1. State the problem: We want to add the mixed numbers $-1\frac{3}{4}$ and $-1\frac{2}{3}$.\n\n2. Convert each mixed number to an improper fraction.\n\nFor $-1\frac{3}{4}$: $$-1\frac{3}{4} = -\left(1 + \frac{3}{4}\right) = -\frac{4}{4} - \frac{3}{4} = -\frac{7}{4}.$$\n\nFor $-1\frac{2}{3}$: $$-1\frac{2}{3} = -\left(1 + \frac{2}{3}\right) = -\frac{3}{3} - \frac{2}{3} = -\frac{5}{3}.$$\n\n3. Add the two improper fractions: $$-\frac{7}{4} + -\frac{5}{3} = -\frac{7}{4} - \frac{5}{3}.$$\n\n4. Find a common denominator between 4 and 3, which is 12. Convert each fraction: $$-\frac{7}{4} = -\frac{21}{12},\quad -\frac{5}{3} = -\frac{20}{12}.$$\n\n5. Add the fractions: $$-\frac{21}{12} - \frac{20}{12} = -\frac{41}{12}.$$\n\n6. Convert the improper fraction back to a mixed number: $$-\frac{41}{12} = -3\frac{5}{12}$$ since $41 = 3 \times 12 + 5$.\n\nFinal answer: $-3\frac{5}{12}$.