Problem 26 - Elementary Algebra

Question

Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{4(x-3)}{5 x}+\frac{2(x+6)}{5 x}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{6}{5} \).

Step 1: Identify the Problem

We are given the expression \( \frac{4(x-3)}{5x} + \frac{2(x+6)}{5x} \) and need to add these fractions.

Step 2: Identify Common Denominator

Both fractions have the same denominator \( 5x \). This means we can directly add the numerators.

Step 3: Add the Numerators

Add the numerators of the two fractions: \( 4(x-3) + 2(x+6) \).

Step 4: Distribute the Numbers

First, expand each term: \( 4(x-3) = 4x - 12 \) and \( 2(x+6) = 2x + 12 \).

Step 5: Combine Like Terms

Combine the numerators: \( 4x - 12 + 2x + 12 = 6x \).

Step 6: Simplify the Fraction

The new numerator is \( 6x \), so the expression becomes \( \frac{6x}{5x} \). Simplify this by cancelling \( x \) in the numerator and the denominator.

Step 7: Final Answer

After simplifying, we are left with \( \frac{6}{5} \).

Key Concepts

Fraction Addition

Fraction addition involves adding the numerators of two or more fractions while keeping the denominator common. This is crucial because the fractions need to be aligned in a way that makes mathematical sense.

When you add fractions, start by ensuring all fractions involved have the same denominator. If they don't, you need to convert them so they do. Once you've ensured the denominators are the same:

• Keep the denominator the same while adding the fractions.
• Add the numerators together to get the new numerator.

In the exercise, we had fractions with a shared denominator, i.e., \( \frac{4(x-3)}{5x} + \frac{2(x+6)}{5x} \). Therefore, the easiest step is to add the numerators directly since the denominators don't need to change. This simplifies the process greatly and ensures clarity in further steps.

Common Denominator

A common denominator is essential for adding fractions because it aligns the fractions to a common base. Without this, addition can't take place nicely, as fractions would represent different portions of a whole.

To identify a common denominator when adding fractions:

• Look at the denominators to see if they're the same already, as in our example with \( 5x \).
• If they're not the same, you can find the least common multiple (LCM) of the denominators.

In the given problem, both fractions already had a common denominator of \( 5x \). This allowed for straightforward addition of the numerators without additional steps or adjustments. Keeping denominators common simplifies the operation and ensures accuracy.

Simplifying Expressions

Simplifying expressions is crucial for making results more readable and easier to understand. In mathematics, the simplest form often provides the clearest insights into the nature of the expression.

When simplifying, consider these steps:

• Distribute any numbers or variables outside parentheses to terms inside. For example, distribute \( 4 \) across \( (x-3) \) and \( 2 \) across \( (x+6) \).
• Combine like terms by adding coefficients of similar variables.
• Reduce fractions by cancelling common factors in the numerator and denominator.
• Check for potential simplification through factorization or cancelling.

In the final step of our worked example, the fraction \( \frac{6x}{5x} \) was simplified by cancelling the common factor \( x \), leaving us with \( \frac{6}{5} \). This step removes unnecessary complexity and ensures the result is easily interpretable. Simplifying expressions is always a good practice in algebra, as it leads to cleaner and more concise results.