Problem 36 - Introductory Algebra for College Students

Question

Simplify complex rational expression by the method of your choice. \(\frac{x+9-\frac{7}{x}}{x-6+\frac{4}{x}}\)

Step-by-Step Solution

Answer

\(\frac{x^2 + 9x - 7}{x^2 - 6x + 4}\)

Step 1: Rewrite the Complex Fractions

This step involves rewriting each of the complex fraction into one single fraction in both the numerator and the denominator. We do this by finding the common denominator and simplifying. The common denominator is \(x\).The numerator becomes \(x^2 + 9x - 7\).The denominator becomes \(x^2 - 6x + 4\).

Step 2: Simplify the Expression

Now, let's divide, we have \(\frac{x^2 + 9x - 7}{x^2 - 6x + 4}\). This fraction cannot be simplified further so it's the final answer.

Step 3: Check the Solution

You could check by substituting \(x\) with any reasonable number, say \(x = 1\), to the simplified and original expression and confirm if they give the same result.

Key Concepts

Simplifying Fractions

Simplifying fractions means reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. With complex rational expressions, simplifying fractions is a crucial step.
To simplify, you follow these straightforward steps:

• First, identify any common factors in the numerator or the denominator.
• Next, divide both the numerator and denominator by these common factors.
• The result is the simplest form of the fraction.

When working with expressions with variables, watch out for like terms or common factors and apply the same process. Reducing complex rational expressions streamlines the problem, making it easier to understand or solve.

Finding Common Denominators

Finding a common denominator is essential when dealing with fractions, especially with complex rational expressions. It refers to making the bottom parts (denominators) of two or more fractions the same. This process is critical for combining or comparing fractions.
Here's how to find a common denominator:

• Determine the Least Common Multiple (LCM) of the denominators. If they are polynomials, find the LCM of polynomial terms.
• Multiply each fraction by the suitable expression that makes their denominators equal.
• Once the denominators are the same, you can easily add, subtract, or simplify the fractions as needed.

In complex rational expressions, a common denominator allows all parts of the expression to interact smoothly, helping simplify the entire expression, as seen in the original exercise.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (such as +, -, *, /). They can represent numbers and operations and are a fundamental part of algebra.
When working with algebraic expressions:

• Identify and group like terms. These are terms that have the same variables raised to the same power.
• Apply algebraic rules, such as distributive, associative, and commutative properties, to simplify or rearrange the expressions.
• Be mindful of the operations involved, especially when dealing with negative signs and division.

In the context of simplifying complex rational expressions, recognizing algebraic expressions helps in rewriting terms and finding common factors. This makes the simplification process smoother and more straightforward.