Problem 78 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{5}{x-1}-\frac{4}{x+6}$$

Step-by-Step Solution

Answer

\(\frac{x+34}{(x-1)(x+6)}\) is the simplest form.

Step 1: Find the Least Common Denominator (LCD)

To add or subtract fractions, they must have a common denominator. In this case, the denominators are \(x-1\) and \(x+6\). Therefore, the least common denominator is \((x-1)(x+6)\).

Step 2: Rewrite Each Fraction

Rewrite each fraction with the common denominator \((x-1)(x+6)\). The first fraction becomes \(\frac{5(x+6)}{(x-1)(x+6)}\), and the second fraction becomes \(\frac{4(x-1)}{(x-1)(x+6)}\).

Step 3: Expand the Numerators

Expand the numerators of the modified fractions: \(5(x+6) = 5x + 30\) and \(4(x-1) = 4x - 4\). So the fractions become \(\frac{5x+30}{(x-1)(x+6)}\) and \(\frac{4x-4}{(x-1)(x+6)}\).

Step 4: Subtract the Fractions

Subtract the second fraction from the first: \(\frac{5x+30}{(x-1)(x+6)} - \frac{4x-4}{(x-1)(x+6)}\). Simplify by subtracting the numerators: \((5x+30) - (4x-4)=x+34\).

Step 5: Write the Final Fraction

The simplified subtraction is \(\frac{x+34}{(x-1)(x+6)}\). Ensure it is fully simplified; in this case, there are no further simplifications possible.

Key Concepts

Least Common Denominator

To effectively add or subtract algebraic fractions, it's crucial that the fractions share the same denominator. This shared denominator is known as the Least Common Denominator (LCD). It is the smallest expression that is divisible by each of the original denominators.
In algebra, determining the LCD involves checking the factors of each denominator and finding their multiplication.

• Analyze each fraction's denominator: here, the denominators are \(x-1\) and \(x+6\).
• The LCD becomes the product \((x-1)(x+6)\).

Finding the LCD simplifies the process of combining fractions by providing a common ground for both fractions. This step is pivotal because missing or incorrect LCD calculation leads to wrong answers in fraction operations.

Simplifying Fractions

Once the fractions are rewritten with a common denominator, it's essential to simplify each fraction. Simplifying involves making the fraction as simple as possible without changing its value.
In algebraic expressions, this usually refers to expanding the numerators so they can be combined easily.

• For example, for the fraction \(\frac{5(x+6)}{(x-1)(x+6)}\), simplify the numerator to be \(5x + 30\).
• Similarly, simplify the numerator of \(\frac{4(x-1)}{(x-1)(x+6)}\) to be \(4x - 4\).

Simplifying fractions allows an easier approach to subtraction or addition later on as it provides the necessary individual components that need to be worked on. Each step furthers the process towards achieving the simplest form of the final answer.

Subtracting Fractions

With the fractions now carrying the same denominator and having simplified numerators, the next goal is subtraction. Unlike adding or subtracting numbers, fractions require extra care in aligning denominators and simplifying numerators.
The steps to subtract fractions with the same denominator are straightforward:

• Subtract the numerators while keeping the common denominator unchanged. For our fractions: \((5x + 30) - (4x - 4) = x + 34\).
• The expression becomes \(\frac{x+34}{(x-1)(x+6)}\).

After subtraction, always look to see if you can simplify the final answer further by factoring numerators or denominators. Here, \(x + 34\) cannot be factored further alongside the denominator, so the fraction is in its simplest form. Properly subtracting fractions ensures accuracy and integrity of the solution.