Problem 37 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{x+1}{2 x-5} \cdot \frac{x}{x+1} $$

Step-by-Step Solution

Answer

\( \frac{x}{2x-5} \)

Step 1: Simplifying the Expression

First, let's write down the expression: \[ \frac{x+1}{2x-5} \cdot \frac{x}{x+1} \]We are dealing with two fractions being multiplied together. To multiply fractions, multiply the numerators together and the denominators together.

Step 2: Cancel Common Factors

Notice that both the numerator of the first fraction and the denominator of the second fraction have a common factor of \( x+1 \). Cancelling \( x+1 \) from both sides:- Numerator: \( x+1 \cdot x \) becomes \( x \)- Denominator: \( (2x-5) \cdot (x+1) \) becomes \( 2x-5 \)Thus, the expression simplifies to:\[ \frac{x}{2x-5} \]

Key Concepts

Simplifying Expressions

When simplifying expressions involving algebraic fractions, our main goal is to reduce the expression to its simplest form. This involves cleaning up the expression so that it's easy to understand and work with. To simplify algebraic fractions, follow these steps:

• Identify common factors in the numerators and denominators of each fraction.
• Look for opportunities to cancel these common factors, thereby reducing the complexity of the expression.
• After cancelling, rewrite the expression with the remaining factors.

As shown in the original exercise, we started with two fractions and simplified them by identifying and cancelling the common factor of \( x+1 \) to reach the final expression: \( \frac{x}{2x-5} \). This results in a simpler expression that is easier to interpret and use in further calculations.

Multiplying Fractions

Multiplying fractions, whether they are algebraic or numeric, follows a straightforward rule. To multiply two fractions together, you multiply the numerators and then multiply the denominators. Here’s a quick step-by-step guide:

• Multiply the numerators of the fractions together to get a new numerator.
• Multiply the denominators together to get a new denominator.
• If possible, simplify the new fraction by cancelling common factors.

For example, in our exercise problem, we had the fractions \( \frac{x+1}{2x-5} \) and \( \frac{x}{x+1} \). First, we multiplied the numerators: \( (x+1) \times x \), and then the denominators: \( (2x-5) \times (x+1) \). With this, we formed the new fraction before simplification.

Canceling Common Factors

Canceling common factors is a crucial step in simplifying expressions and multiplying fractions. It involves removing the same factors from both the numerator and the denominator to simplify the fraction.Here's how to cancel common factors effectively:

• Identify the common factor in both the numerator and denominator.
• Ensure that the factor is multiplied in both places, allowing it to be cancelled out.
• After cancelling, rewrite the fraction with the remaining parts.

In the exercise source, the factor \( x+1 \) appeared in the numerator of the first fraction and the denominator of the second. By cancelling this shared factor, the expression was simplified from \( \frac{(x+1) \cdot x}{(2x-5) \cdot (x+1)} \) to the much simpler form \( \frac{x}{2x-5} \). Canceling factors not only simplifies the expression but also makes further calculations more manageable.