- Identify the common denominator as \((x-6)(x+6) = x^2 - 36\).
- Rewrite the fractions as \(\frac{15x+90}{x^2-36}\) and \(\frac{7x-42}{x^2-36}\).
- Sum the numerators and combine like terms to get \(\frac{22x+48}{x^2-36}\).
- The correct choice is \(\frac{22x + 48}{x^2 - 36}\), resulting in \(\boxed{\frac{22x + 48}{x^2 - 36}}\).
Explanation
- Analysis and Identification of Common Denominator
To find the equivalent expression for \(\frac{15}{x-6} + \frac{7}{x+6}\), we first analyze the structure of the fractions. Both denominators, \(x-6\) and \(x+6\), are distinct linear factors. To add these two fractions, we need to express them over a common denominator. The least common denominator is the product of the two denominators: \((x-6)(x+6) = x^2 - 36\).
- Rewriting Fractions with a Common Denominator
Next, we rewrite each fraction to have this common denominator.
For the first fraction, we multiply the numerator and the denominator by \((x+6)\):
\[\frac{15}{x-6} = \frac{15(x+6)}{(x-6)(x+6)} = \frac{15x + 90}{x^2 - 36}\]
For the second fraction, we multiply the numerator and the denominator by \((x-6)\):
\[\frac{7}{x+6} = \frac{7(x-6)}{(x+6)(x-6)} = \frac{7x - 42}{x^2 - 36}\]
- Combining and Simplifying the Numerators
Now that both fractions share the same denominator, we can add them by summing their numerators:
\[\frac{15x + 90}{x^2 - 36} + \frac{7x - 42}{x^2 - 36} = \frac{(15x + 90) + (7x - 42)}{x^2 - 36}\]
Combining the terms in the numerator, we combine the \(x\)-terms (\(15x + 7x = 22x\)) and the constants (\(90 - 42 = 48\)):
\[\frac{22x + 48}{x^2 - 36}\]
- Conclusion
Comparing our result \(\frac{22x + 48}{x^2 - 36}\) to the given options, we identify it as the fourth option. Therefore, the equivalent expression is \(\frac{22x + 48}{x^2 - 36}\).
Examples
In engineering or physics, you might need to combine measurements taken from two different sensors that measure the same quantity with different sensitivities. If the sensor responses are given by fractional functions of a parameter \(x\), adding these functions is equivalent to combining the data into a single model for analysis. This algebraic process allows you to simplify complex physical models into a more manageable single expression, facilitating easier calculations and interpretation of the combined system data.