Problem 36 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{\frac{4}{3 x+1}+\frac{3}{x}}{\frac{6}{x}-\frac{2}{3 x+1}}$$

Step-by-Step Solution

Answer

The simplest form of the complex fraction is \(\frac{7 - 3x}{2(3x+1)}\).

Step 1: Convert the complex fraction into standard division format

Here, the complex fraction can be seen as a division problem with numerator \(\frac{4}{3x + 1} + \frac{3}{x}\) and denominator \(\frac{6}{x} - \frac{2}{3x + 1}\). Thus, it can be rewritten as: \[\frac{\frac{4}{3x + 1} + \frac{3}{x}}{\frac{6}{x} - \frac{2}{3x + 1}} = \frac{4}{3x + 1} + \frac{3}{x} \div \frac{6}{x} - \frac{2}{3x + 1}\]

Step 2: Change the division into multiplication

We know that division is the same as multiplication by the reciprocal. So we flip the fraction in the denominator and change the operation to multiplication:\[\frac{4}{3x + 1} + \frac{3}{x} \times \frac{x}{6} - \frac{1}{3x + 1}\]

Step 3: Simplify the expression

Now let's combine fraction terms and simplify the expression:First, multiply \(\frac{3}{x} \times \frac{x}{6}\) , it becomes \(\frac{1}{2}\), Second, subtract \(\frac{1}{2}\) from \(\frac{4}{3x + 1}\), it results in \(\frac{4}{3x + 1} - \frac{1}{2}\)

Step 4: Reformat to standard fraction form

Let's put this back into the form of a single fraction. To do this, need to find a common denominator, which is \(2(3x+1)\). Now express each fraction with this common denominator, and subtract:\[\frac{8 - (3x + 1)}{2(3x+1)} = \frac{7 - 3x}{2(3x+1)}\]

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible, often by reducing it to its smallest equivalent form. To simplify a fraction, we look for common factors in the numerator and the denominator and divide both by these factors. This makes the fraction smaller and easier to handle. Here's how you can simplify a complex fraction:

• Find the greatest common factor (GCF) of both the numerator and the denominator.
• Divide both the numerator and the denominator by their GCF.
• If you're dealing with variables, factor them out if they appear in both the numerator and the denominator.

Remember, simplifying a fraction does not change its value, only its appearance. This step is crucial for keeping calculations neat and manageable, especially when dealing with algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They don't include equal signs like equations do. In our exercise, the expression \(\frac{4}{3x + 1} + \frac{3}{x}\) represents an algebraic expression among fractions.To manage algebraic expressions effectively:

• Identify like terms which can be combined or simplified.
• Apply arithmetic operations strictly following the operational hierarchy - PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Manage complex expressions by breaking them down into simpler parts.

Working with algebraic expressions involves not only identifying numbers and variables but also understanding how operations affect the values involved.

Fraction Operations

Fraction operations involve tasks such as addition, subtraction, multiplication, and division of fractions. With complex fractions, these operations can seem daunting, but they are manageable when broken down step-by-step. To simplify complex fractions in our problem:

• Recognize when to add, subtract, multiply or divide fractions.
• Convert division into a multiplication by the reciprocal when dividing fractions.
• Find a common denominator when adding or subtracting fractions with different denominators.
• Step through each operation carefully, maintaining order to arrive at the correct solution.

A complex fraction may look intimidating, but by transforming division into multiplication and finding common denominators, the process becomes more manageable. Practice these operations consistently to master simplifying complex fractions.