To add the fractions [tex]\(-\frac{5}{9}\)[/tex] and [tex]\(3 \frac{3}{8}\)[/tex], we first need to convert the mixed number to an improper fraction.
1. Convert [tex]\(3 \frac{3}{8}\)[/tex] to an improper fraction:
[tex]\[
3 \frac{3}{8} = \frac{3 \times 8 + 3}{8} = \frac{24 + 3}{8} = \frac{27}{8}
\][/tex]
2. Now, we add [tex]\(-\frac{5}{9}\)[/tex] and [tex]\(\frac{27}{8}\)[/tex]:
To add these fractions, we need a common denominator. The least common denominator (LCD) of 9 and 8 is 72.
Convert each fraction to have a denominator of 72:
[tex]\[
-\frac{5}{9} = -\frac{5 \times 8}{9 \times 8} = -\frac{40}{72}
\][/tex]
[tex]\[
\frac{27}{8} = \frac{27 \times 9}{8 \times 9} = \frac{243}{72}
\][/tex]
3. Add the fractions with the common denominator:
[tex]\[
-\frac{40}{72} + \frac{243}{72} = \frac{-40 + 243}{72} = \frac{203}{72}
\][/tex]
4. Therefore, the result of the addition is:
[tex]\[
-\frac{5}{9} + 3 \frac{3}{8} = \frac{203}{72}
\][/tex]
So, the final result is:
[tex]\[
\left( -\frac{5}{9}, 3 \frac{3}{8}, \frac{203}{72} \right)
\][/tex]
