1. State the problem: Subtract and simplify the mixed numbers $7 \dfrac{5}{6}$ and $4 \dfrac{2}{6}$. 2. Convert mixed numbers to improper fractions: $$7 \dfrac{5}{6} = \frac{7 \times 6 + 5}{6} = \frac{42 + 5}{6} = \frac{47}{6}$$ $$4 \dfrac{2}{6} = \frac{4 \times 6 + 2}{6} = \frac{24 + 2}{6} = \frac{26}{6}$$ 3. Subtract the improper fractions: $$\frac{47}{6} - \frac{26}{6} = \frac{47 - 26}{6} = \frac{21}{6}$$ 4. Simplify the fraction $\frac{21}{6}$ by dividing numerator and denominator by their greatest common divisor $3$: $$\frac{21 \div 3}{6 \div 3} = \frac{7}{2}$$ 5. Convert the improper fraction $\frac{7}{2}$ back to a mixed number: $$7 \div 2 = 3 \text{ remainder } 1 \Rightarrow 3 \dfrac{1}{2}$$ **Final answer:** $3 \dfrac{1}{2}$