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Chapter 6: Problem 55

Add or subtract as indicated. If possible, simplify your answer. See Examples I through 6. $$ \frac{3}{2 x+10}+\frac{8}{3 x+15} $$

Short Answer

Expert verified
\(\frac{25}{6(x+5)}\)

Step by step solution

01

Identify and simplify the denominators

The denominators in the expression are \(2x + 10\) and \(3x + 15\). Notice both can be factored. The first denominator can be rewritten as \(2(x + 5)\) and the second as \(3(x + 5)\). This helps in finding a common denominator.
02

Find the least common denominator (LCD)

To add these fractions, we need a common denominator. Since both denominators have the term \((x + 5)\), the least common denominator is \(2 \cdot 3 \cdot (x + 5) = 6(x + 5)\).
03

Rewrite fractions with the LCD

Convert each fraction to have the common denominator \(6(x + 5)\). Multiply the first fraction by \(3\) (to account for the \(3\) in the second denominator) and the second fraction by \(2\) (to account for the \(2\) in the first denominator):\[\frac{3 \cdot 3}{6(x+5)} + \frac{8 \cdot 2}{6(x+5)} = \frac{9}{6(x+5)} + \frac{16}{6(x+5)}\]
04

Add the fractions

Now that they have the same denominator, add the numerators:\[\frac{9 + 16}{6(x+5)} = \frac{25}{6(x+5)}\]
05

Simplify if possible

Check if the fraction can be simplified. In this case, \(\frac{25}{6(x + 5)}\) is already in its simplest form, as there are no common factors between the numerator and the denominator.

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Key Concepts

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

Finding the Least Common Denominator
Adding fractions requires a shared base, called the least common denominator (LCD). This ensures the fractions can be combined properly. Here's how you find it:
  • Factor each denominator. In our example, the denominators are \(2x + 10\) and \(3x + 15\). By factoring, we get \(2(x + 5)\) and \(3(x + 5)\).
  • Determine the LCD by taking the highest power of each factor present in both denominators. Here, they share \((x + 5)\). To make the LCD, multiply the separate numbers \(2\) and \(3\) together, along with the common factor \((x + 5)\).
The least common denominator for the fractions is \(6(x + 5)\). This step ensures the numbers align for straightforward addition or subtraction, avoiding arithmetic errors.
Simplifying Fractions
Simplifying fractions makes them easier to work with and often appears as a final task when performing operations with fractions. This step involves reducing the expression to its simplest form:
  • Check both the numerator and the denominator for common factors.
  • In our example \( \frac{25}{6(x+5)} \), we see the numerator is 25 and the denominator is \(6(x+5)\).
  • Since 25 and 6 don't share any factors other than 1, and \(x + 5\) does not affect the numerical part directly, the fraction is already simplified.
Remember always to check for simplification at the end of fraction operations, as this makes the result cleaner and more understandable.
Factoring Expressions
Factoring is breaking down a complicated expression into simpler, multipliable elements. It's a critical skill in algebra that aids in solving equations and simplifying expressions.
  • Start by identifying common factors in a term. In our expressions \(2x + 10\) and \(3x + 15\), notice both terms have a common element \(x + 5\).
  • Factor out these common terms: \(2x + 10\) becomes \(2(x + 5)\), and \(3x + 15\) becomes \(3(x + 5)\). This step helps identify shared components for further operations like finding the LCD.
Once you've factored expressions, other arithmetic tasks are easier, because you've broken down the problem to its core components. Always take a moment to factor when tackling algebraic fractions. It simplifies the process and often uncovers solutions more straightforwardly.

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Most popular questions from this chapter

If the area of a parallelogram is \(\left(2 x^{2}-17 x+35\right)\) square centimeters and its base is \((2 x-7)\) centimeters, find its height.

Determine whether each division problem is a candidate for the synthetic division process. $$ \left(x^{4}-6\right) \div\left(x^{3}+3 x-1\right) $$

For each statement, find the constant of variation and the variation equation. See Examples 5 and 6. \(y\) varies inversely as the square of \(x ; y=0.052\) when \(x=5\)

For each statement, find the constant of variation and the variation equation. See Examples 5 and 6. \(y\) varies directly as the cube of \(x ; y=32\) when \(x=4\)

Find each square root. See Section 1.3. $$ \sqrt{\frac{25}{121}} $$
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