Problem 63 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{3 x}{x-4}+\frac{2}{x+6}$$

Step-by-Step Solution

Answer

The simplification of the given expression is \(\frac{3x^2+20x-8}{(x-4)(x+6)}\)

Step 1: Identify the denominator

Identify the denominator for each fraction. The denominators, in this case, are \(x-4\) and \(x+6\). Therefore, the least common denominator (LCD) would be the product of \(x-4\) and \(x+6\).

Step 2: Convert the fractions

Convert each of the fractions to have the LCD as the denominator. The first fraction becomes \(\frac{3x(x+6)}{(x-4)(x+6)}\) and the second fraction becomes \(\frac{2(x-4)}{(x-4)(x+6)}\).

Step 3: Combine the fractions

Now, both fractions have the same denominator so they can be combined. This gives the following: \(\frac{3x(x+6) + 2(x-4)}{(x-4)(x+6)}\).

Step 4: Simplify the numerator

Simplify the numerator, treating each part separately, gives \(\frac{3x^2+18x+(2x-8)}{(x-4)(x+6)}\). Further simplification leads to \(\frac{3x^2+20x-8}{(x-4)(x+6)}\).

Key Concepts

Least Common Denominator

When dealing with fractions, finding a common denominator is crucial for combining them. The least common denominator (LCD) is the smallest expression that contains all the individual denominators within a problem. It ensures all fractions have a shared reference point for addition or subtraction.

In algebra, the LCD is found by multiplying the distinct factors of each denominator. In our example, the denominators are \(x-4\) and \(x+6\).
Since these expressions do not share common factors, their product \((x-4)(x+6)\) becomes the LCD. This step ensures that both fractions can be expressed with a common denominator so they can be easily combined.

Combining Fractions

Once fractions have a common denominator, combining them becomes straightforward. With the same denominator, you can simply add or subtract the numerators while retaining the denominator.

• First, rewrite each fraction with the LCD as the new denominator.
• The process involves adjusting the numerator by multiplying it with any factors added to create the LCD.
• For example, for \(\frac{3x}{x-4}\) and \(\frac{2}{x+6}\), we have \(\frac{3x(x+6)}{(x-4)(x+6)}\) and \(\frac{2(x-4)}{(x-4)(x+6)}\).

With the fractions converted, they now share the same denominator \((x-4)(x+6)\), allowing the numerators to be combined effectively into: \[\frac{3x(x+6) + 2(x-4)}{(x-4)(x+6)}\]
This step is necessary before any simplification can occur.

Simplifying Expressions

Simplifying expressions often involves combining like terms and reducing complex expressions to a simpler form.

In our example, start by focusing on the numerator: \[3x(x+6) + 2(x-4)\] Expand the terms within the numerator:

• Multiply out \(3x(x+6)\) resulting in \(3x^2 + 18x\).
• Multiply out \(2(x-4)\) resulting in \(2x - 8\).
• Add the results: \(3x^2 + 18x + 2x - 8\).
• Combine like terms: \(3x^2 + 20x - 8\).

This simplifies the expression to a more condensed format without changing its value. Simplification makes. the expression easier to understand or further manipulate.

Algebraic Fractions

Algebraic fractions involve expressions where the numerator, the denominator, or both, are algebraic expressions. Simplifying and manipulating these fractions can be trickier compared to basic numbers due to the presence of variables.

• Such fractions often require finding common denominators, à la the least common denominator, for addition or subtraction.
• Multiplication or division might involve factoring or canceling common factors for simplification.
• Simplified algebraic fractions provide clearer insights or easier inputs for further equations or inequalities.

These fractions can represent real-world relationships in physics, economics, etc., where dynamics change with variables. Understanding them enhances one's ability to solve complex algebraic problems effectively.