The problem requires us to evaluate the expression $\frac{4}{9} - 1\frac{2}{9} - \frac{5}{9} + 2\frac{4}{9}$.

1. Problem Description

The problem requires us to evaluate the expression $\frac{4}{9} - 1\frac{2}{9} - \frac{5}{9} + 2\frac{4}{9}$.

2. Solution Steps

First, convert the mixed numbers to improper fractions:

$1\frac{2}{9} = \frac{1 \times 9 + 2}{9} = \frac{9+2}{9} = \frac{11}{9}$

$2\frac{4}{9} = \frac{2 \times 9 + 4}{9} = \frac{18+4}{9} = \frac{22}{9}$

Now, substitute these improper fractions into the expression:

$\frac{4}{9} - \frac{11}{9} - \frac{5}{9} + \frac{22}{9}$

Since all fractions have the same denominator, we can combine the numerators:

$\frac{4 - 11 - 5 + 22}{9}$

Now, perform the arithmetic operations in the numerator:

$\frac{4 - 11 - 5 + 22}{9} = \frac{-7 - 5 + 22}{9} = \frac{-12 + 22}{9} = \frac{10}{9}$

Now we convert the improper fraction back to a mixed number:

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

3. Final Answer

$1\frac{1}{9}$