Problem 49 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{5}{2-x}+\frac{x}{2 x-4}\)

Step-by-Step Solution

Answer

\( \frac{x - 10}{2(x-2)} \) is the simplified form of the expression.

Step 1: Identify Denominators

The first step is to identify the denominators of each fraction in the expression. The expression is \( \frac{5}{2-x}+\frac{x}{2x-4} \). The denominators are \( 2-x \) and \( 2x-4 \).

Step 2: Factor and Simplify Denominators

Factor the denominators if possible. Notice that \( 2x-4 \) can be factored as \( 2(x-2) \). Now rewrite the expression: \( \frac{5}{2-x}+\frac{x}{2(x-2)} \).

Step 3: Transform Denominators

Observe that \( 2-x \) is equivalent to \( -(x-2) \). Rewrite the first fraction with this transformation: \( \frac{5}{2-x} = -\frac{5}{x-2} \). The expression becomes: \( -\frac{5}{x-2} + \frac{x}{2(x-2)} \).

Step 4: Find a Common Denominator

The common denominator of the two fractions is \( 2(x-2) \). Transform each fraction so that they both have the common denominator.

Step 5: Adjust Numerators

For the first fraction \( -\frac{5}{x-2} \) convert it to have the common denominator: \( -\frac{5}{x-2} = -\frac{5 \cdot 2}{2(x-2)} = -\frac{10}{2(x-2)} \). The second fraction already has the common denominator: \( \frac{x}{2(x-2)} \).

Step 6: Combine the Fractions

Combine the fractions \( \frac{-10}{2(x-2)} + \frac{x}{2(x-2)} \) by adding the numerators: \( \frac{-10 + x}{2(x-2)} \).

Step 7: Simplify the Expression

Simplify the numerator \( x - 10 \). Since \( x - 10 \) cannot be simplified further and doesn't factor with the denominator, the final expression is \( \frac{x - 10}{2(x-2)} \).

Key Concepts

Common Denominators

When dealing with algebraic fractions, finding a common denominator is an essential step. It allows us to combine different fractions into a single expression. Think of it like finding a common ground or language so different fractions can "speak" to each other.

In our exercise, we had the fractions \( \frac{5}{2-x} \) and \( \frac{x}{2(x-2)} \). These fractions had different denominators, which seem tricky at first. To add these fractions together, we look for a denominator that is shared by both, also known as the common denominator.

• Here, we factored \( 2x-4 \) to get \( 2(x-2) \) and realized that \( 2-x \) is the same as \(-(x-2)\). This allows us to rewrite both fractions with the common denominator \( 2(x-2) \).

Once both fractions have the common denominator, you can proceed with adding or subtracting their numerators. It's like preparing a pie where you first need the same size base before adding different toppings.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is like decluttering, where you aim to make the expression as neat and simple as possible. When simplifying, you always want to ensure you aren’t losing any important information, similar to making sure you don’t throw away something important while cleaning.

In our worked example, we started with two fractions: \( \frac{-10}{2(x-2)} \) and \( \frac{x}{2(x-2)} \). Once they were both rewritten with a common denominator, we could simplify by combining the numerators. This meant calculating \( -10 + x \), so the combined numerator became \( x - 10 \).

The simplified fraction was then \( \frac{x - 10}{2(x-2)} \). At this stage, you check if the expression can be simplified further. This usually involves looking for common factors in the numerator and the denominator that can be cancelled out. However, here \( x - 10 \) and the denominator had no common factors apart from 1, so the simplification process was complete.

Factoring

Factoring is a technique used to break down expressions into their multiplicative components. It's like peeling back the layers of an onion to see what's inside and is a key strategy for simplifying complex expressions.

In algebra, factoring often helps to simplify expressions or solve equations. During this exercise, we encountered the factorization in the denominator \( 2x-4 \). This expression could be rewritten as \( 2(x-2) \).

• To factor expressions, you look for: common factors that can be divided out, patterns such as a difference of squares, or even special products that result from expansion.
• This factoring allowed us to identify a common structure with another denominator \( x-2 \), leading to the unified denominator of \( 2(x-2) \) that was shared across both fractions. This step streamlined the process of combining the fractions.

Factoring is a foundational skill in algebra because it gives us the ability to see connections and simplify expressions efficiently.