Problem 53 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\frac{3}{a-2}-\frac{1}{2-a}$$

Step-by-Step Solution

Answer

The sum is \( \frac{4}{a-2} \).

Step 1: Recognize the Common Denominator

First, observe that the denominators \( a-2 \) and \( 2-a \) can be simplified to a common form. Notice that \( 2-a = -(a-2) \), so they differ by just a sign.

Step 2: Rewrite the Fractions

Rewrite \( \frac{1}{2-a} \) as \( -\frac{1}{a-2} \). This makes the fractions easier to work with: \[ \frac{3}{a-2} - \left(-\frac{1}{a-2}\right) \].

Step 3: Combine the Fractions

With both fractions now having the common denominator \( a-2 \), they can be combined: \[ \frac{3}{a-2} + \frac{1}{a-2} = \frac{3+1}{a-2} \].

Step 4: Simplify the Result

Add the numerators and put them over the common denominator: \[ \frac{4}{a-2} \]. This is the simplified form of the expression.

Key Concepts

Common Denominator

A critical aspect of working with algebraic fractions involves finding a common denominator, which refers to a shared base in the denominators of different fractions. In the given exercise, the denominators are \(a-2\) and \(2-a\). At first glance, these may not seem to have a common denominator, but with closer inspection, you can identify that \(2-a\) is actually the negative of \(a-2\). By rearranging \(2-a\) as \(-(a-2)\), you establish a common denominator easily.

Why is this important? When fractions are added or subtracted, having a common denominator allows the numerators to be directly combined. This common base simplifies the operation and is crucial for fraction manipulation. Finding a common denominator is a fundamental step in both addition and subtraction of fractions and aids in simplifying the result later on. It sets the stage for making the process straightforward and ensures accuracy in handling different terms efficiently.

Simplifying Fractions

Once a common denominator is established, we can move onto simplifying the fractions. Simplifying means combining fractions so they share a single denominator, then resolving any operations in the numerators.

In this specific exercise, after ensuring both fractions have a common denominator of \(a-2\) by rewriting \(\frac{1}{2-a}\) as \(-\frac{1}{a-2}\), you can simplify the process by adding the fractions:

• Combine like terms in the numerators.
• Place the resultant expression over a single denominator.

This yields \(\frac{3}{a-2} + \frac{1}{a-2} = \frac{3+1}{a-2}\), simplifying to \(\frac{4}{a-2}\).

This step is essential because it reduces the complexity of your expression, making it easier to evaluate and understand. Simplifying fractions is a task that underscores the clarity of algebraic expressions, emphasizing efficiency and precision in the process.

Fraction Addition and Subtraction

Performing addition and subtraction with fractions can initially seem daunting, especially when dealing with variables. However, understanding the steps involved can simplify the procedure significantly. After ensuring a common denominator is found and simplifying the fractions, the operation boils down to basic arithmetic on the numerators.

For example, once our fractions share the denominator \(a-2\), you focus solely on the numerators. Instead of subtracting, the exercise turns into an addition because of the sign change in the rewritten fraction. Simply add the numerators \(3+1\), resulting in \(4\), while maintaining the common denominator \(a-2\).

The key elements to remember are:

• Find a common denominator first.
• Simplify the algebraic expressions where possible.
• Perform simple arithmetic on the numerator while keeping the denominator constant.

The result \(\frac{4}{a-2}\) concludes the process of adding or subtracting algebraic fractions and showcases the power of organization and methodical thinking in algebra.