Question

Add \(\frac{2 x-5}{5 x+10}+\frac{x+1}{3 x+6}\). Write your answer in lowest terms. A. \(\frac{11 x-10}{15(x+2)^2}\) B. \(\frac{11 x-10}{15(x+2)}\) C. \(\frac{11 x^2+12 x-20}{(5 x+10)(3 x+6)}\) D. \(\frac{11 x^2+12 x-20}{15(x+2)^2}\)

Short answer

The correct answer is D: \(\frac{11 x - 10}{15(x + 2)^2}\).

Step-by-step solution

- Factor the Denominators

Factor the denominators of each fraction. For \(5x + 10\), factor out the common factor of 5:\(5x + 10 = 5(x + 2)\). For \(3x + 6\), factor out the common factor of 3:\(3x + 6 = 3(x + 2)\).

- Determine the Least Common Denominator (LCD)

The least common denominator is found by taking the product of each unique factor to its highest power: The factors are \(5(x + 2)\) and \(3(x + 2)\).So LCD = \(15(x + 2)^2\).

- Rewrite Each Fraction with the LCD as the Denominator

Rewrite each fraction with the common denominator \(15(x + 2)^2\):For \(\frac{2x-5}{5(x+2)}\), multiply the numerator and denominator by \(3(x + 2)\):\(\frac{(2x - 5) * 3(x + 2)}{15(x + 2)^2} = \frac{3(2x - 5)}{15(x + 2)^2} = \frac{6x - 15}{15(x + 2)^2}\).For \(\frac{x+1}{3(x+2)}\), multiply the numerator and denominator by \(5(x + 2)\):\(\frac{(x + 1) * 5(x + 2)}{15(x + 2)^2} = \frac{5(x + 1)}{15(x + 2)^2} = \frac{5x + 5}{15(x + 2)^2}\).

- Add the Fractions

Combine the numerators over the common denominator:\(\frac{6x - 15}{15(x + 2)^2} + \frac{5x + 5}{15(x + 2)^2} = \frac{(6x - 15) + (5x + 5)}{15(x + 2)^2} = \frac{11x - 10}{15(x+2)^2}\).

- Present the Simplified Result and Select the Correct Answer

The simplified result is \(\frac{11 x - 10}{15(x + 2)^2}\). Comparing with the given options, the correct answer is \textbf{D.}\

Key concepts

adding fractions

Adding fractions involves creating a common denominator, which is necessary for combining the numerators.
To start, make sure the denominators are the same. If they aren’t, find the least common denominator (LCD).
After converting the fractions to have the same denominator, you can combine the numerators directly. Let’s look at an example:
For fractions \( \frac{2}{5} \) and \( \frac{3}{10} \), the LCD is 10.
Rewrite \( \frac{2}{5} \) as \( \frac{4}{10} \). Now, add: \[ \frac{4}{10} + \frac{3}{10} = \frac{7}{10} \].
So, remember: make denominators the same, then add the numerators, and keep the common denominator.

factoring denominators

Factoring denominators is crucial in simplifying expressions and finding the LCD.
Factoring means breaking down the denominator into its simplest components. Consider \(5x+10\): it can be factored as \(5(x+2)\). Similarly, for \(3x+6\), the factor form is \(3(x+2)\).
By factoring, you easily spot common factors, making it simpler to determine the LCD.
Practice finding factors by checking for common multipliers in each term. Factorization is like peeling the layers of an onion - you get to the core components needed for simplification.

least common denominator

The least common denominator (LCD) is the smallest common multiple of the denominators.
Identifying the LCD helps simplify the addition of fractions by ensuring a uniform denominator.
For denominators like \(5(x+2)\) and \(3(x+2)\), the factors are 5, 3, and \(x+2\). The LCD for these is \(15(x+2)^2\).
To find the LCD:

• Factor each denominator into primes or smallest factors
• Identify all unique factors
• Use the highest power of each factor found

This process ensures the least common multiple is used, simplifying further steps.

simplifying algebraic expressions

Simplifying algebraic expressions facilitates easier manipulation and understanding.
Follow these steps:

• Combine like terms
• Factor when possible
• Reduce fractions by canceling common factors

Take \( \frac{6x-15}{15(x+2)^2} + \frac{5x+5}{15(x+2)^2} \): the common denominator allows you to combine numerators directly:
\[ \frac{(6x-15) + (5x+5)}{15(x+2)^2} = \frac{11x -10 }{15(x+2)^2} \].
Simplify final results by reducing the expression to its lowest terms. Always check for further factorability to simplify fully.