Let's solve the expression step by step:

$-\frac{1}{2} + 1 \frac{1}{9} + -\frac{1}{3}$

First, convert the mixed number $1 \frac{1}{9}$ into an improper fraction:

$1 \frac{1}{9} = \frac{10}{9}$

Now, the expression becomes:

$-\frac{1}{2} + \frac{10}{9} - \frac{1}{3}$

To add and subtract these fractions, we need a common denominator. The least common denominator (LCD) for 2, 9, and 3 is 18. Now, convert each fraction:

$-\frac{1}{2} = -\frac{9}{18}, \quad \frac{10}{9} = \frac{20}{18}, \quad -\frac{1}{3} = -\frac{6}{18}$

Now, substitute these fractions back into the expression:

$-\frac{9}{18} + \frac{20}{18} - \frac{6}{18}$

Combine the fractions:

$\frac{-9 + 20 - 6}{18} = \frac{5}{18}$

So, the final result is:

$\frac{5}{18}$

Would you like further details or have any questions?

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Here are 5 related questions you might consider:

1. How do you find the least common denominator for a set of fractions?
2. What is the process for converting a mixed number to an improper fraction?
3. How do you add or subtract fractions with different denominators?
4. What are some strategies for simplifying fractions?
5. How can you determine if a fraction is already in its simplest form?

Tip: Always check to see if your final fraction can be simplified further by finding the greatest common divisor of the numerator and denominator.