1. **State the problem:** Simplify the expression $$1\frac{4}{7} \div \frac{2}{3} - 1\frac{5}{7}$$. 2. **Convert mixed numbers to improper fractions:** $$1\frac{4}{7} = \frac{11}{7}, \quad 1\frac{5}{7} = \frac{12}{7}$$. 3. **Rewrite the expression:** $$\frac{11}{7} \div \frac{2}{3} - \frac{12}{7}$$. 4. **Division of fractions rule:** To divide by a fraction, multiply by its reciprocal: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$. 5. **Apply the rule:** $$\frac{11}{7} \times \frac{3}{2} - \frac{12}{7}$$. 6. **Multiply fractions:** $$\frac{11 \times 3}{7 \times 2} - \frac{12}{7} = \frac{33}{14} - \frac{12}{7}$$. 7. **Find common denominator for subtraction:** The denominators are 14 and 7; convert $$\frac{12}{7}$$ to $$\frac{24}{14}$$. 8. **Subtract fractions:** $$\frac{33}{14} - \frac{24}{14} = \frac{33 - 24}{14} = \frac{9}{14}$$. **Final answer:** $$\boxed{\frac{9}{14}}$$.