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Chapter 7: Problem 31

Multiply or divide as indicated. $$ \frac{5 x-10}{12} \div \frac{4 x-8}{8} $$

Short Answer

Expert verified
The simplified result is \( \frac{5}{6} \).

Step by step solution

01

Understand the Problem

We need to divide the rational expression \( \frac{5x-10}{12} \) by the rational expression \( \frac{4x-8}{8} \). Remember that dividing by a fraction is equivalent to multiplying by its reciprocal.
02

Reciprocal of the Divisor

Find the reciprocal of the divisor \( \frac{4x-8}{8} \). The reciprocal is found by flipping the numerator and the denominator, resulting in \( \frac{8}{4x-8} \).
03

Rewrite the Division as Multiplication

Transform the original division problem into a multiplication problem using the reciprocal from Step 2: \[ \frac{5x-10}{12} \times \frac{8}{4x-8} \].
04

Factor the Numerators and Denominators

Factor the expressions where possible. For \( 5x-10 \), it can be factored as \( 5(x-2) \). Similarly, \( 4x-8 \) can be factored as \( 4(x-2) \). Now the expression looks like: \[ \frac{5(x-2)}{12} \times \frac{8}{4(x-2)} \].
05

Simplify by Canceling Common Factors

In the expression \( \frac{5(x-2)}{12} \times \frac{8}{4(x-2)} \), the \( (x-2) \) terms in the numerator of the first fraction and the denominator of the second fraction cancel each other out. The expression becomes: \[ \frac{5}{12} \times \frac{8}{4} \].
06

Multiply the Remaining Fractions

Multiply the resulting fractions: \( \frac{5}{12} \times \frac{8}{4} \). First, simplify \( \frac{8}{4} \) to \( 2 \). The expression now becomes \( \frac{5}{12} \times 2 \). Multiply to get \( \frac{10}{12} \).
07

Simplify the Resulting Fraction

Simplify the fraction \( \frac{10}{12} \). Both the numerator and the denominator can be divided by 2, resulting in \( \frac{5}{6} \).

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Key Concepts

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

Multiplying Rational Expressions
Multiplying rational expressions involves a straightforward process. When you multiply two rational expressions, you multiply their numerators together and their denominators together. Consider this as nothing more than multiplying two fractions.
  • Write down the expression you want to multiply.
  • Factorize any expressions where possible. This makes simplification easier later on.
  • Multiply the numerators together.
  • Multiply the denominators together.
  • Simplify the resulting expression by canceling common factors.
Let's illustrate this with an example: if you're multiplying \( \frac{a}{b} \times \frac{c}{d} \), the result is \( \frac{ac}{bd} \). Remember to always look for opportunities to factor expressions. This step can significantly simplify your work.
Dividing Rational Expressions
Dividing rational expressions is just like dividing any other fraction. The key idea is to multiply by the reciprocal of the divisor. Let's break it down:
  • Identify the two rational expressions you're working with.
  • Convert the division to multiplication by taking the reciprocal of the divisor. Flip the numerator and the denominator of the divisor to do this.
  • Once converted, follow the same steps as in multiplication. Multiply the numerators together and the denominators together.
  • Simplify the expression by canceling common factors.
For example, with \( \frac{a}{b} \div \frac{c}{d} \), you would transform this into \( \frac{a}{b} \times \frac{d}{c} \). From there, multiply across numerators and denominators and simplify as necessary.
Simplifying Rational Expressions
Simplifying rational expressions is about making them as straightforward as possible. The aim is to reduce the algebraic fractions to their simplest form while ensuring the expression is mathematically equivalent to the original. Here's how:
  • Factor the numerators and the denominators completely. Look for common factors that can be canceled.
  • Cancel any common factors from the numerator and the denominator. When you cancel, you essentially divide these parts of the fraction.
  • Re-write the simplified expression. Make sure all steps preserve the original expression's value.
As an example, consider \( \frac{ax}{bx} \). If \( x \) is a common factor, it can be canceled, simplifying it to \( \frac{a}{b} \). Simplification often aids in reducing complexity and helps in evaluating or understanding expressions more efficiently.

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Most popular questions from this chapter

Which of the following are equivalent to \(\frac{\frac{1}{x}}{\frac{3}{y}} ?\) a. \(\frac{1}{x} \div \frac{3}{y}\) b. \(\frac{1}{x} \cdot \frac{y}{3}\) c. \(\frac{1}{x} \div \frac{y}{3}\)

Perform each indicated operation. In ice hockey, penalty killing percentage is a statistic calculated as \(1-\frac{G}{P},\) where \(G=\) opponent's power play goals and \(P=\) opponent's power play opportunities. Simplify this expression.

Perform each indicated operation. See Section 1.3. $$ \frac{8}{15} \div \frac{5}{8} $$

One of the great algebraists of ancient times a man named Diophantus. Litle is known of his life other than the lived and worked in Alexandria. Some historians believe he lived during the first century of the Christian era, about the time of Nero. The only che to his personal life is the following epigram found in a collection called the Palatine A nthology. God granted him youth for a sixth of his life and added a twelfth part to this. He clothed his cheeks in down. He lit him the light of wedlock after a seventh part and five years after his marriage. He granted him a son. Alas, lateborn wretched child. After attaining the measure of half his father's life, cruel fate overtook him, thus leaving Diophantus during the last four years of his life only such consolation as the science of numbers. How old was Diophantus at his death?" We are looking for Diophantus' age when he died, so let x represent that age. If we sum the parts of his life, we should get the total age. Parts of his life \(\left\\{\begin{array}{l}{\frac{1}{6} x+\frac{1}{12} x \text { is the time of his youth. }} \\ {\frac{1}{7} x \text { is the time between his youth and when }} \\ {\text { he married. }} \\ {5 \text { years is the time between his marriage }} \\ {\text { and the birth of his son. }} \\\ {\frac{1}{2} x \text { is the time Diophantus had with his son. }} \\ {4 \text { years is the time between his son's death }} \\ {\text { and his own. }}\end{array}\right.\) The sum of these parts should equal Diophantus' age when he died. $$ \frac{1}{6} \cdot x+\frac{1}{12} \cdot x+\frac{1}{7} \cdot x+5+\frac{1}{2} \cdot x+4=x $$ How old was Diophantus when his son was born? How old was the son when he died?

Simplify. $$ \frac{x}{1-\frac{1}{1+\frac{1}{x}}} $$
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