To solve this, first convert all the mixed numbers into improper fractions.

4 1/2 = 9/2
1 1/2 = 3/2

Now plug the numbers into the equation:

9/2 4/9 + 4/9 3/2

Now multiply the numerators and denominators of the fractions:

(9 4) / (2 9) + (4 3) / (9 2)

36 / 18 + 12 / 18

Now add the fractions:

36/18 + 12/18 = 48/18

Reduce the fraction by dividing the numerator and denominator by their greatest common factor, which is 6:

48/18 = 8/3

So, 4 1/2 4/9 + 4/9 1 1/2 = 8/3 or 2 2/3.