The expression you provided is:

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

To combine these two terms, we need to find a common denominator. The denominators are \(x\) and \(x-3\), so the common denominator will be \(x(x-3)\).

Now, let's rewrite each fraction with the common denominator:

\[
\frac{9}{x} = \frac{9(x-3)}{x(x-3)} = \frac{9x - 27}{x(x-3)}
\]

\[
\frac{-5}{x-3} = \frac{-5x}{x(x-3)}
\]

Now, we can combine the two terms:

\[
\frac{9x - 27}{x(x-3)} + \frac{-5x}{x(x-3)} = \frac{9x - 27 - 5x}{x(x-3)} = \frac{4x - 27}{x(x-3)}
\]

So the simplified expression is:

\[
\frac{4x - 27}{x(x-3)}
\]

Would you like a detailed explanation of how to handle this kind of operation or any specific part of this solution?

Here are 5 related questions to deepen your understanding:
1. How would the expression change if the numerator of the second fraction was positive?
2. What happens to the expression if \(x = 3\)?
3. Can you find the domain of the simplified expression?
4. What would happen if both numerators were constants?
5. How would you solve for \(x\) if this expression were set equal to a number?

**Tip:** Always watch out for restrictions in the domain when working with rational expressions. In this case, \(x \neq 0\) and \(x \neq 3\).

Learn how to simplify the rational expression 9/x + (-5)/(x-3) by finding a common denominator and combining terms. This problem involves key algebraic concepts such as rational expressions and domain restrictions. Suitable for Grades 9-10.

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