Problem 1 - Introductory Algebra for College Students

Question

multiply as indicated. $$\frac{4}{x+3} \cdot \frac{x-5}{9}$$

Step-by-Step Solution

Answer

The simplified result is \(\frac{4(x - 5)}{9(x + 3)}\).

Step 1: Multiply the Numerators

Begin by multiplying the numerators together. This is done by simply multiplying \(4\) and \(x - 5\) together.

Step 2: Multiply the Denominators

Next, multiply the denominators together, which involves multiplying \(x + 3\) and \(9\).

Step 3: Write The Multiplication Result

Write the multiplication result as a new fraction. The new fraction will have the result from Step 1 as the numerator and the result from Step 2 as the denominator.

Step 4: Simplify If Possible

Finally, simplify the fraction if possible. In this case, there are no common factors in the numerator and the denominator, so the fraction cannot be simplified.

Key Concepts

Understanding Numerators

When multiplying fractions, the numerators play a crucial role. The numerator is the top part of a fraction, which indicates how many parts we have out of a whole. In our exercise, the numerators are 4 and \(x - 5\). Multiplying these together involves taking each term of the first numerator and multiplying it by each term of the second numerator. Here, you simply multiply 4 by \(x - 5\), resulting in \(4(x - 5)\). This is a straightforward arithmetic operation, but it's essential to distribute properly if the numerator involves more complex expressions. Practice this often to gain confidence in handling expressions with variables and coefficients.

Denominators and Their Role

Denominators determine into how many parts the whole is divided. In a fraction, this is the number below the numerator. For our problem, the denominators are \(x + 3\) and 9. To find the denominator of the product, you multiply these two together, resulting in \((x + 3) \times 9\) or \(9(x + 3)\). It is crucial to keep in mind the set of rules for multiplication. If variables are involved, ensure you've accounted for them all in the product, whether they are constants or polynomial expressions. This understanding is foundational for future operations with more complex fractions.

Simplifying Fractions

Simplifying fractions is the process of making them as simple as possible or easier to understand. To simplify, you need to find the greatest common factor (GCF) of both the numerator and the denominator and divide each by this number. However, in our given problem, \(4(x - 5)\) over \(9(x + 3)\) cannot be simplified further since there are no common terms to divide by. Simplifying fractions helps determine whether a multiplication result can be made less complex. Always check for any common factors; if they exist, simplifying makes calculations more manageable and the results clearer.