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:** $$1\frac{4}{7} = \frac{7 \times 1 + 4}{7} = \frac{11}{7}$$ $$1\frac{5}{7} = \frac{7 \times 1 + 5}{7} = \frac{12}{7}$$ 3. **Rewrite the expression:** $$\frac{11}{7} \div \frac{2}{3} - \frac{12}{7}$$ 4. **Division of fractions rule:** Dividing by a fraction is the same as multiplying by its reciprocal: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$ 5. **Apply the rule:** $$\frac{11}{7} \times \frac{3}{2} = \frac{11 \times 3}{7 \times 2} = \frac{33}{14}$$ 6. **Rewrite the expression:** $$\frac{33}{14} - \frac{12}{7}$$ 7. **Find common denominator:** The denominator 14 is a multiple of 7, so convert $\frac{12}{7}$ to fourteenths: $$\frac{12}{7} = \frac{12 \times 2}{7 \times 2} = \frac{24}{14}$$ 8. **Subtract the fractions:** $$\frac{33}{14} - \frac{24}{14} = \frac{33 - 24}{14} = \frac{9}{14}$$ **Final answer:** $$\boxed{\frac{9}{14}}$$