Problem 155 - College Algebra Essentials

Question

Perform the indicated operation or operations. $$\text { Simplify: } \frac{2}{\frac{3}{x}-1}$$

Step-by-Step Solution

Answer

Simplified form of given equation is \( \frac{2x}{3-x} \)

Step 1: Rewrite the equation

By nature, division by fractions results in multiplication by their reciprocal. We must first convert this equation into an easier form. The problem thus becomes: \[2 \times \frac{x}{3-x}\]

Step 2: Cancel like terms

In this step, multiply numerator and denominator. Notice there are no common factors to simplify, so the solution is: \[ \frac{2x}{3-x} \]

Key Concepts

Fraction Operations

When tackling problems involving fractions, understanding how to manipulate them is crucial. Operations on fractions primarily include addition, subtraction, multiplication, and division.
In the context of division, as seen in the exercise, we often deal with complex fractions, which occur when there is a fraction in either the numerator, denominator, or both.
To simplify fraction division, replace the division with multiplication by the reciprocal of the divisor. This involves flipping the divisor fraction upside down (exchanging the numerator and the denominator).

For example, in the exercise, you start with \( \frac{2}{\frac{3}{x} - 1} \). This is a division operation where the term \( \frac{3}{x} - 1 \) is a fraction in the denominator. It's crucial to simplify it by taking the reciprocal of the denominator and multiplying by the numerator of the main fraction. This leads us to multiply by \( \frac{x}{3-x} \), simplifying the division operation efficiently.

Simplifying Expressions

Simplifying expressions is all about reducing them to their simplest form without changing their value, making them easier to work with. When dealing with fractions, it is important to look for opportunities to reduce or cancel terms where possible.
In mathematical operations like the exercise example, simplification happens after multiplication.
After rewriting the original division as a multiplication, we have the expression \( 2 \times \frac{x}{3-x} \), which simplifies to \( \frac{2x}{3-x} \).
Simplifying involves checking for common factors in the numerator and the denominator. However, in this example, there are no common factors between \( 2x \) and \( 3-x \), so the expression remains as is. Simplifying expressions properly can often involve expanding or factoring, depending on the form of the expression.

Reciprocal of Fractions

Understanding reciprocals is fundamental when working with division of fractions. The reciprocal of a fraction is simply another fraction created by swapping the numerator and the denominator positions.
For any fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
In the given exercise, finding the reciprocal is a key step. The original problem involved a complex denominator: \( \frac{3}{x} - 1 \).
To find the reciprocal, you look at the fractional part, which is \( \frac{3}{x} \), flip it to become \( \frac{x}{3} \), and adjust for the entire denominator to treat it appropriately in the division process.

This flips your approach from dividing by the fraction to multiplying by its reciprocal, simplifying the process significantly. Using reciprocals effectively helps transform division into multiplication, which is often more straightforward to compute, thereby streamlining fraction operations.