/*! 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 36 Add or subtract. Write answer in... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 36

Add or subtract. Write answer in lowest terms. \(\frac{x+2}{5 x}+\frac{6 x+3}{3 x}\)

Short Answer

Expert verified
\frac{11 x+7}{5 x}\

Step by step solution

01

Find the Common Denominator

To add the fractions \(\frac{x+2}{5 x} + \frac{6 x+3}{3 x}\), the denominators must be the same. The denominators are 5x and 3x. The least common multiple of 5x and 3x is 15x.
02

Convert Fractions to Have a Common Denominator

Rewrite each fraction to have the common denominator 15x. For \(\frac{x+2}{5 x}\), multiply both numerator and denominator by 3 to get \(\frac{3(x+2)}{15 x} = \frac{3x+6}{15x}\). For \(\frac{6 x+3}{3 x}\), multiply both numerator and denominator by 5 to get \(\frac{5(6x+3)}{15 x} = \frac{30x+15}{15x}\).
03

Add the Fractions

Now that both fractions have the common denominator 15x, add the numerators: \(\frac{3x+6}{15x} + \frac{30x+15}{15x} = \frac{(3x+6) + (30x+15)}{15x} = \frac{33x+21}{15x}\).
04

Simplify the Fraction

To simplify \(\frac{33x + 21}{15x}\), factor out any common terms. Both the numerator and the denominator can be divided by 3: \(\frac{33x + 21}{15x} = \frac{3(11x + 7)}{3(5x)} = \frac{11x + 7}{5x}\).

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.

least common multiple
When adding algebraic fractions, we first need a common denominator. This involves finding the Least Common Multiple (LCM) of the denominators. The LCM of two numbers is the smallest number that both numbers divide into without leaving a remainder.
common denominator
To add fractions, we make the denominators the same by finding a common denominator. Our goal is to rewrite each fraction so they have the same bottom part. In our exercise, the denominators are 5x and 3x. The LCM of these is 15x. This means we will rewrite each fraction so they both have a denominator of 15x.
simplifying fractions
After rewriting fractions to have the same denominator and combining them, we must simplify the result. Simplifying means making the fraction as simple as possible. Usually, this involves dividing the numerator and the denominator by their Greatest Common Divisor (GCD). In our exercise, after we add the fractions, we get \ \(\frac{33x+21}{15x}\). Both 33x+21 and 15x can be divided by 3, simplifying to \ \(\frac{11x+7}{5x}\).
factoring algebraic expressions
Factoring algebraic expressions makes it easier to simplify fractions. Factoring involves breaking down an expression into its simplest elements. For example, 33x+21 can be factored into 3(11x+7). This helps us to see common factors more clearly. In our solution, factoring allows us to simplify the fraction by canceling out common terms from the numerator and denominator.

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

Solve each problem involving direct or inverse variation. If \(z\) varies inversely as \(x^{2},\) and \(z=9\) when \(x=\frac{2}{3},\) find \(z\) when \(x=\frac{5}{4}\)

Solve each equation for \(k\) $$ 25=9 k $$

Solve each problem involving direct or inverse variation. If \(x\) varies directly as \(y,\) and \(x=27\) when \(y=6,\) find \(x\) when \(y=2\)

Rewrite each rational expression with the indicated denominator. $$ \frac{36 r}{r^{2}-r-6}=\frac{?}{(r-3)(r+2)(r+1)} $$

Solve each problem involving direct or inverse variation. If \(d\) varies directly as \(r,\) and \(d=200\) when \(r=40,\) find \(d\) when \(r=60\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.