1. **State the problem:** Solve the inequality $$\frac{3x - 2}{4} \leq \frac{4x + 1}{3}$$. 2. **Write the inequality:** $$\frac{3x - 2}{4} \leq \frac{4x + 1}{3}$$. 3. **Clear denominators by multiplying both sides by 12 (the least common multiple of 4 and 3):** $$12 \times \frac{3x - 2}{4} \leq 12 \times \frac{4x + 1}{3}$$ 4. **Simplify each side:** $$3 \times (3x - 2) \leq 4 \times (4x + 1)$$ 5. **Expand both sides:** $$9x - 6 \leq 16x + 4$$ 6. **Bring all terms involving $x$ to one side and constants to the other:** $$9x - 16x \leq 4 + 6$$ 7. **Simplify:** $$-7x \leq 10$$ 8. **Divide both sides by $-7$ to isolate $x$, remembering to reverse the inequality sign because we divide by a negative number:** $$\cancel{-7}x \geq \frac{10}{\cancel{-7}}$$ 9. **Final simplified inequality:** $$x \geq -\frac{10}{7}$$ **Answer:** $$x \geq -\frac{10}{7}$$