Problem 119 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{\frac{1}{x+3}+\frac{2}{x-3}}{2-\frac{1}{x-3}} $$

Step-by-Step Solution

Answer

Simplified expression: \( \frac{3(x+1)(x-3)}{(x+3)(2x-7)} \).

Step 1: Simplify the Numerator

First, we need to add the fractions in the numerator. The expression is \( \frac{1}{x+3} + \frac{2}{x-3} \). To add these fractions, we need a common denominator, which is \((x+3)(x-3)\). The numerator becomes:\[ \frac{1 \cdot (x-3) + 2 \cdot (x+3)}{(x+3)(x-3)} = \frac{x-3 + 2x + 6}{(x+3)(x-3)} = \frac{3x + 3}{(x+3)(x-3)} \]

Step 2: Simplify the Denominator

Now, simplify the expression in the denominator, \( 2 - \frac{1}{x-3} \). We need a common denominator here as well, which is \(x-3\):\[ \frac{2(x-3) - 1}{x-3} = \frac{2x - 6 - 1}{x-3} = \frac{2x - 7}{x-3} \]

Step 3: Form New Fraction from Simplified Parts

Replace the original fraction's numerator and denominator with the results from the previous steps to form a new single fraction:\[ \frac{\frac{3x + 3}{(x+3)(x-3)}}{\frac{2x - 7}{x-3}} \]

Step 4: Simplify the Complex Fraction

To simplify \( \frac{\frac{3x + 3}{(x+3)(x-3)}}{\frac{2x - 7}{x-3}} \), multiply by the reciprocal of the denominator:\[ \frac{3x + 3}{(x+3)(x-3)} \times \frac{x-3}{2x-7} = \frac{(3x + 3)(x-3)}{(x+3)(2x-7)} \]

Step 5: Simplify Further if Possible

We check if we can simplify \( \frac{(3x + 3)(x-3)}{(x+3)(2x-7)} \). First factor \(3x+3\) as \(3(x+1)\):\[ \frac{3(x+1)(x-3)}{(x+3)(2x-7)} \]No further simplification is possible.

Key Concepts

Algebraic Fractions

An algebraic fraction is a fraction where both the numerator and the denominator are algebraic expressions. Rather than simple integers, these can include variables, polynomials, or a combination of both. In the context of our expression, \(\frac{\frac{1}{x+3}+\frac{2}{x-3}}{2-\frac{1}{x-3}}\), both the numerator and the denominator are algebraic fractions themselves. This makes the overall expression a complex fraction, presenting another layer of simplification.

• Algebraic fractions follow the same principles as regular fractions for simplification.
• You must find a common denominator to add or subtract fractions.
• The ultimate goal is to make the fraction as simple as possible.

Understanding how to manipulate these expressions is crucial in algebra, as they form the basis for solving more complex equations.

Numerator and Denominator

In any fraction, the top number is the numerator, and the bottom number is the denominator. They work together to define the value of the fraction. With algebraic fractions, these components can be algebraic expressions themselves, such as polynomials or simple variables. In the problem at hand:

• The initial numerator is \(\frac{1}{x+3} + \frac{2}{x-3}\).
• The denominator is \(2 - \frac{1}{x-3}\).

To simplify, you must handle the numerators and denominators separately. First, simplify each part to make them easier to work with before addressing them as a whole. This involves finding common denominators within each portion of the fraction for uniformity and ease of further operations.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions, whether they are simple numeric fractions or more complicated algebraic ones. This process ensures that the fractions share the same bottom number, allowing seamless addition or subtraction. In our exercise:

• For the numerator, \(\frac{1}{x+3} + \frac{2}{x-3}\), the common denominator is \((x+3)(x-3)\).
• The expression becomes unified, \(\frac{3x + 3}{(x+3)(x-3)}\), for simpler handling.

Similarly, simplify the expression in the denominator using the same principle. By finding a common denominator, any complex algebraic fraction becomes much easier to manage and ultimately simplify.

Complex Fractions

Complex fractions are fractions in which the numerator, the denominator, or both are themselves fractions. These require careful handling to simplify. The given problem \(\frac{\frac{3x + 3}{(x+3)(x-3)}}{\frac{2x - 7}{x-3}}\) is an example of a complex fraction. To simplify:

• Convert the complex fraction by multiplying the numerator by the reciprocal of the denominator.
• This results in a single equivalent fraction: \(\frac{(3x + 3)(x-3)}{(x+3)(2x-7)}\).

Handling complex fractions involves additional steps, but the underlying principles remain the same. Look to break them down using multiplication after invert-reciprocal operations and never forget to check if further simplification is possible.