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Chapter 8: Problem 79

Perform the operations and simplify, if possible. See Example 9. $$\frac{x+2}{x+5}-\frac{x-3}{x+7}$$

Short Answer

Expert verified
The simplified result is \( \frac{7x + 29}{(x+5)(x+7)} \).

Step by step solution

01

Identify Common Denominator

To subtract the fractions \( \frac{x+2}{x+5} \) and \( \frac{x-3}{x+7} \), we first need to find a common denominator. The denominators are \( x+5 \) and \( x+7 \). The common denominator is their product: \( (x+5)(x+7) \).
02

Rewrite Each Fraction with Common Denominator

Rewrite each fraction with the common denominator. For \( \frac{x+2}{x+5} \), multiply the numerator and denominator by \( x+7 \):\[ \frac{(x+2)(x+7)}{(x+5)(x+7)} \].For \( \frac{x-3}{x+7} \), multiply the numerator and denominator by \( x+5 \):\[ \frac{(x-3)(x+5)}{(x+7)(x+5)} \].
03

Simplify the Numerators

Expand the numerators:- Expand \( (x+2)(x+7) \):\( x^2 + 7x + 2x + 14 = x^2 + 9x + 14 \).- Expand \( (x-3)(x+5) \):\( x^2 + 5x - 3x - 15 = x^2 + 2x - 15 \).
04

Subtract the Numerators

Subtract the expanded numerators while maintaining the common denominator:\[ \frac{(x^2 + 9x + 14) - (x^2 + 2x - 15)}{(x+5)(x+7)} \].Simplify the numerator:\( (x^2 + 9x + 14) - (x^2 + 2x - 15) = x^2 + 9x + 14 - x^2 - 2x + 15 = 7x + 29 \).
05

Write the Final Expression

Combine the results to get the simplified form:\[ \frac{7x + 29}{(x+5)(x+7)} \].This is the simplified result of the operation.

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Key Concepts

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

Common Denominator
When dealing with fractions in algebra, finding a common denominator is crucial, especially when you want to perform operations like addition or subtraction. The common denominator helps you compare or combine fractions by providing a uniform base.
For the algebraic fractions \( \frac{x+2}{x+5} \) and \( \frac{x-3}{x+7} \), the denominators \( x+5 \) and \( x+7 \) don't match, so we can't directly subtract them.
  • To find the common denominator, multiply the individual denominators together: \( (x+5)(x+7) \).
  • This product ensures both fractions use the same base, making further operations possible.
Finding the common denominator is like setting up a shared language for the fractions, making it easier to subtract them.
Simplifying Expressions
Once a common denominator is found, each fraction needs to be rewritten to reflect this new denominator. Simply adjusting one without changing the other would give incorrect results.
To do this properly:
  • Multiply the numerator and denominator of each fraction by the necessary terms to achieve the common denominator.
  • For \( \frac{x+2}{x+5} \), multiply both terms by \( x+7 \): \( \frac{(x+2)(x+7)}{(x+5)(x+7)} \).
  • For \( \frac{x-3}{x+7} \), multiply both terms by \( x+5 \): \( \frac{(x-3)(x+5)}{(x+7)(x+5)} \).
Rewriting these expressions with a common denominator paves the way for further simplification by aligning the structures of the fractions. At this point, expressions can be expanded and ultimately simplified.
Subtracting Fractions
Subtracting fractions requires an aligned base, which we've achieved through a common denominator. With that in place, move to subtract the numerators while maintaining the common denominator. Expand each expression in the numerators:
  • \( (x+2)(x+7) \) expands to \( x^2 + 9x + 14 \).
  • \( (x-3)(x+5) \) expands to \( x^2 + 2x - 15 \).
Subtract these two results:
  • \( (x^2 + 9x + 14) - (x^2 + 2x - 15) \).
  • Combine like terms to simplify: \( 7x + 29 \).
The remaining step involves placing this simplified result over the common denominator: \( \frac{7x + 29}{(x+5)(x+7)} \).
This final expression reflects the fully simplified version of our original subtraction problem, thanks to aligned operations and careful arithmetic.

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Most popular questions from this chapter

Environmental Cleanup. Suppose the cost (in dollars) of removing \(p \%\) of the pollution in a river is given by the rational function $$f(p)=\frac{50,000 p}{100-p} \text { where } 0 \leq p<100$$ Find the cost of removing each percent of pollution. a. \(50 \%\) b. \(80 \%\)

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In the following problems, simplify each expression by performing the indicated operations and solve each equation. $$\frac{5}{3 x+12}-\frac{1}{9}=\frac{x-1}{3 x}$$

Perform the operations and simplify, if possible. See Example \(8 .\) $$\frac{2}{5 a}+\frac{3}{2 a}$$
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