Problem 57 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{x-3}{x-1} \div \frac{x^{2}}{x-1} \div \frac{x-3}{x} $$

Step-by-Step Solution

Answer

\(\frac{x}{x^{2}} = \frac{1}{x} \) after simplifying.

Step 1: Understand the Given Expression

The expression given is a complex fraction. It consists of a main division operation, where the first term \(\frac{x-3}{x-1}\) is divided by the second term \(\frac{x^2}{x-1}\) and then this result is further divided by the third term \(\frac{x-3}{x}\).

Step 2: Apply Division Rule for Fractions

Recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the expression as: \[\left(\frac{x-3}{x-1}\right) \times \left(\frac{x-1}{x^2}\right) \times \left(\frac{x}{x-3}\right)\].

Step 3: Simplify the Expression

With the rewritten expression, look to simplify by canceling common terms in the numerators and denominators:- The \(x-1\) cancels each other out from the first and second fractions.- The \(x-3\) cancels out from the first and third fractions.

Key Concepts

Fraction Operations

Fraction operations often involve more than just simple addition or subtraction. Unlike whole number operations, fractions are numbers that express a part of a whole. To effectively perform operations like division, we need to understand how to manipulate these values so that the operation becomes simpler and more intuitive. Here, division may not seem straightforward at first, but it follows a logical structure that simplifies the process.

• When dealing with fractions, always remember that you can perform operations on both the numerator and the denominator independently.
• Simplifying fractions by finding common terms in the numerator and denominator can help achieve simpler forms.
• Operations such as division, multiplication, addition, or subtraction often rely on changing one of the numbers (often the denominator) to make calculations easier.

Understanding these key points about fraction operations helps build a solid foundation for tackling more complicated expressions, as seen in complex mathematical problems.

Division of Fractions

Dividing fractions can seem tricky at first, but once you understand the fundamental concept, it becomes much easier. The most important thing to remember is that dividing by a fraction is equivalent to multiplying by its reciprocal. This is the key to simplifying division operations with fractions.

When you encounter a division of fractions, as seen in the original exercise, follow these handy steps:

• Identify the fractions you need to divide.
• Take the reciprocal (or "flip") of the fraction you are dividing by.
• Change the division sign to a multiplication sign.

In the given expression, we're dividing two fractions and then another fraction afterwards. By turning the division into multiplication using reciprocals, you simplify the process extensively. Therefore, instead of dividing complex fractions, you're now multiplying them, which is a more straightforward operation.

Reciprocal Multiplication

Reciprocal multiplication is an integral concept when dealing with the division of fractions. A reciprocal of a number or fraction is essentially the 'flipped' version. For a fraction \( \frac{a}{b} \), the reciprocal would be \( \frac{b}{a} \). This operation allows you to turn division into multiplication, simplifying the calculation.

Here's why reciprocal multiplication is useful:

• It converts division problems into multiplication, which are generally easier to handle and solve.
• Working with reciprocals helps to quickly simplify expressions by canceling out terms that appear in both the numerators and denominators.
• Understanding reciprocal multiplication is crucial for solving problems in algebra that involve rational expressions and equations.

For our problem, after identifying and writing the reciprocal of the second and third terms, the expression is rewritten as a multiplication problem. This transformation simplifies solving the expression, revealing any common terms that can be canceled out, leading to a much more manageable solution.