Problem 45 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The simplified form of the given rational complex expression is 0

Step 1: Identify the Inner Fraction

The inner fraction in this case is \(\frac{1}{x}\). This one should be simplified first.

Step 2: Eliminate the Inner Fraction

To remove the fraction within a fraction, multiply the inner fraction and the surrounding expression by the inner fraction's denominator. Therefore, \(x*\frac{1}{1-\frac{1}{x}}*x - x\) should be rewritten as: \(\frac{x}{x-\frac{x}{x}}-x\), which then simplifies to \(\frac{x}{1}-x\).

Step 3: Simplify the Expression

Now, continue simplifying the fraction \(\frac{x}{1}-x\) which results in the final expression: \(x-x\).

Step 4: Calculation

Calculate the final expression: \(x-x\)

Step 5: Final Simplification

The final simplified form of the rational complex expression is \(0\)

Key Concepts

Simplifying Rational Expressions

Simplifying rational expressions is a fundamental concept in algebra. In essence, it involves reducing a complex fraction to its simplest form, making it easier to work with. This can be particularly useful when you're dealing with algebraic expressions that have multiple fractions within them.

• To simplify complex rational expressions, you'll often need to focus on identifying and reducing fractions within fractions.
• Make sure you substitute equivalent expressions where possible to simplify each part of the fraction.
• Simplification helps in understanding and solving algebraic equations with greater ease.

Approaching the task methodically allows the simplification process to be both accurate and efficient, ultimately leading to a clearer understanding of the problem at hand.

Inner Fraction

An inner fraction is a smaller fraction that is part of a larger complex fraction. Recognizing the inner fraction is the first step towards simplifying a complex rational expression.

• In our example, \(1 - \frac{1}{x}\) contains the inner fraction \(\frac{1}{x}\).
• This inner fraction must be addressed before tackling the entire expression.
• Identifying such elements within an expression ensures that you focus on simplifying step by step.

Understanding and isolating the inner fraction lays the groundwork for further fraction elimination and simplification.

Fraction Elimination

Fraction elimination involves removing the inner fraction from the complex expression. This is done by multiplying the entire expression by the denominator of the inner fraction to simplify it.

• In the given exercise, the inner fraction \(\frac{1}{x}\) is eliminated by multiplying the expression by \(x\).
• This multiplication clears the fraction within the larger fraction, which then allows for further simplification of the expression.
• Once this step is complete, the expression becomes easier to manage.

Effective elimination of fractions within fractions often leads to a simpler expression that is much easier to work with.

Algebraic Simplification

Algebraic simplification is the final step in handling rational expressions. It involves reducing the expression to its simplest form by combining like terms and performing basic arithmetic operations.

• In our example, after fraction elimination, we simplify \(\frac{x}{1} - x\) to obtain \(x - x\).
• By performing the arithmetic operation, we find that \(x - x = 0\).
• This simplification provides the final expression which represents the initial complex rational expression in its simplest possible form.

Algebraic simplification is essential for obtaining the simplest version of an expression, making the solution clear and concise.