1. **State the problem:** Simplify the expression $$\left(\frac{3}{2} + \left(\frac{4}{5} - \left(\frac{5}{3} \div \frac{7}{6}\right)\right)\right) \times \left(\frac{9}{4} \div \frac{3}{2}\right)$$. 2. **Recall the rules:** - Division of fractions: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$. - Addition and subtraction require common denominators. 3. **Simplify the inner division:** $$\frac{5}{3} \div \frac{7}{6} = \frac{5}{3} \times \frac{6}{7} = \frac{5 \times 6}{3 \times 7} = \frac{30}{21}$$ Simplify $$\frac{30}{21}$$ by dividing numerator and denominator by 3: $$\frac{\cancel{30}^{10}}{\cancel{21}^{7}} = \frac{10}{7}$$. 4. **Simplify inside the brackets:** $$\frac{4}{5} - \frac{10}{7} = \frac{4 \times 7}{5 \times 7} - \frac{10 \times 5}{7 \times 5} = \frac{28}{35} - \frac{50}{35} = \frac{28 - 50}{35} = \frac{-22}{35}$$. 5. **Add $$\frac{3}{2}$$ to the result:** Find common denominator for $$\frac{3}{2}$$ and $$\frac{-22}{35}$$, which is 70: $$\frac{3}{2} = \frac{3 \times 35}{2 \times 35} = \frac{105}{70}$$ $$\frac{-22}{35} = \frac{-22 \times 2}{35 \times 2} = \frac{-44}{70}$$ Add: $$\frac{105}{70} + \frac{-44}{70} = \frac{105 - 44}{70} = \frac{61}{70}$$. 6. **Simplify the second division:** $$\frac{9}{4} \div \frac{3}{2} = \frac{9}{4} \times \frac{2}{3} = \frac{9 \times 2}{4 \times 3} = \frac{18}{12}$$ Simplify $$\frac{18}{12}$$ by dividing numerator and denominator by 6: $$\frac{\cancel{18}^{3}}{\cancel{12}^{2}} = \frac{3}{2}$$. 7. **Multiply the two results:** $$\frac{61}{70} \times \frac{3}{2} = \frac{61 \times 3}{70 \times 2} = \frac{183}{140}$$. 8. **Final answer:** $$\frac{183}{140}$$ or as a mixed number $$1 \frac{43}{140}$$.