Problem 120 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{\frac{1}{x}+\frac{2}{x}}{\frac{1}{x-1}+\frac{x}{2}} $$

Step-by-Step Solution

Answer

\(\frac{6(x-1)}{x(x^2-x+2)}\)

Step 1: Add fractions in the numerator

The numerator of the expression is \( \frac{1}{x} + \frac{2}{x} \). Since both terms have the same denominator, we can simply add the numerators to get: \[ \frac{1+2}{x} = \frac{3}{x}. \]

Step 2: Simplify the denominator

The denominator of the expression is \( \frac{1}{x-1} + \frac{x}{2} \). To add these, we need a common denominator, which is \((x-1) \cdot 2 = 2(x-1)\). Rewriting each fraction: \[ \frac{1}{x-1} = \frac{2}{2(x-1)}, \quad \frac{x}{2} = \frac{x(x-1)}{2(x-1)}. \] Add these fractions: \[ \frac{2 + x(x-1)}{2(x-1)} = \frac{x^2-x+2}{2(x-1)}. \]

Step 3: Write the simplified fraction

Combine the simplified numerator and denominator into a single fraction: \[ \frac{\frac{3}{x}}{\frac{x^2-x+2}{2(x-1)}}. \]

Step 4: Simplify the complex fraction

To simplify \( \frac{\frac{3}{x}}{\frac{x^2-x+2}{2(x-1)}} \), we multiply by the reciprocal of the denominator: \[ \frac{3}{x} \cdot \frac{2(x-1)}{x^2-x+2}. \] This results in: \[ \frac{3 \cdot 2(x-1)}{x(x^2-x+2)} = \frac{6(x-1)}{x(x^2-x+2)}. \]

Key Concepts

Common Denominator

When simplifying rational expressions, one crucial step is finding a common denominator. This is a necessary step before you can add or subtract fractions. A common denominator is essentially a shared multiple of the denominators of each fraction involved.

• To find a common denominator, determine the least common multiple (LCM) of all the denominators.
• Rewrite each fraction so they share this common denominator.
• In our problem, the denominators in the denominator of the original expression are \( x-1 \) and \( 2 \). Their common denominator is the product \( 2(x-1) \).

After finding this common denominator, you can then add the fractions together, combining their numerators over your newfound common denominator. This sets the stage for simplifying more complex expressions.

Complex Fractions

Complex fractions are fractions where the numerator, denominator, or both are themselves fractions. Simplifying these involves several steps. A common method involves rewriting the complex fraction as a multiplication problem.

• Identify the main numerator and denominator of the complex fraction.
• Simplify each separately if needed, as you saw with the expression's numerator and denominator originally.
• Re-write the complex fraction as a division problem. For example, \( \frac{\frac{3}{x}}{\frac{x^2-x+2}{2(x-1)}} \) becomes a fraction multiplication: \( \frac{3}{x} \times \frac{2(x-1)}{x^2-x+2} \).

By doing this, you transform the complex fraction into something much simpler, aiding in more effortless calculation and leading towards a more straightforward form.

Algebraic Addition

Adding algebraic fractions utilizes the same principle as adding numerical fractions; the key lies in the denominators. Once a common denominator is established, addition can easily proceed by adjusting the numerators accordingly.

• If the fractions already share a common denominator, simply add the numerators.
• If not, find a common denominator first, as in aligning \( \frac{1}{x} + \frac{2}{x} \) into \( \frac{3}{x} \), giving the expression a unified base that allows direct addition.
• The process ensures precise simplification, allowing you to maintain the expression integrity.

Algebraic addition may seem complex at first, but with a step-by-step approach, it becomes highly manageable and even intuitive over time!