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Chapter 6: Problem 27

Add or subtract as indicated. Simplify each answer. See Examples \(3 a\) and \(3 b\) $$ \frac{4}{3 x}+\frac{3}{2 x} $$

Short Answer

Expert verified
The simplified form is \( \frac{17}{6x} \).

Step by step solution

01

Identify the Least Common Denominator (LCD)

The given expression is \( \frac{4}{3x} + \frac{3}{2x} \). Here, the denominators are \(3x\) and \(2x\). The least common denominator (LCD) needs to include each factor of the denominators. The prime factors are \(3\), \(2\), and \(x\). Therefore, the LCD is \(6x\).
02

Adjust Denominators to the LCD

Re-write each fraction with the LCD as the new denominator. Multiply the numerator and the denominator of the first fraction by \(2\) and the second fraction by \(3\) so that both fractions have a denominator of \(6x\). This becomes \( \frac{4 \cdot 2}{3x \cdot 2} = \frac{8}{6x} \) and \( \frac{3 \cdot 3}{2x \cdot 3} = \frac{9}{6x} \).
03

Add the Fractions

Once the fractions have the same denominator, you can add them by adding their numerators. This results in \( \frac{8}{6x} + \frac{9}{6x} = \frac{8+9}{6x} = \frac{17}{6x} \).
04

Simplify the Result

Check if the resulting fraction can be simplified. The numerator 17 and the denominator 6x do not have any common factors, so the expression is already in its simplest form.

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Key Concepts

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

Understanding the Least Common Denominator
When working with rational expressions, finding a common ground for the denominators is crucial. This common ground is known as the least common denominator (LCD). To solve problems involving the addition or subtraction of fractions, each fraction must be expressed with the same denominator. Here鈥檚 how you can find it:
  • Identify the denominators of the fractions you are asked to add or subtract. In our example, these are \(3x\) and \(2x\).
  • Factor each denominator to its prime components. For example, \(3x\) can be broken down into \(3\) and \(x\); similarly, \(2x\) becomes \(2\) and \(x\).
  • Choose the highest power of each factor present in any of the denominators. In our case, we have the factors \(3\), \(2\), and \(x\).
  • Multiply these selected factors to determine the LCD. Thus, the LCD for our example is \(6x\).
Steps like these ensure that the denominators become equal, allowing you to easily add or subtract fractions later.
Mastering the Art of Adding Fractions
Once you have established a common denominator for all fractions, adding them together becomes significantly more straightforward. Here鈥檚 what you need to remember:Convert each fraction using the least common denominator:
  • Multiply the numerator and the denominator of each fraction such that both fractions meet the LCD of \(6x\).
  • For instance, the fraction \(\frac{4}{3x}\) needs both its numerator and denominator multiplied by \(2\), making it \(\frac{8}{6x}\).
  • Similarly, \(\frac{3}{2x}\) requires multiplying both elements by \(3\), converting it to \(\frac{9}{6x}\).
Add the adjusted fractions:
  • Since both fractions now have the same denominator, simply add their numerators. This results in \(\frac{8 + 9}{6x}\).
  • The sum of the numerators here is \(17\), leading to the combined fraction \(\frac{17}{6x}\).
This approach lets you seamlessly navigate through fraction addition, as long as they share common denominators.
Simplifying Expressions for Clarity
After performing addition or subtraction on fractions, the final expression may still require simplification to achieve its most compact form. Simplification involves:
  • Inspecting the numerator and the denominator for any common factors.
  • In our example, the resulting expression \(\frac{17}{6x}\) has numerators and denominators that are already as simple as they can be, because there are no shared factors between \(17\) and \(6x\).
If there were common factors:
  • They would be divided out to reduce the fraction to its lowest terms. This process often involves dividing both the numerator and the denominator by their greatest common divisor.
Understanding and applying the concept of simplification ensures clarity and adherence to mathematical standards, which is essential in creating more manageable and intelligible expressions.

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Most popular questions from this chapter

Kraft Foods is a provider of many of the best-known food brands in our supermarkets. Among their wellknown brands are Kraft, Oscar Mayer, Maxwell House, and Oreo. Kraft Foods' annual revenues since 2005 can be modeled by the polynomial function \(R(x)=0.06 x^{3}+0.02 x^{2}+1.67 x+32.33,\) where \(R(x)\) is revenue in billions of dollars and \(x\) is the number of years since \(2005 .\) Kraft Foods' net profit can be modeled by the function \(P(x)=0.07 x^{3}-0.42 x^{2}+0.7 x+2.63,\) where \(P(x)\) is the net profit in billions of dollars and \(x\) is the number of years since \(2005 .\) (Source: Based on information from Kraft Foods) a. Suppose that a market analyst has found the model \(P(x)\) and another analyst at the same firm has found the model \(R(x) .\) The analysts have been asked by their manager to work together to find a model for Kraft Foods' profit margin. The analysts know that a company's profit margin is the ratio of its profit to its revenue. Describe how these two analysts could collaborate to find a function \(m(x)\) that models Kraft Foods' net profit margin based on the work they have done independently. b. Without actually finding \(m(x),\) give a general description of what you would expect the answer to be.

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