To solve the expression
[tex]\[ \frac{- \frac{4}{3} + \left(- \frac{3}{4} + 7 + 1\right) - \left(\frac{-7 - \frac{7}{5} - \frac{11}{2}}{7}\right)}{\frac{5}{3}}, \][/tex]
we will work through it step by step, starting with the innermost parentheses.
1. Evaluate [tex]\(\left(- \frac{3}{4} + 7 + 1\right)\)[/tex]:
[tex]\[
- \frac{3}{4} + 7 + 1 = -0.75 + 7 + 1 = 7.25
\][/tex]
2. Evaluate [tex]\(\left(-7 - \frac{7}{5} - \frac{11}{2}\right)\)[/tex]:
[tex]\[
\text{First, compute each term:}
\][/tex]
[tex]\[
-7, \quad - \frac{7}{5} = -1.4, \quad \text{and} \quad - \frac{11}{2} = -5.5
\][/tex]
[tex]\[
\text{Summing these terms:}\quad -7 - 1.4 - 5.5 = -13.9
\][/tex]
3. Now, substitute back into the larger expression:
[tex]\[
- \frac{4}{3} + (7.25) - \left(\frac{-13.9}{7}\right)
\][/tex]
4. Simplify [tex]\(\left(\frac{-13.9}{7}\right)\)[/tex]:
[tex]\[
\frac{-13.9}{7} = -1.9857
\][/tex]
5. Substitute this back into the outer expression:
[tex]\[
- \frac{4}{3} + 7.25 - (-1.9857)
\][/tex]
[tex]\[
- \frac{4}{3} = -1.3333
\][/tex]
[tex]\[
-1.3333 + 7.25 + 1.9857 = 7.9024
\][/tex]
6. Finally, evaluate the outer fraction:
[tex]\[
\frac{7.9024}{\frac{5}{3}}
\][/tex]
[tex]\[
\frac{7.9024}{5 / 3} = 7.9024 \times \frac{3}{5} = 4.7414
\][/tex]
So, the result of the expression is:
[tex]\[ \boxed{4.7414} \][/tex]
