1. **State the problem:** Solve the equation $$3 - \frac{X}{2} + \frac{X}{4} - \frac{X}{8} = 4 - \frac{X}{16}$$ for $X$. 2. **Combine like terms on the left side:** The terms involving $X$ are $$-\frac{X}{2} + \frac{X}{4} - \frac{X}{8}$$. 3. **Find a common denominator for the $X$ terms:** The denominators are 2, 4, and 8. The least common denominator is 8. Rewrite each term: $$-\frac{X}{2} = -\frac{4X}{8}, \quad \frac{X}{4} = \frac{2X}{8}, \quad -\frac{X}{8} = -\frac{X}{8}$$ 4. **Sum the $X$ terms:** $$-\frac{4X}{8} + \frac{2X}{8} - \frac{X}{8} = \frac{-4X + 2X - X}{8} = \frac{-3X}{8}$$ 5. **Rewrite the equation:** $$3 - \frac{3X}{8} = 4 - \frac{X}{16}$$ 6. **Bring all $X$ terms to one side and constants to the other:** $$- \frac{3X}{8} + \frac{X}{16} = 4 - 3$$ 7. **Find common denominator for $X$ terms on the left:** 16 Rewrite: $$- \frac{6X}{16} + \frac{X}{16} = 1$$ 8. **Combine $X$ terms:** $$- \frac{6X}{16} + \frac{X}{16} = -\frac{5X}{16}$$ So, $$-\frac{5X}{16} = 1$$ 9. **Solve for $X$:** Multiply both sides by 16: $$-5X = 16$$ Divide both sides by -5: $$X = -\frac{16}{5}$$ **Final answer:** $$X = -\frac{16}{5}$$