Problem 26 - Algebra for College Students

Question

Simplify each complex fraction. $$\frac{2-\frac{3}{a-2}}{4-\frac{1}{a+2}}$$

Step-by-Step Solution

Answer

\(\frac{(2a-7)(a+2)}{(a-2)(4a+7)}\)

Step 1 - Simplify the Numerator

The numerator is given by \(2 - \frac{3}{a-2}\). To simplify this part, find a common denominator. The common denominator here is \(a-2\). Rewrite the numerator as \(\frac{2(a-2) - 3}{a-2}\).

Step 2 - Simplify the Denominator

The denominator is given by \(4 - \frac{1}{a+2}\). Similar to the numerator, find a common denominator. The common denominator here is \(a+2\). Rewrite the denominator as \(\frac{4(a+2) - 1}{a+2}\).

Step 3 - Simplify the Entire Fraction

You now have the fraction as: \(\frac{\frac{2(a-2) - 3}{a-2}}{\frac{4(a+2) - 1}{a+2}}\). Simplify this complex fraction by multiplying the numerator by the reciprocal of the denominator. This results in: \[\frac{2(a-2) - 3}{a-2} \times \frac{a+2}{4(a+2) - 1}\].

Step 4 - Further Simplify the Expression

Combine and simplify the expressions. The numerator becomes: \(2a - 4 - 3 = 2a - 7\) and the denominator becomes: \(4a + 8 - 1 = 4a + 7\). So the fraction becomes: \[\frac{2a - 7}{a-2} \times \frac{a+2}{4a + 7}\].

Step 5 - Combine and Simplify

Combine and simplify by canceling out any common terms if possible. In this case, there are no common factors that cancel out, so the simplified fraction stays as: \(\frac{(2a-7)(a+2)}{(a-2)(4a+7)}\).

Key Concepts

Numerator and Denominator Simplification

When simplifying complex fractions, a good first step involves breaking down the numerator and denominator separately.
Let's start by simplifying the numerator, which is given by:

• Numerator: \(2 - \frac{3}{a-2}\)

To do so, find a common denominator for the terms involved. The common denominator here is \(a-2\). This allows us to rewrite the numerator as: \(\frac{2(a-2) - 3}{a-2}\).

On the other hand, the denominator of the original fraction is:

• Denominator: \(4 - \frac{1}{a+2}\)

Similar to the numerator, we need to find a common denominator for these terms. Here, the common denominator is \(a+2\). Therefore, rewrite the denominator as: \(\frac{4(a+2) - 1}{a+2}\).
Simplifying both numerators and denominators individually helps in understanding and processing complex fractions better.

Finding Common Denominators

Finding common denominators is crucial when simplifying terms involving fractions.
This involves identifying a common base that can unify the different parts of the fraction. For our complex fraction, we identified the common denominators for the numerator and the denominator as follows:

• For \(2 - \frac{3}{a-2}\), the common denominator is \(a-2\).
• For \(4 - \frac{1}{a+2}\), the common denominator is \(a+2\).

By rewriting these parts with a common denominator, we transform the expressions into another form that is easier to work with.
For the numerator, we get \(\frac{2(a-2) - 3}{a-2}\), and for the denominator: \(\frac{4(a+2) - 1}{a+2}\).
Finding common denominators not only helps in simplifying complex fractions but also makes subsequent steps such as multiplying or dividing much more manageable.

Multiplying by Reciprocal

Once we have simplified both the numerator and denominator separately, we can put them together into a single complex fraction.
Our expression is now: \(\frac{\frac{2(a-2) - 3}{a-2}}{\frac{4(a+2) - 1}{a+2}}\).
The next step is to simplify this further by multiplying by the reciprocal. This means turning the denominator of our complex fraction upside down and multiplying it by the numerator. Effectively, this is akin to performing division of fractions.

Apply this operation:

• \(\frac{2(a-2) - 3}{a-2} \times \frac{a+2}{4(a+2) - 1}\).

Multiplying by reciprocal simplifies the complex fraction to a single fraction that is easier to deal with.
After this step, we only need to simplify the terms inside the numerator and denominator to see if they can be further reduced. This leads us to the simplified fraction: \(\frac{(2a-7)(a+2)}{(a-2)(4a+7)}\). Remember, multiplying by the reciprocal is a powerful technique to simplify complex fractions. It breaks down an otherwise complicated fraction into a multiplication problem.