To find an equivalent expression for [tex]\(\frac{15}{x-6} + \frac{7}{x+6}\)[/tex], we need to combine these fractions into a single fraction with a common denominator.
### Step-by-Step Solution:
1. Identify the denominators:
- The first fraction is [tex]\(\frac{15}{x-6}\)[/tex].
- The second fraction is [tex]\(\frac{7}{x+6}\)[/tex].
2. Find the common denominator:
- The common denominator of [tex]\((x-6)\)[/tex] and [tex]\((x+6)\)[/tex] is their product, [tex]\((x-6)(x+6)\)[/tex]. This simplifies to [tex]\(x^2 - 36\)[/tex].
3. Rewrite each fraction with the common denominator:
- The first fraction [tex]\(\frac{15}{x-6}\)[/tex] is rewritten as:
[tex]\[
\frac{15(x+6)}{(x-6)(x+6)} = \frac{15x + 90}{x^2 - 36}
\][/tex]
- The second fraction [tex]\(\frac{7}{x+6}\)[/tex] is rewritten as:
[tex]\[
\frac{7(x-6)}{(x-6)(x+6)} = \frac{7x - 42}{x^2 - 36}
\][/tex]
4. Add the two fractions:
- Combine the numerators over the common denominator:
[tex]\[
\frac{15x + 90}{x^2 - 36} + \frac{7x - 42}{x^2 - 36} = \frac{(15x + 90) + (7x - 42)}{x^2 - 36}
\][/tex]
- Combine the terms in the numerator:
[tex]\[
(15x + 90) + (7x - 42) = 15x + 7x + 90 - 42 = 22x + 48
\][/tex]
- So the final expression is:
[tex]\[
\frac{22x + 48}{x^2 - 36}
\][/tex]
### Conclusion:
The expression that is equivalent to [tex]\(\frac{15}{x-6} + \frac{7}{x+6}\)[/tex] is:
[tex]\[
\frac{22x + 48}{x^2 - 36}
\][/tex]
Therefore, the correct answer is:
D. [tex]\(\frac{22x + 48}{x^2 - 36}\)[/tex]
