Problem 13 - Algebra 2

Question

Simplify each expression. $$ \frac{x+\frac{x}{3}}{x-\frac{x}{6}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{8}{5} \).

Step 1: Identify the Common Denominator

To simplify the inner fractions, start by finding a common denominator for the fractions inside the larger expression. The expression has two inner fractions: \( \frac{x}{3} \) and \( \frac{x}{6} \). The common denominator for these fractions is 6.

Step 2: Rewrite the Inner Fractions

Rewrite each fraction with the common denominator. For \( \frac{x}{3} = \frac{2x}{6} \) and \( \frac{x}{6} \) remains the same. Rewrite the main fraction using these expressions: \[\frac{x + \frac{2x}{6}}{x - \frac{x}{6}}.\]

Step 3: Combine the Numerator and Denominator

Combine the terms in the numerator and the denominator by rewriting them with the common denominator. For the numerator, \( x + \frac{2x}{6} = \frac{6x}{6} + \frac{2x}{6} = \frac{8x}{6} \). For the denominator, \( x - \frac{x}{6} = \frac{6x}{6} - \frac{x}{6} = \frac{5x}{6} \). So the expression becomes: \[\frac{\frac{8x}{6}}{\frac{5x}{6}}.\]

Step 4: Simplify the Fraction

To simplify \( \frac{\frac{8x}{6}}{\frac{5x}{6}} \), multiply both the numerator and the denominator by 6 to eliminate the fractions. This results in \( \frac{8x}{5x} \).

Step 5: Simplify Further

Cancel the common factor \( x \) from the numerator and the denominator, resulting in \( \frac{8}{5} \).

Step 6: Final Result

The simplified form of the given expression is \( \frac{8}{5} \).

Key Concepts

Common Denominator

When it comes to simplifying algebraic expressions, the concept of a common denominator is crucial, especially when you are dealing with fractions. A common denominator is essentially a shared multiple of the denominators in two or more fractions. Finding a common denominator allows you to combine or subtract fractional terms smoothly.

• It provides a common ground where different fractions can be compared or combined.
• The common denominator should be at least as large as the greatest of the denominators you're working with.

In the given exercise, fractions like \( \frac{x}{3} \) and \( \frac{x}{6} \) needed to be expressed with the same denominator. Here, the smallest common denominator is 6.
By rewriting \( \frac{x}{3} \) as \( \frac{2x}{6} \), both fractions inside the larger algebraic expression share the denominator of 6, making it easier to combine them and proceed with the simplification process.

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form. Once you have a common denominator, combining fractions becomes straightforward. The focus shifts to simplifying the resulting expression. Let's break it down:

• First, combine terms using the common denominator to create a single term for the numerator and denominator individually.
• Next, cancel out the common density in both the numerator and denominator.

In this exercise, the expression \( \frac{x + \frac{2x}{6}}{x - \frac{x}{6}} \) is rewritten with fractions \( \frac{8x}{6} \) and \( \frac{5x}{6} \).
Multiplying through by 6 eliminates the fractional terms, transforming it to \( \frac{8x}{5x} \). The last step involves canceling out the common factor \( x \) in both the numerator and denominator, leaving you with the simplified fraction \( \frac{8}{5} \). This is a classic example of simplifying fractions where extra factors are removed.

Algebraic Fractions

Algebraic fractions involve variables in their expressions, which can often make simplification seem difficult. But the core idea remains similar to numeric fractions: simplify by finding a common denominator, combine like terms, and reduce to the simplest form.

• Algebraic fractions share similar rules with numeric fractions but incorporate variables.
• This process often includes factoring and canceling out common factors.

In algebraic terms, simplifying the expression \( \frac{\frac{8x}{6}}{\frac{5x}{6}} \) involves managing both the numerical and variable components. Variables like \( x \) are common in both the numerator and denominator, which allows us to cancel them out, leaving the simplified form \( \frac{8}{5} \).
Understanding how to manage algebraic fractions aids in various aspects of math, from solving equations to integrating these concepts into more complex expressions. The approach to working with algebraic fractions simplifies broader algebraic manipulations, making it less daunting.