Let's simplify the given mathematical expression step by step. Here is the expression:

$2\frac{1}{6} \cdot \left( 1\frac{1}{3}x - 1\frac{5}{6} \right) + 2\frac{3}{4} \cdot \left( 1\frac{1}{5}x - 2\frac{5}{6} \right) - \frac{917x}{90} + \frac{631}{72}$

Step 1: Convert mixed fractions to improper fractions.

• $2\frac{1}{6} = \frac{13}{6}$
• $1\frac{1}{3} = \frac{4}{3}$
• $1\frac{5}{6} = \frac{11}{6}$
• $2\frac{3}{4} = \frac{11}{4}$
• $1\frac{1}{5} = \frac{6}{5}$
• $2\frac{5}{6} = \frac{17}{6}$

Now the expression becomes:

$\frac{13}{6} \cdot \left( \frac{4}{3}x - \frac{11}{6} \right) + \frac{11}{4} \cdot \left( \frac{6}{5}x - \frac{17}{6} \right) - \frac{917x}{90} + \frac{631}{72}$

Step 2: Expand the brackets.

• Expand $\frac{13}{6} \cdot \left( \frac{4}{3}x - \frac{11}{6} \right)$: $\frac{13}{6} \cdot \frac{4}{3}x = \frac{52}{18}x = \frac{26}{9}x, \quad \frac{13}{6} \cdot \frac{11}{6} = \frac{143}{36}$ So, $\frac{13}{6} \cdot \left( \frac{4}{3}x - \frac{11}{6} \right) = \frac{26}{9}x - \frac{143}{36}$.

• Expand $\frac{11}{4} \cdot \left( \frac{6}{5}x - \frac{17}{6} \right)$: $\frac{11}{4} \cdot \frac{6}{5}x = \frac{66}{20}x = \frac{33}{10}x, \quad \frac{11}{4} \cdot \frac{17}{6} = \frac{187}{24}$ So, $\frac{11}{4} \cdot \left( \frac{6}{5}x - \frac{17}{6} \right) = \frac{33}{10}x - \frac{187}{24}$.

Now the expression becomes: $\left( \frac{26}{9}x - \frac{143}{36} \right) + \left( \frac{33}{10}x - \frac{187}{24} \right) - \frac{917x}{90} + \frac{631}{72}$

Step 3: Combine $x$-terms.

• Write all $x$-terms with a common denominator: $\frac{26}{9}x = \frac{260}{90}x, \quad \frac{33}{10}x = \frac{297}{90}x, \quad -\frac{917x}{90} = -\frac{917}{90}x$ Combine: $\frac{260}{90}x + \frac{297}{90}x - \frac{917}{90}x = \frac{260 + 297 - 917}{90}x = \frac{-360}{90}x = -4x$

Step 4: Combine constant terms.

• Write all constant terms with a common denominator: $-\frac{143}{36}, \quad -\frac{187}{24}, \quad \frac{631}{72}$ Find a common denominator (72): $-\frac{143}{36} = -\frac{286}{72}, \quad -\frac{187}{24} = -\frac{561}{72}, \quad \frac{631}{72} = \frac{631}{72}$ Combine: $-\frac{286}{72} - \frac{561}{72} + \frac{631}{72} = \frac{-286 - 561 + 631}{72} = \frac{-216}{72} = -3$

Final Expression:

$-4x - 3$

Let me know if you need further clarification or detailed steps for any part! Here are five related questions:

1. How do you find the least common denominator when combining fractions?
2. Why is it important to convert mixed fractions into improper fractions during simplification?
3. Can you explain how to distribute coefficients in expressions with parentheses?
4. What are common mistakes to watch for when simplifying terms involving fractions?
5. How do you check if the final simplified expression is correct?

Tip: Always double-check your common denominators to ensure accurate simplifications of fractions.