1. We are asked to find the value of $2 \frac{1}{4} - \frac{1}{6}$.\n2. Convert the mixed number $2 \frac{1}{4}$ to an improper fraction. Since $2 = \frac{8}{4}$, we have $2 \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4}$.\n3. Now the expression is $\frac{9}{4} - \frac{1}{6}$.\n4. To subtract these fractions, find a common denominator. The least common denominator (LCD) of 4 and 6 is 12.\n5. Convert each fraction to have denominator 12: $$\frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12}$$ and $$\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}$$.\n6. Subtract the fractions: $$\frac{27}{12} - \frac{2}{12} = \frac{27 - 2}{12} = \frac{25}{12}$$.\n7. Convert the improper fraction back to a mixed number: $\frac{25}{12} = 2 \frac{1}{12}$.\n8. Therefore, the final answer is $2 \frac{1}{12}$.