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The given expression is:

\[
\frac{9x - 11}{(x - 3)(x + 5)}
\]

To decompose this into partial fractions, we aim to express the fraction in the form:

\[
\frac{9x - 11}{(x - 3)(x + 5)} = \frac{A}{x - 3} + \frac{B}{x + 5}
\]

### Steps to solve:

1. Start with:
   \[
   \frac{9x - 11}{(x - 3)(x + 5)} = \frac{A}{x - 3} + \frac{B}{x + 5}
   \]

2. Multiply both sides by \((x - 3)(x + 5)\) to clear the denominator:
   \[
   9x - 11 = A(x + 5) + B(x - 3)
   \]

3. Expand both sides:
   \[
   9x - 11 = A(x) + A(5) + B(x) - B(3)
   \]
   \[
   9x - 11 = Ax + 5A + Bx - 3B
   \]

4. Combine like terms:
   \[
   9x - 11 = (A + B)x + (5A - 3B)
   \]

5. Now, equate the coefficients of \(x\) and the constant terms:
   - For the \(x\)-terms: \(A + B = 9\)
   - For the constant terms: \(5A - 3B = -11\)

### Solve the system of equations:

1. From \(A + B = 9\), we have \(B = 9 - A\).

2. Substitute \(B = 9 - A\) into the second equation \(5A - 3B = -11\):
   \[
   5A - 3(9 - A) = -11
   \]
   \[
   5A - 27 + 3A = -11
   \]
   \[
   8A - 27 = -11
   \]
   \[
   8A = 16
   \]
   \[
   A = 2
   \]

3. Now, substitute \(A = 2\) into \(A + B = 9\):
   \[
   2 + B = 9 \implies B = 7
   \]

### Final decomposition:
\[
\frac{9x - 11}{(x - 3)(x + 5)} = \frac{2}{x - 3} + \frac{7}{x + 5}
\]

Would you like further details on any part of this process?

Here are some related questions to expand on this:

1. How do partial fractions apply to solving integrals?
2. Can partial fractions be used with higher degree polynomials?
3. What happens if the denominator contains repeated factors?
4. How can partial fractions help solve differential equations?
5. Can this decomposition be verified by adding the fractions back?

**Tip:** Partial fractions are particularly useful for breaking down complex rational expressions when performing integration.

Decompose the expression 9x-11 / ((x-3)(x+5))

Learn how to decompose the rational expression 9x-11 over (x-3)(x+5) into partial fractions. This detailed step-by-step solution covers key algebraic concepts like linear equations and partial fraction decomposition. Suitable for students in Grades 10-12.

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