 2x + 4 \) can be factored as \( 2(x + 2) \). For the second fraction, the denominator \( 3x + 6 \) can be factored as \( 3(x + 2) \).
2. Identify the least common denominator (LCD) of the two fractions. The denominators are \( 2(x + 2) \) and \( 3(x + 2) \). The LCD is \( 6(x + 2) \), which is the product of the unique factors from both denominators.
3. Rewrite each fraction with the LCD as the denominator. Multiply the numerator and denominator of the first fraction \( \frac{3}{2(x + 2)} \) by 3, resulting in \( \frac{9}{6(x + 2)} \). Multiply the numerator and denominator of the second fraction \( \frac{2}{3(x + 2)} \) by 2, resulting in \( \frac{4}{6(x + 2)} \).
4. Combine the two fractions into a single fraction since they now have the same denominator. Add the numerators: \( \frac{9}{6(x + 2)} + \frac{4}{6(x + 2)} = \frac{9 + 4}{6(x + 2)} \).
5. Simplify the numerator of the resulting fraction. Combine \( 9 + 4 \) to get \( 13 \), so the final expression is \( \frac{13}{6(x + 2)} \).

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Understanding how to manipulate these expressions is crucial for performing operations like addition and subtraction. In this case, the expressions involve linear polynomials, which can be simplified or combined by finding a common denominator.

Finding a Common Denominator

To add or subtract rational expressions, it is essential to find a common denominator. This involves identifying the least common multiple (LCM) of the denominators involved. For the given problem, the denominators are 2x + 4 and 3x + 6, which can be factored to facilitate finding the LCM.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. In this exercise, recognizing that 2x + 4 can be factored as 2(x + 2) and 3x + 6 as 3(x + 2) is key to simplifying the rational expressions before performing the addition.