1. The problem asks to solve the expression $1 \frac{1}{7} - \frac{3}{4}$.\n2. Convert the mixed number to an improper fraction: $1 \frac{1}{7} = \frac{8}{7}$.\n3. Find a common denominator for $\frac{8}{7}$ and $\frac{3}{4}$. The least common denominator (LCD) of 7 and 4 is 28.\n4. Convert both fractions to have denominator 28: $\frac{8}{7} = \frac{8 \times 4}{7 \times 4} = \frac{32}{28}$ and $\frac{3}{4} = \frac{3 \times 7}{4 \times 7} = \frac{21}{28}$.\n5. Subtract the fractions: $\frac{32}{28} - \frac{21}{28} = \frac{32 - 21}{28} = \frac{11}{28}$.\n6. The result of the subtraction is $\frac{11}{28}$.\n\nFinal answer: $\frac{11}{28}$