Problem 63 - Algebra for College Students

Question

Simplify each complex fraction. $$ 2-\frac{x}{3-\frac{2}{x}} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{-x^2 + 6x - 4}{3x - 2}\).

Step 1: Simplify the Nested Fraction

First, look at the inner fraction \(3 - \frac{2}{x}\). Rewrite this as a single fraction by finding a common denominator, which is \(x\). This gives us \[ 3 = \frac{3x}{x}.\] Now, rewrite the whole inner expression as \[ 3 - \frac{2}{x} = \frac{3x - 2}{x}.\]

Step 2: Substitute Back into the Original Expression

Now replace \(3 - \frac{2}{x}\) with \(\frac{3x - 2}{x}\) in the original expression: \[ 2 - \frac{x}{\frac{3x - 2}{x}}.\] This becomes \[ 2 - \frac{x \cdot x}{3x - 2} = 2 - \frac{x^2}{3x - 2}.\]

Step 3: Simplify the Complex Fraction

Now we have a simpler complex fraction: \(2 - \frac{x^2}{3x - 2}\). To simplify, find a common denominator for the terms in this expression. The common denominator is \(3x - 2\). So, express \(2\) as \(\frac{2(3x - 2)}{3x - 2}\).

Step 4: Combine into a Single Fraction

Rewrite \(2\) as \(\frac{6x - 4}{3x - 2}\). Substitute back into the expression: \[ \frac{6x - 4}{3x - 2} - \frac{x^2}{3x - 2}.\] Now, combine these to form a single fraction: \[ \frac{6x - 4 - x^2}{3x - 2}.\]

Step 5: Simplify the Numerator

Simplify the numerator \(6x - 4 - x^2\) to get \(-x^2 + 6x - 4\). So the expression becomes \[ \frac{-x^2 + 6x - 4}{3x - 2}.\]

Step 6: Check for Further Simplification

Check if the numerator \(-x^2 + 6x - 4\) can be factored further, but it cannot be easily simplified further. So, the simplified form of the given complex fraction is \[ \frac{-x^2 + 6x - 4}{3x - 2}.\]

Key Concepts

Nested Fractions

Nested fractions are fractions that contain a fraction within another fraction, creating layers within the expression. These can sometimes be tricky to simplify because we need to carefully unravel the structure to make the calculations easier.
In the exercise, you began with the expression \(2 - \frac{x}{3-\frac{2}{x}}\). Here, the term \(3 - \frac{2}{x}\) is a nested fraction. It nests the smaller fraction \(\frac{2}{x}\) within the larger one.
To tackle this, look at the innermost part first. Simplify it into a single fraction at each level, starting from the innermost fraction outwards. This involves making a fraction into a single denominator, which can then be more easily subtracted, added, multiplied, or divided as needed.

Common Denominator

When dealing with fractions, finding a common denominator is key to making calculations straightforward. A common denominator allows us to rewrite fractions so they have the same base, which makes combining them possible.
In the given problem, when you simplified \(3 - \frac{2}{x}\), you found the common denominator to be \(x\). This is because the number 3 can be expressed as \(\frac{3x}{x}\).
This approach kept your focus on maintaining consistency across the fraction expressions, allowing you to recombine termen using the shared base. This step is crucial in transforming a complicated expression into something far simpler to manage.

Simplification Techniques

Simplifying complex fractions often involves a series of smaller simplification steps. In this problem, after dealing with nested fractions, you had to simplify the larger expression into a single fraction.
To do this:

• Identify terms that can be expressed using a common base.
• Rewrite expressions to align them with this base.
• Combine terms by performing the necessary arithmetic operations.

For example, in \(2 - \frac{x}{\frac{3x - 2}{x}}\), further rewriting lets you express \(2\) as \(\frac{6x - 4}{3x - 2}\). Having fractions share a common base allowed you to combine them into a single, simpler fraction effectively.

Factoring Polynomials

Factoring polynomials is a useful skill when simplifying expressions. While it did not lead to further simplification in this case, it's crucial you know how to check for it.
Once you simplified the numerator \(-x^2 + 6x - 4\), examine it closely to see if it can break down further. Sometimes, expressions can be factored even if not immediately obvious.
Why is factoring important?

• It can further reduce the expression's complexity.
• Identifying common factors can give insights into equivalent simpler forms.

In our exercise, the expression didn't break down further which is why factoring was not applicable, but always keep this tool in mind as part of your simplification toolkit.