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Chapter 8: Problem 9

Add or subtract, as indicated. $$\frac{n}{n+6}-\frac{9}{n+6}$$

Short Answer

Expert verified
The simplified expression is: \(\frac{n-9}{n+6}\)

Step by step solution

01

Identify the common denominator

We need to find the common denominator of the two fractions, which is already given as (n + 6).
02

Combine the numerators

Since both denominators are the same, we can combine the numerators: \(n-9\)
03

Write the simplified expression

Now, write the simplified expression by placing the new numerator over the common denominator: \[\frac{n-9}{n+6}\] The simplified expression is: \[\frac{n-9}{n+6}\]

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Key Concepts

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

Fraction Operations
Fraction operations are a key part of algebra and mathematics. When dealing with fractions, the main operations are addition, subtraction, multiplication, and division. In this exercise, we are focused on subtraction. Subtraction of fractions is straightforward if the denominators are the same.
Here's a simple way to handle fraction operations:
  • When adding or subtracting fractions with the same denominator, you just operate on the numerators.
  • For fractions with different denominators, find a common denominator first, then adjust numerators accordingly.
In this exercise, the same denominator simplifies the process, letting us directly subtract the numerators without any additional adjustments.
Common Denominator
A common denominator is crucial when adding or subtracting fractions. It ensures that fractions are combined on the same terms, making them compatible for the intended operation.
In this problem, both fractions already have a common denominator of \( n + 6 \). This means we don't need to convert anything, as both fractions are ready to be directly subtracted.
  • Having a common denominator allows you to focus on the numerators for addition or subtraction.
  • A common denominator is particularly useful when dealing with multiple fractions, as it simplifies the process significantly.
Thus, checking for and confirming a common denominator is always an essential step before moving forward with operations on algebraic fractions.
Numerator Simplification
Simplifying numerators is often the final step in fraction operations after ensuring a common denominator is present. This step allows us to transform the operation into a clean, simplified expression.
In the exercise given, since the denominator \( n + 6 \) is shared between the two fractions, we subtract the numerators: \( n - 9 \). This produces the simplified expression:
\[\frac{n - 9}{n + 6}\]
  • Numerator simplification often involves basic operations: addition, subtraction, or factoring, depending on the situation.
  • After this step, check your work to ensure no further simplification is possible.
By managing the numeric side of the fraction first, we achieve a simpler, more manageable expression that maintains the correct value.

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Most popular questions from this chapter

Solve each equation. $$\frac{1}{r-1}+\frac{2}{5 r-3}=\frac{37}{5 r^{2}-8 r+3}$$

Solve each equation. $$\frac{w}{3}=\frac{2 w-5}{12}$$

Solve each equation. $$\frac{h}{h^{2}+2 h-8}+\frac{4}{h^{2}+8 h-20}=\frac{4}{h^{2}+14 h+40}$$

Identify the least common denominator of each group of rational expression, and rewrite each as an equivalent rational expression with the LCD as its denominator. \(\frac{11}{g-3}, \frac{4}{9-g^{2}}\)

Identify the least common denominator of each group of rational expression, and rewrite each as an equivalent rational expression with the LCD as its denominator. \(\frac{10}{4 k-1}, \frac{k}{1-16 k^{2}}\)
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