Problem 32 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{-x+2}{18}}{\frac{8}{9}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{-(x-2)}{16} \).

Step 1: Identify the complex fraction

A complex fraction is a fraction in which the numerator or the denominator (or both) contain fractions. In this case, the fraction is \( \frac{\frac{-x+2}{18}}{\frac{8}{9}} \).

Step 2: Simplify the complex fraction by multiplying by the reciprocal

To simplify the complex fraction, multiply the numerator by the reciprocal of the denominator. The reciprocal of \( \frac{8}{9} \) is \( \frac{9}{8} \). Thus,\[ \frac{-x+2}{18} \times \frac{9}{8} \].

Step 3: Multiply the fractions

When multiplying the fractions, multiply the numerators together and the denominators together:\[ \frac{(-x+2) \times 9}{18 \times 8} \].

Step 4: Simplify the resulting fraction

First, perform the multiplication:The numerator becomes \(((-x) + 2) \times 9 = -9x + 18\).The denominator becomes \(18 \times 8 = 144\).Thus, we have:\[ \frac{-9x + 18}{144} \].

Step 5: Factor and reduce when possible

Factor the expression in the numerator:\(-9x + 18 = -9(x - 2)\).The fraction becomes:\[ \frac{-9(x - 2)}{144} \].Simplify by finding the greatest common divisor of the numerator and the denominator, which is 9:\[ \frac{-(x-2)}{16} \].

Key Concepts

Reciprocal

The concept of a reciprocal is fundamental in simplifying complex fractions. Reciprocals are inverses of a number or fraction. To find the reciprocal of a fraction, you simply swap the numerator and the denominator.

For example, the reciprocal of \( \frac{8}{9} \) is \( \frac{9}{8} \). This is because reciprocals multiply together to equal one. That means \( \frac{8}{9} \times \frac{9}{8} = 1 \). In this exercise, the reciprocal of the denominator \( \frac{8}{9} \) is used to transform the complex fraction into a simple multiplication operation. This makes the process of simplification much more straightforward.

When you multiply the entire fraction by this reciprocal, the denominator becomes one and essentially vanishes, allowing you to focus solely on multiplying the numerator by the reciprocal.

Simplification

Simplification is the process of reducing a fraction or an expression to its simplest form. Complex fractions can look daunting at first but breaking them down into simpler parts can make them more manageable.

Here are the steps typically involved in simplification:

• Identify Complex Parts: Recognize the numerator and denominator that contain fractions themselves. In this example, \( \frac{\frac{-x+2}{18}}{\frac{8}{9}} \).
• Apply Reciprocal: Multiply by the reciprocal to eliminate the fraction in the denominator, which transforms the problem into a multiplication operation.
• Perform Multiplication: Multiply the numerators and denominators separately. For this exercise, the result of multiplying is \( \frac{(-9x + 18)}{144} \).
• Simplify Further: If possible, reduce the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Simplification ensures clarity and leads to expressions that are easier to compute or analyze in further operations.

Factorization

Factorization involves breaking down an expression into a product of simpler variables or numbers. This is particularly useful when simplifying the result of a complex fraction.

Once you have multiplied through by the reciprocal and have a single fraction, factorization plays its role. Take the numerator \(-9x + 18\), which can be factored into \(-9(x - 2)\). Factorization helps to reveal common factors that can be canceled out with the denominator, which in this scenario is 144.

In this exercise, both the numerator and denominator share a factor of 9. By dividing both by this factor:

• Numerator: Factor -9 out to get \(-9(x - 2)\).
• Denominator: Find the GCD of 144, which is 9 considering the simplified form.

The fraction \( \frac{-9(x - 2)}{144} \) simplifies to \( \frac{-(x-2)}{16} \). Factorization, thus, removes redundancies, providing a more efficient and cleaner result.