Problem 24 - Algebra for College Students

Question

Simplify each complex fraction. $$\frac{x-\frac{x+6}{x+2}}{x-\frac{4 x+15}{x+2}}$$

Step-by-Step Solution

Answer

\(\frac{x - 2}{x - 5}\)

Step 1: Simplify the Numerator

First, simplify the numerator, which is \(x - \frac{x+6}{x+2}\). Combine the fractions over a common denominator: \(\frac{x(x+2) - (x+6)}{x+2}\).

Step 2: Simplify the Numerator Expression

Expand and simplify the expression in the numerator: \(\frac{x^2 + 2x - x - 6}{x+2} = \frac{x^2 + x - 6}{x+2}\).

Step 3: Simplify the Denominator

Next, simplify the denominator, which is \(x - \frac{4x+15}{x+2}\). Combine the fractions over a common denominator: \(\frac{x(x+2) - (4x+15)}{x+2}\).

Step 4: Simplify the Denominator Expression

Expand and simplify the expression in the denominator: \(\frac{x^2 + 2x - 4x - 15}{x+2} = \frac{x^2 - 2x - 15}{x+2}\).

Step 5: Apply the Division Rule for Complex Fractions

Rewrite the complex fraction as a product: \(\frac{\frac{x^2 + x - 6}{x+2}}{\frac{x^2 - 2x - 15}{x+2}} = \frac{x^2 + x - 6}{x^2 - 2x - 15}\).

Step 6: Factor Numerator and Denominator

Factor both the numerator and the denominator. The numerator \(x^2 + x - 6\) factors to \((x + 3)(x - 2)\), and the denominator \(x^2 - 2x - 15\) factors to \((x + 3)(x - 5)\).

Step 7: Simplify the Fraction

Cancel out the common factor \(x + 3\): \(\frac{(x + 3)(x - 2)}{(x + 3)(x - 5)} = \frac{x - 2}{x - 5}\).

Key Concepts

Numerator Simplification

Simplifying the numerator is a crucial first step in dealing with complex fractions. In this exercise, our numerator is expressed as: \(x - \frac{x+6}{x+2}\). To combine these terms, we need a common denominator.

• Multiply the term without a fraction by the denominator of the fraction you want to combine it with.
• For our example, multiply \(x\) by \(x+2\), resulting in: \(x(x+2)\).

Once they share a common denominator, we rewrite the expression as: \(\frac{x(x+2) - (x+6)}{x+2}\).
Then simplify by expanding and combining like terms. Expand \(x(x+2)\) to get \(x^2 + 2x\). Then subtract: \(x^2 + 2x - x - 6\). Combine to get the simplified numerator: \(\frac{x^2 + x - 6}{x+2}\).

Denominator Simplification

Next, we apply similar steps to simplify the denominator. The denominator starts as: \(x - \frac{4x + 15}{x+2}\). As with the numerator, we need a common denominator.

• Multiply \(x\) by \(x+2\): \(x(x+2)\).
• Rewrite the expression with the common denominator: \(\frac{x(x+2) - (4x + 15)}{x+2}\).

Expand \(x(x+2)\) to get \(x^2 + 2x\), then subtract: \(x^2 + 2x - 4x - 15\). Combine like terms: \(x^2 - 2x - 15\). This simplifies our denominator to: \(\frac{x^2 - 2x - 15}{x+2}\).

Factoring Polynomials

Factoring polynomials is essential in simplifying complex fractions. Factoring involves breaking down a polynomial into its multiplicative components. For this exercise, we need to factor both the numerator and the denominator.

• The numerator, \(x^2 + x - 6\), factors to \((x + 3)(x - 2)\).
• The denominator, \(x^2 - 2x - 15\), factors to \((x + 3)(x - 5)\).

Factoring helps us find and cancel out common terms, which simplifies the fraction further.

Common Denominators

Understanding common denominators is central when combining fractions. A common denominator allows you to add or subtract fractions easily. For both the numerator and the denominator in our example, we used the shared denominator \(x+2\).
Steps:

• Identify the least common denominator (LCD). In this case, it is \(x+2\).
• Combine the fractions, ensuring they share this common denominator.

This method enables consistent and structured steps toward simplifying any complex fraction.