1. Stating the problem: Solve for $x$ in the equation $$\frac{6}{11} - \left(\frac{8}{5} + \frac{9}{11} - 2\right) \div \left[\frac{7}{30} - \left(\frac{2}{15} + \frac{7}{5} - \frac{6}{4}\right)\right] = \left(\frac{18}{99} + \frac{32}{132} + \frac{8}{66}\right) \div x$$ 2. Simplify inside the parentheses step-by-step: - Calculate $\frac{8}{5} + \frac{9}{11} - 2$: $$\frac{8}{5} + \frac{9}{11} - 2 = \frac{8}{5} + \frac{9}{11} - \frac{22}{11} = \frac{8}{5} - \frac{13}{11}$$ Find common denominator $55$: $$\frac{8}{5} = \frac{88}{55}, \quad \frac{13}{11} = \frac{65}{55}$$ So, $$\frac{88}{55} - \frac{65}{55} = \frac{23}{55}$$ - Calculate $\frac{2}{15} + \frac{7}{5} - \frac{6}{4}$: Convert to common denominator $60$: $$\frac{2}{15} = \frac{8}{60}, \quad \frac{7}{5} = \frac{84}{60}, \quad \frac{6}{4} = \frac{90}{60}$$ Sum: $$8/60 + 84/60 - 90/60 = (8 + 84 - 90)/60 = 2/60 = \frac{1}{30}$$ - Calculate $\frac{7}{30} - \frac{1}{30} = \frac{6}{30} = \frac{1}{5}$ 3. Now the left side numerator: $$\frac{6}{11} - \frac{23}{55}$$ Common denominator $55$: $$\frac{6}{11} = \frac{30}{55}$$ So, $$\frac{30}{55} - \frac{23}{55} = \frac{7}{55}$$ 4. The left side is now: $$\frac{7}{55} \div \frac{1}{5} = \frac{7}{55} \times 5 = \frac{35}{55} = \frac{7}{11}$$ 5. Simplify the right side numerator: $$\frac{18}{99} + \frac{32}{132} + \frac{8}{66}$$ Find common denominator $396$: $$\frac{18}{99} = \frac{72}{396}, \quad \frac{32}{132} = \frac{96}{396}, \quad \frac{8}{66} = \frac{48}{396}$$ Sum: $$72 + 96 + 48 = 216$$ So, $$\frac{216}{396} = \frac{18}{33} = \frac{6}{11}$$ 6. The equation is now: $$\frac{7}{11} = \frac{6}{11} \div x = \frac{6}{11} \times \frac{1}{x} = \frac{6}{11x}$$ 7. Solve for $x$: $$\frac{7}{11} = \frac{6}{11x}$$ Multiply both sides by $11x$: $$7x = 6$$ Divide both sides by 7: $$x = \frac{6}{7}$$ Final answer: $$\boxed{\frac{6}{7}}$$