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Chapter 9: Problem 23

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

Short Answer

Expert verified
The simplified fraction is \( \frac{x-2}{2(x+2)} \).

Step by step solution

01

Identify Common Factors

First, determine whether there are common factors in both the numerator and the denominator. The numerator is \(3x - 6\), and the denominator is \(6x + 12\).
02

Factorize the Numerator

Factor out the greatest common divisor in the numerator. \(3x - 6\) can be factored as \(3(x - 2)\).
03

Factorize the Denominator

Factor out the greatest common divisor in the denominator. \(6x + 12\) can be factored as \(6(x + 2)\).
04

Simplify the Fraction

Divide both the numerator and the denominator by the common factors. The fraction \( \frac{3(x-2)}{6(x+2)} \) simplifies as follows: \( \frac{3}{6} \) simplifies to \( \frac{1}{2} \), so the final fraction is \( \frac{(x-2)}{2(x+2)} \).

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Key Concepts

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

greatest common divisor
When simplifying fractions, finding the greatest common divisor (GCD) is very helpful. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. To illustrate, in the exercise, we need to simplify the fraction \(\frac{3x-6}{6x+12}\). First, we need the GCD of the terms in the numerator and denominator. For the numerator \(3x - 6\), both terms share a factor of 3. Similarly, for the denominator \(6x + 12\), both terms share a factor of 6. Factoring out these common divisors will allow us to simplify the fraction easily.
numerator and denominator
A fraction consists of two main parts: the numerator and the denominator. These terms refer to the top and bottom parts of the fraction, respectively. For example, in the exercise \(\frac{3x-6}{6x+12}\), \(3x - 6\) is the numerator, and \(6x + 12\) is the denominator. When reducing fractions, it is essential to work on both parts. Checking for common factors and simplifying both the numerator and the denominator will help reduce the fraction to its simplest form.
factoring polynomials
Factoring polynomials is a method used to break down expressions into simpler components called factors. In the exercise, we factor the numerator and the denominator separately. The numerator \(3x - 6\) can be factored as \(3(x - 2)\) because we find the common factor of 3. Similarly, the denominator \(6x + 12\) can be factored as \(6(x + 2)\) because the common factor is 6. This factoring process makes subsequent steps of fraction reduction easier, as we can cancel out common terms.
fraction reduction
After factoring, the next step is to reduce the fraction by canceling out common terms. Once the numerator and denominator are factored \( \frac{3(x-2)}{6(x+2)} \), we simplify by dividing both parts by their common factor. In this case, \( \frac{3}{6} \) simplifies to \( \frac{1}{2} \). Therefore, the original fraction simplifies to \( \frac{(x-2)}{2(x+2)} \). Reducing fractions ensures we have the simplest form, making them easier to work with in further calculations or algebraic manipulations.

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Most popular questions from this chapter

In Exercises \(52-55,\) simplify the given expression as completely as possible. Final answers should be expressed with positive exponents only. $$x^{3} x^{4}\left(x^{3}\right)^{4}$$

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{11}{x^{2}-9}-\frac{7}{2 x+6}=\frac{2}{x+3}$$

Each of Exercises \(37-50\) is a formula either from mathematics or the physical or social sciences. Solve each of the formulas for the indicated variable. $$\left.\frac{x-\mu}{s}<2 \text { for } \mu \quad \text { (Assume } s>0\right)$$

Each of Exercises \(37-50\) is a formula either from mathematics or the physical or social sciences. Solve each of the formulas for the indicated variable. $$\frac{1}{f}=\frac{1}{f_{1}}+\frac{1}{f_{2}} \quad \text { for } f_{1}$$

Solve each of the problems algebraically. Be sure to label what the variable represents. In a certain voting district the ratio of Republicans to Democrats is 4 to \(7 .\) If there are 2520 Democrats, how many Republicans are there?
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