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Chapter 7: Problem 33

Add or subtract. Write answer in lowest terms. \(\frac{x+1}{6}+\frac{3 x+3}{9}\)

Short Answer

Expert verified
\frac{x+1}{2}

Step by step solution

01

Find a Common Denominator

Identify the denominators in both fractions: 6 and 9. The least common multiple (LCM) of 6 and 9 is 18. Rewrite each fraction with 18 as the common denominator.
02

Rewrite Fractions

Rewrite \(\frac{x+1}{6}\) and \(\frac{3 x+3}{9}\) with the common denominator of 18. \(\frac{x+1}{6} = \frac{3(x+1)}{18} = \frac{3x+3}{18}\). Then, rewrite \(\frac{3 x + 3}{9}\) as \(\frac{2(3x+3)}{18} = \frac{6x+6}{18}\).
03

Combine the Fractions

Add the two fractions with a common denominator: \(\frac{3x+3}{18} + \frac{6x+6}{18}\). This results in: \(\frac{(3x+3) + (6x+6)}{18} = \frac{9x + 9}{18}\).
04

Simplify the Fraction

Factor out the greatest common factor (GCF) from the numerator: \(\frac{9(x+1)}{18}\). Simplify by dividing both the numerator and denominator by 9: \(\frac{x+1}{2}\).

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Key Concepts

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

common denominator
When adding or subtracting fractions, it's essential to have a common denominator. The denominator is the bottom part of a fraction, and a common denominator means both fractions have the same bottom number. This makes calculations easier. For example, if we need to add \(\frac{x+1}{6}\) and \(\frac{3x+3}{9}\), we look for a common denominator. We identify that the least common multiple (LCM) of 6 and 9 is 18. Then, we rewrite each fraction with the denominator of 18 so that they are easier to add together. Always remember: same denominators simplify the process!
simplifying fractions
Simplifying a fraction is making it as simple as possible. This involves reducing it to the lowest terms by factoring out the greatest common factor (GCF) from both the numerator and the denominator. For instance, after adding \(\frac{3x+3}{18}\) and \(\frac{6x+6}{18}\), we get \(\frac{9x+9}{18}\). The numerator 9x + 9 can be factored to \(\frac{9(x+1)}{18}\). We then divide both the numerator and the denominator by 9, simplifying the fraction to \(\frac{x+1}{2}\). Simplifying fractions makes them easier to understand and work with.
least common multiple
The least common multiple (LCM) is the smallest multiple that two or more numbers share. Finding the LCM is crucial when working with fractions that have different denominators. For example, to find the LCM of 6 and 9, look at their multiples. The multiples of 6 are 6, 12, 18, and so on. The multiples of 9 are 9, 18, and so on. The smallest common multiple is 18. This means we rewrite fractions such as \(\frac{x+1}{6}\) and \(\frac{3x+3}{9}\) to have 18 as their denominators. Having the LCM makes adding and subtracting fractions straightforward.

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Most popular questions from this chapter

The percent of deaths caused by smoking is modeled by the rational expression $$\frac{x-1}{x}$$ where \(x\) is the number of times a smoker is more likely than a nonsmoker to die of lung cancer. This is called the incidence rate. (Source: Walker, A., Observation and lnference: An Introduction to the Methods of Epidemiology, Epidemiology 91影视 Inc.) For example, \(x=10\) means that a smoker is 10 times more likely than a nonsmoker to die of lung cancer. Find the percent of deaths if the incidence rate is the given number. (a) 5 (b) 10 (c) 20 (d) Can the incidence rate equal \(0 ?\) Explain.

Rewrite each rational expression with the indicated denominator. $$ \frac{4 m}{m^{2}+m-2}=\frac{?}{(m-1)(m-3)(m+2)} $$

Solve each equation for \(k\) $$ 180=\frac{k}{20} $$

Concept Check If you are given the following problem, what must be the polynomial that is represented by the question mark? $$\frac{4 y+12}{2 y-10} \div \frac{?}{y^{2}-y-20}=\frac{2(y+4)}{y-3}$$

Multiply or divide as indicated. Write each answer in lowest terms. See Section 1.1 $$ \frac{3}{7} \cdot \frac{4}{5} $$
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