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Chapter 6: Problem 19

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

Short Answer

Expert verified
\(\frac{4}{x}\)

Step by step solution

01

Identify the complex fraction

The given complex fraction is \(\frac{\frac{2}{x} - 3}{\frac{2 - 3x}{2}}\). Identify the numerator and the denominator.
02

Simplify the numerator

The numerator is \(\frac{2}{x} - 3\). This can be written as a single fraction: \(\frac{2 - 3x}{x}\).
03

Simplify the denominator

The denominator is \(\frac{2 - 3x}{2}\).
04

Simplify by dividing the fractions

Divide the simplified numerator by the simplified denominator: \(\frac{\frac{2 - 3x}{x}}{\frac{2 - 3x}{2}} = \frac{2 - 3x}{x} \times \frac{2}{2 - 3x}\).
05

Simplify the expression

Multiply the fractions: \(\frac{(2-3x) \times 2}{x \times (2-3x)} = \frac{2 \times 2}{x} = \frac{4}{x}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

complex fractions
Complex fractions are fractions where either the numerator, the denominator, or both contain one or more fractions themselves. They might look confusing at first, but breaking them down step by step makes them much more manageable. The main goal when simplifying complex fractions is to convert them into a single, simpler fraction.
algebraic simplification
Algebraic simplification involves reducing expressions to their simplest form. This often requires combining like terms, factoring, and performing basic arithmetic operations. In our example, we simplified the numerator \(\frac{2}{x} - 3\) to write it as a single fraction \(\frac{2 - 3x}{x}\). Similarly, the denominator \(\frac{2 - 3x}{2}\) was already in its simplest form.
fraction division
Dividing fractions involves multiplying by the reciprocal (or inverse). For instance, when we have \(\frac{\frac{2 - 3x}{x}}{\frac{2 - 3x}{2}}\), we can rewrite it as multiplying by the reciprocal. This changes the division into a multiplication problem: \(\frac{2 - 3x}{x} \times \frac{2}{2 - 3x}\). This step is crucial since it allows us to proceed to the multiplication step and further simplify the expression.
multiplying fractions
When multiplying fractions, multiply the numerators together and the denominators together. In the simplified problem, we had \(\frac{2 - 3x}{x} \times \frac{2}{2 - 3x}\). Multiplying these fractions gives us \(\frac{(2-3x) \times 2}{x \times (2-3x)} = \frac{2 \times 2}{x} = \frac{4}{x}\). Notice how \(2 - 3x\) cancels out in the numerator and denominator, which simplifies our final expression further.

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Most popular questions from this chapter

In these subtraction problems, the rational expression that follows the subtraction sign has a numerator with more than one term. Be careful with signs and find each difference. See Examples \(6-10 .\) 75\. \(\frac{5}{x^{2}-9}-\frac{x+2}{x^{2}+4 x +3}\)

Let \(P\), \(Q\), and \(R\) be rational expressions defined as follows. $$P=\frac{6}{x+3}, \quad Q=\frac{5}{x+1}, \quad R=\frac{4 x}{x^{2}+4 x+3}$$ Perform the operations and express \(P+Q-R\) in lowest terms.

Simplify each complex fraction. Use either method. $$ \frac{\frac{y+8}{y-4}}{\frac{y^{2}-64}{y^{2}-16}} $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{6}{5}-\frac{1}{9}}{\frac{2}{5}+\frac{5}{3}} $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{t+5}{t-8}}{\frac{t^{2}-25}{t^{2}-64}} $$
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