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Magazine Chapter 6: Problem 24
Simplify each complex fraction. $$\frac{x-\frac{x+6}{x+2}}{x-\frac{4 x+15}{x+2}}$$
Short Answer
Expert verified
\(\frac{x - 2}{x - 5}\)
Step by step solution
01
Simplify the Numerator
First, simplify the numerator, which is \(x - \frac{x+6}{x+2}\). Combine the fractions over a common denominator: \(\frac{x(x+2) - (x+6)}{x+2}\).
02
Simplify the Numerator Expression
Expand and simplify the expression in the numerator: \(\frac{x^2 + 2x - x - 6}{x+2} = \frac{x^2 + x - 6}{x+2}\).
03
Simplify the Denominator
Next, simplify the denominator, which is \(x - \frac{4x+15}{x+2}\). Combine the fractions over a common denominator: \(\frac{x(x+2) - (4x+15)}{x+2}\).
04
Simplify the Denominator Expression
Expand and simplify the expression in the denominator: \(\frac{x^2 + 2x - 4x - 15}{x+2} = \frac{x^2 - 2x - 15}{x+2}\).
05
Apply the Division Rule for Complex Fractions
Rewrite the complex fraction as a product: \(\frac{\frac{x^2 + x - 6}{x+2}}{\frac{x^2 - 2x - 15}{x+2}} = \frac{x^2 + x - 6}{x^2 - 2x - 15}\).
06
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 \((x + 3)(x - 5)\).
07
Simplify the Fraction
Cancel out the common factor \(x + 3\): \(\frac{(x + 3)(x - 2)}{(x + 3)(x - 5)} = \frac{x - 2}{x - 5}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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: \(\frac{x^2 + x - 6}{x+2}\).
- 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: \(x(x+2)\).
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: \(\frac{x(x+2) - (4x + 15)}{x+2}\).
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 \((x + 3)(x - 5)\).
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:
Steps:
- Identify the least common denominator (LCD). In this case, it is \(x+2\).
- Combine the fractions, ensuring they share this common denominator.
