We are asked to simplify the expression:

$\frac{1}{3x} - \frac{4}{x} + \frac{5}{9x}$

Step 1: Find a common denominator

The least common denominator (LCD) of the terms involves the denominators $3x$, $x$, and $9x$. The LCD of these three is $9x$.

Step 2: Rewrite each fraction with the common denominator

• The first term is $\frac{1}{3x}$. To have a denominator of $9x$, multiply both the numerator and denominator by 3:

$\frac{1}{3x} = \frac{3}{9x}$

• The second term is $\frac{4}{x}$. To have a denominator of $9x$, multiply both the numerator and denominator by 9:

$\frac{4}{x} = \frac{36}{9x}$

• The third term is already $\frac{5}{9x}$, so it remains unchanged.

Step 3: Combine the fractions

Now, we can combine the fractions over the common denominator of $9x$:

$\frac{3}{9x} - \frac{36}{9x} + \frac{5}{9x} = \frac{3 - 36 + 5}{9x}$

Simplify the numerator:

$3 - 36 + 5 = -28$

So, the simplified expression is:

$\frac{-28}{9x}$

Final Answer:

$\frac{-28}{9x}$

Permissible values:

For the expression to be valid, the denominator cannot be zero. Therefore, $x \neq 0$.

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Would you like more details or have any questions? Here are some related questions for you:

1. How do you find the least common denominator (LCD) for rational expressions?
2. What happens if you try to simplify fractions with different variables in the denominators?
3. How do you deal with negative signs when simplifying expressions?
4. What is the significance of excluding zero in the permissible values of $x$?
5. How can you check your simplified expression to ensure it's correct?

Tip: When combining fractions, always start by finding the least common denominator (LCD) to simplify the process.