/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Add or subtract. Write answer in... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 16

Add or subtract. Write answer in lowest terms. \(\frac{4 x}{x+2}-\frac{2+3 x}{x+2}\)

Short Answer

Expert verified
\(\frac{x - 2}{x + 2}\)

Step by step solution

01

Identify the common denominator

Both fractions already have the same denominator, which is \(x + 2\).
02

Combine the numerators

Since the denominators are the same, subtract the numerator of the second fraction from the numerator of the first fraction: \(\frac{4x - (2 + 3x)}{x + 2}\).
03

Simplify the numerator

Distribute the negative sign and combine like terms in the numerator: \(4x - 2 - 3x = x - 2\). This gives us the fraction \(\frac{x - 2}{x + 2}\).
04

Verify if the fraction is in lowest terms

The numerator \(x - 2\) and the denominator \(x + 2\) have no common factors other than 1. Therefore, the fraction is already in its lowest terms.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Adding and Subtracting Fractions
When dealing with fractions, especially rational expressions, addition and subtraction follow similar rules as with regular fractions. Begin by ensuring both fractions have the same denominator. If they do, you can directly add or subtract the numerators while maintaining the common denominator.

If the denominators are different, you must first find a common denominator. This often involves finding the least common multiple (LCM) of the denominators. Once the denominators are the same, combine the numerators accordingly.

For example, in the given exercise, the fractions \(\frac{4x}{x+2} - \frac{2+3x}{x+2}\) already have the same denominator, making the addition or subtraction straightforward. Simply focus on the numerators.
Common Denominator
A common denominator is essential for adding or subtracting fractions. It is the shared denominator that allows you to combine the fractions. Think of the denominator as a common ground that holds the fractions together.

To find a common denominator, follow these steps:
  • Determine the denominators of the fractions.
  • Find the LCM of these denominators.
  • Adjust each fraction so that both have this LCM as their new denominator.
In the exercise, because both fractions \(\frac{4x}{x+2}\) and \(\frac{2+3x}{x+2}\) already have \((x+2)\) as their denominator, there's no need to adjust them. This simplifies the process as we can proceed directly to combining the numerators.
Simplifying Fractions
Once you've combined the fractions by adding or subtracting their numerators, the next step is to simplify the result, if possible. Simplifying means reducing the fraction to its lowest terms.

This involves:
  • Factoring both the numerator and the denominator.
  • Canceling out any common factors.
For example, in the final step of the exercise, the numerator of the resulting fraction is \({x-2}\) and the denominator is \({x+2}\). These two terms have no common factors other than 1. Thus, the fraction \(\frac{x-2}{x+2}\) is already in its simplest form.

Always verify that your final answer is simplified to help you avoid any unnecessary complexities and to make further calculations easier.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine whether each equation represents direct or inverse variation. $$ y=\frac{5}{x} $$

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{6 s^{2}+17 s+10}{s^{2}-4} \cdot \frac{s^{2}-2 s}{6 s^{2}+29 s+20}$$

Solve each variation problem. For a constant area, the length of a rec tangle varies inversely as the width. The length of a rectangle is \(27 \mathrm{ft}\) when the width is \(10 \mathrm{ft}\). Find the width of a rectangle with the same area if the length is \(18 \mathrm{ft}\).

The average number of vehicles waiting in line to enter a sports arena parking area is approximated by the rational expression $$\frac{x^{2}}{2(1-x)}$$ where \(x\) is a quantity between 0 and 1 known as the traffic intensity. (Source: Mannering, E., and W. Kilareski, Principles of Highway Engineering and Traffic Control, John Wiley and Sons.) To the nearest tenth, find the average number of vehicles waiting if the traffic intensity is the given number. (a) 0.1 (b) 0.8 (c) 0.9 (d) What happens to waiting time as traffic intensity increases?

Solve each variation problem. The force required to compress a spring varies directly as the change in the length of the spring. If a force of 12 lb is required to compress a certain spring 3 in., how much force is required to compress the spring 5 in.?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.