Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operations and simplify.\(\frac{x+2}{5(x-3)} \div \frac{x-2}{5(x-3)}\)

Step-by-Step Solution

Answer

The simplified version of the original expression is \(\frac{x+2}{x-2}\).

Step 1: Rewrite the Division as Multiplication

The first operation to apply here is to convert the division to a multiplication operation by finding the reciprocal (flipping the numerator and the denominator) of the second fraction.\[ \frac{x+2}{5(x-3)} \div \frac{x-2}{5(x-3)} = \frac{x+2}{5(x-3)} \times \frac{5(x-3)}{x-2}\]

Step 2: Cancel the Common Factors

Both fractions now have a shared term, '5(x-3)', in their numerator and denominator which can be cancelled out.\[ \frac{x+2}{5(x-3)} \times \frac{5(x-3)}{x-2} = \frac{x+2}{x-2}\]

Step 3: Simplify the Resulting Fraction

Check for any common factors between the numerator and the denominator of the final fraction that can be simplified. The final fraction \(\frac{x+2}{x-2}\) is already simplified as there are no common factors to cancel out.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a critical skill in mathematics that allows us to rearrange expressions and equations to make them easier to work with. In the context of rational expressions, the key goal is often to simplify or solve the expression. In this exercise, we began by transforming a division problem into a multiplication problem, which is a common algebraic technique.

To perform this transformation, we find the reciprocal of the second fraction involved in the division. When you flip the numerator and the denominator of a fraction, that fraction becomes its reciprocal:

• The fraction \(\frac{x-2}{5(x-3)}\) becomes \(\frac{5(x-3)}{x-2}\).

Converting division to multiplication simplifies operations because multiplication of fractions is often more straightforward. Once fractions are in multiplication form, you can more easily see how to simplify the expression by canceling out any common factors. This process is a prime example of using algebraic manipulation to solve problems effectively.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form by eliminating common factors between the numerator and the denominator. In our example, once we had rewritten the division as a multiplication, we noted that both fractions shared the same term, \(5(x-3)\), which could be canceled out. By removing these common factors, we simplified the operation greatly.

The process of canceling involves:

• Identifying the common terms in both the numerator and the denominator of the multiplication equation.
• Crossing them out, which effectively reduces the complexity of the expression.

After canceling, our new simplified expression was \(\frac{x+2}{x-2}\). There were no further common factors in this fraction, so it remained in its simplest form. Simplifying fractions minimizes the chance of errors and makes further calculations easier.

Division of Fractions

Dividing fractions sounds tricky, but it's all about transforming the division into a multiplication. When dividing by a fraction, you multiply by its reciprocal. This simple step changes the way we approach the problem and often makes the solution evident.

The steps include:

• Taking the reciprocal of the divisor fraction. For instance, \(\frac{x-2}{5(x-3)}\) becomes \(\frac{5(x-3)}{x-2}\).
• Changing the division sign to a multiplication sign, thus transforming \(\frac{x+2}{5(x-3)} \div \frac{x-2}{5(x-3)}\) into \(\frac{x+2}{5(x-3)} \times \frac{5(x-3)}{x-2}\).

Once in multiplication format, it becomes easier to spot and cancel any common terms between the fractions, making the expression simpler to solve. Understanding division as multiplication with the reciprocal is a foundational skill that facilitates deeper understanding in algebra.