Problem 87 - Intermediate Algebra

Question

Simplify. \(\frac{\frac{x}{6}}{\frac{2 x}{3}+\frac{1}{2}}\)

Step-by-Step Solution

Answer

\( \frac{x}{4x + 3} \)

Step 1: Simplify the denominator

First, we need to simplify the denominator of the fraction, which is \( \frac{2x}{3} + \frac{1}{2} \). To do this, find a common denominator. The common denominator of 3 and 2 is 6.

Step 2: Adjust each term in the denominator to have the same denominator

Convert each fraction in the denominator to have 6 as the denominator:\( \frac{2x}{3} = \frac{2x \times 2}{3 \times 2} = \frac{4x}{6} \)\( \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \)

Step 3: Combine the fractions in the denominator

Add the fractions \( \frac{4x}{6} \) and \( \frac{3}{6} \) to form the unified denominator:\[ \frac{4x}{6} + \frac{3}{6} = \frac{4x + 3}{6} \]

Step 4: Divide the numerator by the simplified denominator

Now that the denominator is simplified to \( \frac{4x + 3}{6} \), divide the numerator, \( \frac{x}{6} \), by this expression.This is equivalent to multiplying by the reciprocal of the denominator:\[ \frac{x}{6} \div \frac{4x + 3}{6} = \frac{x}{6} \times \frac{6}{4x + 3} \]

Step 5: Simplify the resulting expression

We can now multiply the fractions:\[ \frac{x}{6} \times \frac{6}{4x + 3} = \frac{x \times 6}{6 \times (4x + 3)} = \frac{x}{4x + 3} \]The 6 in the numerator and denominator cancel out.

Key Concepts

Common Denominator

When working with fractions, finding a common denominator is essential, especially when you need to add or subtract them. A common denominator is a shared multiple of the denominators of the fractions involved.

Here's how you find one:

• Identify the denominators of the fractions.
• Determine the least common multiple (LCM) of these denominators.
• Adjust each fraction so that its denominator matches this LCM.

In our example, the fractions in the denominator are \( \frac{2x}{3} \) and \( \frac{1}{2} \). The LCM of 3 and 2 is 6. This is why both fractions are rewritten to have 6 as their denominator, making it easy to add them. By having a common denominator, you can seamlessly combine the fractions, just like with integers.

Reciprocals

Reciprocals are frequently used in division problems involving fractions. The reciprocal of a fraction is simply flipping the numerator and the denominator.

For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). When dividing by a fraction, multiplying by its reciprocal simplifies the process:

• The division \( \frac{a}{b} \div \frac{c}{d} \) becomes \( \frac{a}{b} \times \frac{d}{c} \).
• This means swapping and then multiplying gives the same result.

In our exercise, after simplifying the denominator to \( \frac{4x + 3}{6} \), we use the reciprocal \( \frac{6}{4x + 3} \). This allows easy multiplication with the numerator \( \frac{x}{6} \), ultimately simplifying the entire rational expression.

Fractions

Fractions are a way to represent parts of a whole. They are written as \( \frac{a}{b} \) where \( a \) is the numerator and \( b \) is the denominator. The key with fractions is understanding how to manipulate them for operations like addition, subtraction, multiplication, and division.

These operations often involve:

• Finding a common denominator for addition and subtraction.
• Multiplying numerators and denominators separately.
• Using reciprocals for division by fractions.

In the given problem, we have \ \frac{x}{6} \, a fraction, divided by another fraction \( \frac{4x + 3}{6} \). Manipulating fractions properly ensures you achieve the simplest form of your expression, as seen by following steps carefully to reach \( \frac{x}{4x + 3} \).

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using various algebra techniques. This includes distributing, combining like terms, and factoring, to name a few.

Considerations while simplifying include:

• Carefully observing the operations to be executed.
• Applying reciprocal and common denominator strategies.
• Canceling common factors in numerators and denominators.

In the simplification task provided, algebraic manipulation is visible when we integrate addends to form a single fraction in the denominator and when cancelling out terms during the multiplication of fractions. Focusing on these aspects step by step helps in revealing the simplest form, which here ends as \( \frac{x}{4x + 3} \). Understanding each part of the expression allows you to manipulate it effectively, whatever the context.