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Chapter 9: Problem 21

Reduce each of the following fractions as completely as possible. $$\frac{3 x-6}{6 x-12}$$

Short Answer

Expert verified
\(\frac{1}{2}\)

Step by step solution

01

Factorize the Numerator and Denominator

Identify the greatest common factor (GCF) of both the numerator and the denominator. For the fraction \(\frac{3x - 6}{6x - 12}\), the GCF of the numerator (3x - 6) is 3, and for the denominator (6x - 12), it is 6. Hence, we can factor them as: \[\frac{3(x - 2)}{6(x - 2)} \]
02

Cancel Out Common Factors

Since the factor \(x - 2\) is present in both the numerator and denominator, it can be canceled out. After canceling, we are left with: \[\frac{3}{6}\]
03

Simplify the Remaining Fraction

Simplify the fraction \(\frac{3}{6}\) by dividing both the numerator and the denominator by their GCF, which is 3: \[\frac{3 \div 3}{6 \div 3} = \frac{1}{2}\]

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Key Concepts

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

Greatest Common Factor
The Greatest Common Factor (GCF) is the largest number that can evenly divide two or more numbers. For example, to find the GCF of 3x - 6 and 6x - 12:
  • List down numerical and variable factors of each term.
  • The terms 3x - 6 both have a numerical factor of 3.
  • The terms 6x - 12 have a numerical factor of 6.
Understanding the GCF helps us to factor expressions for simplification. In our exercise, the GCF for 3x - 6 is 3, and for 6x - 12, it is 6. This step makes the fraction reduction more manageable.
Factoring Polynomials
Factoring polynomials means expressing a polynomial as a product of its factors. This can simplify expressions and solve equations. From our example:
  • 3x - 6 can be written as 3(x - 2).
  • 6x - 12 can be written as 6(x - 2).
Factoring helps us see common factors that can be canceled out, making fractions easier to reduce. So, by writing the terms in factored form, \[\frac{3(x - 2)}{6(x - 2)} \], we see that (x - 2) cancels out, simplifying the process.
Simplifying Fractions
Simplifying fractions means reducing them to their lowest terms. Here are the steps:
  • After factoring both the numerator and the denominator, cancel out the common terms.
  • This left us with \[\frac{3}{6} \].
  • Find the GCF of 3 and 6, which is 3.
  • Divide both numerator and denominator by the GCF. \[ \frac{3 \div 3}{6 \div 3} = \frac{1}{2} \]
This results in the fraction's simplest form. Simplifying fractions ensures they are easy to work with and understand.

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Most popular questions from this chapter

Perform the indicated operations and simplify. $$\frac{9 x^{2}}{25 y^{6}} \div \frac{x y}{15}$$

Determine whether the values \(x=3,8,-5,\) and 10 satisfy the equation $$ \frac{x+3}{2}-\frac{2 x+5}{4}=\frac{1}{4} $$ Try some other values of \(x .\) Why do you think there are so many solutions to this equation? What type of equation is this? How would you prove this?

In Exercises \(1-36,\) solve each of the equations or inequalities explicitly for the indicated variable. $$\frac{x+y}{3}=\frac{x}{3}+\frac{y}{4}-\frac{x}{6}+2 \quad \text { for } y$$

In Exercises \(1-36,\) solve each of the equations or inequalities explicitly for the indicated variable. $$u(v+w)-3=5 \text { for } u$$

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{5}{4 t}-\frac{2}{t}-\frac{1}{2}=-\frac{1}{8}$$
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