Problem 85 - Elementary Algebra

Question

Use this approach to help with the additions and subtractions. $$\frac{5}{x-3}+\frac{1}{3-x}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{4}{x-3}\).

Step 1: Standardize the Denominator

Notice that the denominators \(x-3\) and \(3-x\) are equivalent except for a negative sign. We can standardize them by rewriting the second fraction. Since \(3-x = -(x-3)\), the fraction \(\frac{1}{3-x}\) becomes \(-\frac{1}{x-3}\).

Step 2: Combine the Fractions

Now that both fractions have a common denominator of \(x-3\), they can be combined. Perform the subtraction: \[ \frac{5}{x-3} - \frac{1}{x-3} = \frac{5 - 1}{x-3} = \frac{4}{x-3}. \]

Step 3: Simplify the Expression (If Necessary)

In this case, the expression \(\frac{4}{x-3}\) is already in its simplest form. There are no further simplifications to be made.

Key Concepts

Common Denominator

When dealing with rational expressions, especially those involving fractions, having a common denominator is crucial. Let's break down why this is important. A common denominator means that the denominators of different fractions are the same, which allows the numerators to be directly involved in arithmetic operations like addition or subtraction.
Imagine you want to subtract two fractions:

• Before you can directly subtract the numerators, both fractions must share the same denominator.
• If they don't, you'll need to transform them to have a common denominator. This might involve rewriting one or both of the fractions.
• In our example, the denominators were initially \( x-3 \) and \( 3-x \). Notice these are negatives of each other because \( 3-x = -(x-3) \).
• By multiplying the numerator or adding a negative sign, you can standardize the denominator of the second fraction to be \( x-3 \).

Once both fractions share this common denominator, you can move on to subtracting them.

Subtracting Fractions

Subtracting fractions, especially with rational expressions, can be straightforward once the denominators are the same. Here's how we handle it.

• Once both fractions have a common denominator, you can subtract the numerators directly because the denominator is consistent.
• Consider the example \( \frac{5}{x-3} - \frac{1}{x-3} \). Now that we have \( x-3 \) as the denominator in both, simply subtract the numerators: \( 5 - 1 = 4 \).
• The result is a new single fraction, \( \frac{4}{x-3} \), with the common denominator.

It's similar to subtracting regular numbers but with the denominator remaining constant. This method ensures no values are lost or miscalculated due to differences in denominators.

Simplifying Expressions

Once you have completed the subtraction, it’s time to check if the resulting fraction can be simplified.

• Simplifying a fraction involves reducing it to its simplest form, where the numerator and denominator have no common factors other than 1.
• In the expression \( \frac{4}{x-3} \), the numerator is 4 and the denominator is \( x-3 \). Since 4 and \( x-3 \) share no common factors*, the expression is already as simple as it can be.
• Always check both parts of the fraction. For numeric parts, look for greatest common divisors. For variable parts, factorize if possible.

Simplification helps in ensuring that expressions are in their most concise and understandable form. Keeping expressions simple is crucial, particularly when they are part of larger mathematical problems.