1. **State the problem:** Solve the equation $$|4x - 5| = |x - 1|$$ for $x$. 2. **Recall the property of absolute values:** For any real numbers $A$ and $B$, if $$|A| = |B|,$$ then either $$A = B$$ or $$A = -B.$$ This means we need to consider two cases: 3. **Case 1:** $$4x - 5 = x - 1$$ Solve for $x$: $$4x - x = -1 + 5$$ $$3x = 4$$ $$x = \frac{4}{3}$$ 4. **Case 2:** $$4x - 5 = -(x - 1)$$ $$4x - 5 = -x + 1$$ $$4x + x = 1 + 5$$ $$5x = 6$$ $$x = \frac{6}{5}$$ 5. **Check solutions:** Both $x = \frac{4}{3}$ and $x = \frac{6}{5}$ satisfy the original equation because absolute values are always non-negative and the equality holds. **Final answer:** $$x = \frac{4}{3} \quad \text{or} \quad x = \frac{6}{5}$$