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Chapter 7: Problem 46

Simplify complex rational expression. \(\frac{1}{1-\frac{1}{x+1}}-1\)

Short Answer

Expert verified
The simplified form of the given complex rational expression is \(\frac{1}{x}\).

Step by step solution

01

Simplify the Denominator

Firstly, simplify the expression in the denominator of the original expression. This involves rewriting \(1-\frac{1}{x+1}\) as \(\frac{x+1}{x+1} - \frac{1}{x+1}\) which simplifies further to \(\frac{x}{x+1}\). The original expression therefore simplifies to \(\frac{1}{\frac{x}{x+1}} -1\).
02

Simplify the Complex Fraction

Simplify the complex fraction by multiplying the numerator and the denominator by \(x+1\) to get rid of the denominator in the denominator. This results in \(\frac{(x+1)}{x} -1\).
03

Simplify Further

Continue the simplification by realising that \(\frac{(x+1)}{x} -1\) can be rewritten as \(\frac{x+1}{x} - \frac{x}{x}\), which simplifies to \(\frac{1}{x}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Fractions
One fundamental aspect of working with complex rational expressions is simplifying fractions. To simplify a fraction means to rewrite it in its simplest form, where the numerator and the denominator have no common factors other than 1. This simplification process makes the fraction easier to understand and work with. When simplifying fractions within complex rational expressions, like the one in the exercise, ensure to:
  • Identify any common factors in the numerator and the denominator.
  • Rewrite the expression so that these common factors cancel out.
  • Check the simplified fraction to ensure it's as simple as possible.
Simplifying fractions often requires knowledge of basic algebraic operations, as it lays the groundwork for working with more complex expressions.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operators (such as addition or subtraction). The given exercise involves an algebraic expression with variables, specifically a complex rational expression. Understanding how to manipulate algebraic expressions is key to simplifying them. This includes:
  • Recognizing patterns or similar terms.
  • Performing operations like addition or subtraction with these terms.
  • Applying distributive properties and other algebraic principles.
In the exercise, dealing with the inner algebraic expression in the denominator required recognizing that it's a scenario where subtraction of fractions is needed. This demands converting them to a common denominator. Familiarity with such algebraic manipulations is crucial.
Simplification Process
The simplification process involves a series of steps that make expressions as basic and manageable as possible. This process can be broken down into smaller tasks, each serving a role in reducing the complexity of the expression. In our exercise, the simplification process involves:
  • Firstly addressing the denominator, converting it to a single simpler fraction.
  • Simplifying the complex fraction by eliminating complex denominators through multiplication.
  • Reassessing and simplifying the resulting expression to its most reduced form.
Each step builds upon the previous one, transforming the expression methodically. Patience and attention to detail are crucial as missing any of these steps might lead to incorrect simplification. Understanding this process thoroughly equips you with the ability to tackle a wide range of similar algebraic problems.

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Most popular questions from this chapter

Describe how to identify the corresponding sides in similar triangles.

Describe two similarities between the following problems: $$ \frac{3}{8}+\frac{1}{8} \text { and } \frac{x}{x^{2}-1}+\frac{1}{x^{2}-1} $$

What is the relationship among time traveled, distance traveled, and rate of travel?

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I used \(\frac{a}{d}=\frac{b}{e}\) to show that corresponding sides of similar triangles are proportional, but I could also use \(\frac{a}{b}=\frac{d}{e}\) or \(\frac{d}{a}=\frac{e}{b}\)

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3}{x}+\frac{4}{x-6}$$
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