1. **State the problem:** We need to add the fractions $\frac{4}{7}$ and $\frac{3}{2}$. 2. **Formula and rules:** To add fractions, they must have a common denominator. The formula is: $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$ where $a$, $b$, $c$, and $d$ are integers and $b, d \neq 0$. 3. **Find the common denominator:** The denominators are 7 and 2. The least common denominator (LCD) is the least common multiple (LCM) of 7 and 2, which is 14. 4. **Convert each fraction:** $$\frac{4}{7} = \frac{4 \times 2}{7 \times 2} = \frac{8}{14}$$ $$\frac{3}{2} = \frac{3 \times 7}{2 \times 7} = \frac{21}{14}$$ 5. **Add the fractions:** $$\frac{8}{14} + \frac{21}{14} = \frac{8 + 21}{14} = \frac{29}{14}$$ 6. **Simplify if possible:** $\frac{29}{14}$ is an improper fraction and cannot be simplified further since 29 is prime and does not share factors with 14. 7. **Final answer:** $$\frac{4}{7} + \frac{3}{2} = \frac{29}{14}$$