Problem 59 - Algebra and Trigonometry

Question

Subtract: \(\frac{3}{x-4}-\frac{2}{x+2}\)

Step-by-Step Solution

Answer

The answer is \(\frac{x+14}{(x-4)(x+2)}\)

Step 1: Find the Least Common Denominator (LCD)

Since the denominators of the fractions are \(x-4\) and \(x+2\), the least common denominator (LCD) is their multiplication = \((x-4)(x+2)\).

Step 2: Change the Fractions to Have the Common Denominator

Multiply each fraction by the necessary element to achieve the LCD: \(\frac{3}{x-4}\) becomes \(\frac{3(x+2)}{(x-4)(x+2)}\) and \(\frac{2}{x+2}\) becomes \(\frac{2(x-4)}{(x-4)(x+2)}\).

Step 3: Perform the Subtraction

Subtract the two new fractions after changing them to have the common denominator, i.e., \(\frac{3(x+2)}{(x-4)(x+2)} - \frac{2(x-4)}{(x-4)(x+2)}\).

Step 4: Simplify the Numerator

Simplify the numerator by distributing 3 into \(x+2\) and 2 into \(x-4\) then carry out the subtraction. Hence you will get \(\frac{3x+6-(2x-8)}{(x-4)(x+2)}\) which simplifies to \(\frac{x+14}{(x-4)(x+2)}\) after combining like terms.

Key Concepts

Least Common Denominator (LCD)

When dealing with algebraic fractions, finding the least common denominator (LCD) is crucial. The LCD is the smallest expression that can be divided exactly by each of the separate denominators. In our exercise, we are working with the denominators \(x-4\) and \(x+2\). To find the LCD, we multiply these two binomial expressions together.

The result is our desired common denominator \((x-4)(x+2)\). By doing this, we create a single common base that we can work with for both fractions, simplifying the overall subtraction process. This concept is essential for combining fractions because it allows us to work with just one common denominator rather than managing them separately.

Fraction Subtraction

Subtracting fractions, particularly algebraic ones, involves several logical steps. First, ensure that both fractions have the same denominator. Using the least common denominator (LCD) that we've found is crucial for this. After adjusting the fractions to share the same denominator, the next step is straightforward:

• Subtract the numerators of the fractions while keeping the LCD as the denominator.

In our problem, after converting both fractions, we have:
\[ \frac{3(x+2)}{(x-4)(x+2)} - \frac{2(x-4)}{(x-4)(x+2)} \]

Here, instead of separately dealing with the denominators, they remain constant and our task is to focus solely on subtracting the numerators. This fundamental technique streamlines the process, turning a potentially complicated expression into something manageable.

Simplifying Expressions

Once you've combined or subtracted the algebraic fractions, simplifying the expression becomes crucial. Simplification often involves distributing terms and combining like terms in the numerator. Let's examine the steps involved here.

For our exercise, distribute the coefficients in the numerators:

• Distribute \(3\) through \((x+2)\)
• Distribute \(2\) through \((x-4)\)

This gives us:
\[ 3x + 6 - (2x - 8) \]

Notice the subtraction sign before \(2(x-4)\); it affects each term within the expression. Combining like terms is often the next step.

Simplifying this, we get:
\[ x + 14 \]

The process of simplifying helps in reducing complex expressions to their most concise form, providing clarity and often simplifying further calculations or integrations.