Problem 48 - Algebra 2

Question

Simplify each complex fraction. \(\frac{\frac{2}{x+4}+2}{1+\frac{3}{x+4}}\)

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \(6\frac{x+3}{x+7}\)

Step 1: Identify the Least Common Denominator

First, identify the least common denominator (LCD) of the fractions involved. The fractions are \(\frac{2}{x+4}\) and \(\frac{3}{x+4}\) and the LCD is \(x+4\).

Step 2: Multiply Through by the LCD

Multiply every term in the numerator and the denominator by the LCD. This gives: \((\frac{2}{x+4} \cdot (x+4)) + (2 \cdot (x+4))\) over \((1 \cdot (x+4)) + (\frac{3}{x+4} \cdot (x+4))\) which simplifies to \(2+2x+4x+16\) over \(x+4+3\).

Step 3: Simplify The Expression

Simplify the expression by collecting like terms, resulting in \(\frac{6x+18}{x+7}\).

Step 4: Reduce To Lowest Terms

Try to reduce the fraction to lowest terms. In this case, the fraction can be simplified by factoring out a 6 from the numerator: \(6\frac{x+3}{x+7}\)

Key Concepts

Least Common Denominator

When working with fractions, finding the least common denominator (LCD) is crucial for combining them. Fractions cannot be directly added or subtracted unless they share the same denominator. Here, the fractions involved are \(\frac{2}{x+4}\) and \(\frac{3}{x+4}\), both with a denominator of \(x+4\). This means that \(x+4\) itself is the LCD. Identifying the LCD simplifies the process of combining fractions as it allows us to transform them into equivalent fractions with the same denominator.

• Identify denominators of different fractions.
• Select the common denominator for all.

Once you have found the LCD, you can then perform operations, like addition or subtraction, confidently knowing that the fractions are on equal footing due to their shared denominator.

Simplifying Fractions

Simplifying fractions is all about making fractions easier to understand or use by reducing them to their simplest form. Once you have a complex fraction, like \(\frac{6x+18}{x+7}\), simplification involves removing any common factors from the numerator and denominator.

• Combine like terms if possible.
• Look for common factors that can be divided out.

The goal is to reach the most straightforward version of the fraction, which is as concise as it can be without losing its fundamental value. Often, simplification can also make calculations less error-prone, because you are dealing with smaller and friendlier numbers.

Factoring

Factoring is a method used in algebra to express an equation or an expression as a product of its factors. It's like breaking numbers down into their building blocks. For the numerator \(6x+18\), factoring involves finding a common component that divides all terms involved. Here, we notice both terms are divisible by 6.

• Look for the greatest common factor (GCF) of all terms.
• Rewrite the expression as a product of factors.

For example, \(6x+18\) can be rewritten as \(6(x+3)\), pulling out the common factor 6. Factoring is an essential tool in simplifying fractions as it can expose possible reductions, helping further simplify expressions.

Algebraic Expressions

Algebraic expressions are compositions of numbers, variables, and operation symbols that represent a mathematical object. They take the form of expressions like \(2x + 3\) or more complex forms, like in this case for the complex fraction. When simplifying complex fractions, dealing with algebraic expressions is an integral step.

• The terms can include constants (numbers) and variables (like \(x\)).
• Tools like distribution or combining like terms help manipulate them.

Understanding how to work with these expressions is essential, as it allows you to rearrange and simplify complex fractions effectively. Whether factoring them or simplifying them using operations, mastering algebraic expressions makes handling complex fractions much more manageable.