- Factor the denominator \(x^2-9\) into \((x-3)(x+3)\).
- Express the entire sum with the common denominator \((x-3)(x+3)\) by multiplying the second fraction by \(\frac{x-3}{x-3}\).
- Add the numerators \(3 + 5(x-3)\) to get \(5x-12\).
- Conclude that the simplified form is \(\frac{5x-12}{(x+3)(x-3)}\). \(\boxed{\frac{5x-12}{(x+3)(x-3)}}\)
Explanation
- Analyzing the Expression and Factoring Denominators
The goal is to simplify the algebraic expression \(\frac{3}{x^2-9} + \frac{5}{x+3}\). First, notice that the expression consists of two rational terms with different denominators. To add them, we need to find a common denominator. The denominator of the first term, \(x^2-9\), is a difference of squares and can be factored as \((x-3)(x+3)\). Therefore, the expression is \(\frac{3}{(x-3)(x+3)} + \frac{5}{x+3}\).
- Finding a Common Denominator
To combine the fractions, we need both terms to have the same denominator, which is \((x-3)(x+3)\). We can achieve this by multiplying the numerator and the denominator of the second term, \(\frac{5}{x+3}\), by \((x-3)\). This gives: \(\frac{5(x-3)}{(x+3)(x-3)} = \frac{5x-15}{(x+3)(x-3)}\). Now the expression becomes \(\frac{3}{(x-3)(x+3)} + \frac{5x-15}{(x+3)(x-3)}\).
- Combining and Simplifying the Fractions
With a common denominator of \((x-3)(x+3)\), we can now add the numerators: \(\frac{3 + (5x - 15)}{(x-3)(x+3)}\). Simplifying the numerator, we combine the constant terms: \(3 - 15 = -12\), which leaves us with \(5x - 12\). Thus, the simplified expression is \(\frac{5x - 12}{(x-3)(x+3)}\). Comparing this to the given options, it corresponds to the fourth choice.
Examples
When managing inventory for a project, you might have two different batches of supplies that need to be categorized into the same system of bins. Let's say you have 3 units of an item in bins that group by sets of squares of a number minus 9, and 5 units in bins that group by sets of 3 more than the number. Combining these into a single simplified inventory tracking formula allows for more efficient monitoring. Just like simplifying rational expressions, this helps streamline complex logistics into a manageable, singular mathematical expression.