Problem 32 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{x+10}{x-4} \cdot \frac{x-4}{x-1} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: $ \frac{x+10}{x-4} \cdot \frac{x-4}{x-1} $ Answer: $\frac{x+10}{x-1}$

Step 1: Identify Common Factors

First, we can notice that the numerator of the second fraction and the denominator of the first fraction have a common factor, \((x-4)\). This will be helpful when simplifying the expression.

Step 2: Cancel Out Common Factors

Now, we can cancel out the common factor \((x-4)\) in the numerator and denominator: $$ \frac{x+10}{x-4} \cdot \frac{x-4}{x-1} = \frac{x+10}{\cancel{x-4}} \cdot \frac{\cancel{x-4}}{x-1} $$

Step 3: Perform the Multiplication

After canceling out the common factors, we can perform the multiplication and obtain the final simplified expression: $$ \frac{x+10}{1} \cdot \frac{1}{x-1} = \frac{x+10}{x-1} $$ So the result is: $$ \frac{x+10}{x-1} $$

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that makes handling equations much more manageable. When you simplify an expression, you aim to write it in the most reduced form. Doing this involves combining like terms and eliminating any unnecessary parts. The process helps in understanding complex expressions by making them simpler and more straightforward.

In the context of rational expressions, simplification means cancelling out any common factors that appear in both the numerator and the denominator. For example, in the expression \(\frac{x+10}{x-4} \cdot \frac{x-4}{x-1}\), the \((x-4)\) appears in both the numerator of the second fraction and the denominator of the first. This tells us it can be cancelled out from the expression, making it easier to simplify further.

Common Factors

Common factors are elements or numbers shared by the two or more terms in an expression. Finding these common factors is crucial for simplifying expressions effectively. In each rational expression, the goal is to identify these elements so that they can be cancelled.

In the example problem \(\frac{x+10}{x-4} \cdot \frac{x-4}{x-1}\), the expression \((x-4)\) is a common factor present in both the numerator of the second fraction and the denominator of the first. By recognizing this, we notice that \((x-4)\) can be removed from these terms. This is because any non-zero number divided by itself equals 1, hence \((x-4)/(x-4) = 1\). This step significantly reduces the complexity of the expression and leads to the simplified result \(\frac{x+10}{x-1}\).

Multiplication and Division of Fractions

The multiplication and division of fractions involves straightforward steps. For multiplication, you multiply numerators with numerators and denominators with denominators. However, a smart initial step is simplifying first. This reduces the effort and calculation involved.

In division, we often talk of multiplying by the reciprocal of the divisor. But in some instances, like our problem, it's easier to spot and eliminate common factors before multiplying. Once the expression is simplified, carrying out the multiplication becomes just putting those two numerators and two denominators together.

• Identify any common factors.
• Cancel the common factors to simplify.
• Perform the multiplication by combining simplified terms.

By following these steps, the given exercise simplified to \(\frac{x+10}{x-1}\), highlighting how simplification pre-emptively made multiplication far easier.