1. **State the problem:** Solve the equation $$\frac{5x + 2}{4} - \frac{x - 2}{3} = \frac{x - 5}{16}$$ for $x$. 2. **Identify the formula and rules:** To solve equations with fractions, find the least common denominator (LCD) to clear fractions by multiplying both sides. 3. **Find the LCD:** The denominators are 4, 3, and 16. The LCD is 48. 4. **Multiply both sides by 48:** $$48 \times \left(\frac{5x + 2}{4} - \frac{x - 2}{3}\right) = 48 \times \frac{x - 5}{16}$$ 5. **Distribute multiplication:** $$48 \times \frac{5x + 2}{4} - 48 \times \frac{x - 2}{3} = 48 \times \frac{x - 5}{16}$$ 6. **Simplify each term:** $$12(5x + 2) - 16(x - 2) = 3(x - 5)$$ 7. **Expand each term:** $$60x + 24 - 16x + 32 = 3x - 15$$ 8. **Combine like terms on the left:** $$ (60x - 16x) + (24 + 32) = 3x - 15$$ $$44x + 56 = 3x - 15$$ 9. **Bring variables to one side and constants to the other:** $$44x - 3x = -15 - 56$$ $$41x = -71$$ 10. **Divide both sides by 41:** $$x = \frac{-71}{41}$$ 11. **Final answer:** $$\boxed{x = -\frac{71}{41}}$$