Problem 59 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{5}{3 x-2}+\frac{6}{4 x+5} $$

Step-by-Step Solution

Answer

\(\frac{38x + 13}{(3x - 2)(4x + 5)}\) is the simplest form.

Step 1: Identify the Least Common Denominator (LCD)

The denominators of the expressions are \(3x - 2\) and \(4x + 5\). Since they have no common factors, the least common denominator is the product of the two: \((3x - 2)(4x + 5)\).

Step 2: Write Each Expression with the LCD

To add the fractions, each must have the same denominator. Multiply the numerator and denominator of \( \frac{5}{3x-2} \) by \(4x+5\), and \( \frac{6}{4x+5} \) by \(3x-2\). This gives: \[ \frac{5(4x+5)}{(3x-2)(4x+5)} + \frac{6(3x-2)}{(4x+5)(3x-2)} \].

Step 3: Distribute and Expand the Numerators

Expand the expressions in the numerators: For the first fraction, \(5(4x+5)\) becomes \(20x + 25\). For the second fraction, \(6(3x-2)\) becomes \(18x - 12\). The expression is now:\[ \frac{20x + 25}{(3x - 2)(4x + 5)} + \frac{18x - 12}{(3x - 2)(4x + 5)} \].

Step 4: Add the Numerators

Since the denominators are now the same, add the numerators together:\[(20x + 25) + (18x - 12) = 38x + 13\]. This results in:\[ \frac{38x + 13}{(3x - 2)(4x + 5)} \].

Step 5: Simplify the Expression

Check the numerator \(38x + 13\) for any common factors or further simplification. Since there are no common factors, the expression is already in its simplest form.

Key Concepts

Least Common Denominator

When working with rational expressions, it's important to find a common ground between different fractions. This brings us to the concept of the Least Common Denominator or LCD. The LCD is crucial when adding or subtracting fractions because it allows us to combine them into a single expression. In our exercise, the denominators were \(3x - 2\) and \(4x + 5\). They have no common factors, so we create the least common denominator by multiplying these two distinct expressions. Thus, the LCD becomes \((3x - 2)(4x + 5)\). By doing this, we set a common base that enables us to handle the algebraic fractions as one entity.

Algebraic Fractions

Algebraic fractions are similar to the fractions we know from arithmetic, but they contain variables along with numbers. Just like numerical fractions, they consist of a numerator and a denominator. In the given exercise, our initial algebraic fractions are \(\frac{5}{3x-2}\) and \(\frac{6}{4x+5}\). Each fraction represents a part of a whole, divided by its respective denominator.

• Handling algebraic fractions often requires finding the LCD to simplify operations like addition or subtraction.
• To work with these fractions effectively, we need to align their denominators, making calculations straightforward.

By multiplying each fraction by a form of one (in this case, the other fraction's denominator divided by itself), we achieve a common denominator and can proceed with addition.

Simplification of Expressions

Simplifying expressions is a vital step in solving algebra problems, ensuring that the answer is as neat and concise as possible. After finding a common denominator and aligning the algebraic fractions, we focus on simplifying the resulting expression. Simplification involves various steps, including distributing, combining like terms, and checking for common factors. In our exercise, after combining the fractions over a common denominator, we had the expression \(\frac{38x + 13}{(3x - 2)(4x + 5)}\). Here, simplification meant verifying whether the numerator, \(38x + 13\), had any common factors with itself or with the denominator. Since there were no common factors, the expression was already simplified.

• Always look to simplify your expressions as a final step.
• Simplifying can sometimes reveal an answer that appears more complex at first glance.

Remember, the goal is to make the final expression as simple and as easy to understand as possible.