1. **Problem:** Solve the equation $$|1 - 2m| = m + 2$$ and verify the solutions. 2. **Formula and rules:** The absolute value equation $$|A| = B$$ implies two cases if $$B \geq 0$$: - Case 1: $$A = B$$ - Case 2: $$A = -B$$ If $$B < 0$$, there are no solutions because absolute value is always non-negative. 3. **Apply to the problem:** Here, $$A = 1 - 2m$$ and $$B = m + 2$$. First, check if $$m + 2 \geq 0$$, so $$m \geq -2$$ for solutions to exist. 4. **Case 1:** $$1 - 2m = m + 2$$ Solve for $$m$$: $$1 - 2m = m + 2$$ $$1 - 2m - m = 2$$ $$1 - 3m = 2$$ $$-3m = 1$$ $$m = -\frac{1}{3}$$ Check if $$m \geq -2$$: $$-\frac{1}{3} \geq -2$$ is true, so this is a valid solution. 5. **Case 2:** $$1 - 2m = -(m + 2)$$ Solve for $$m$$: $$1 - 2m = -m - 2$$ $$1 - 2m + m = -2$$ $$1 - m = -2$$ $$-m = -3$$ $$m = 3$$ Check if $$m \geq -2$$: $$3 \geq -2$$ is true, so this is a valid solution. 6. **Verification:** - For $$m = -\frac{1}{3}$$: $$|1 - 2(-\frac{1}{3})| = |1 + \frac{2}{3}| = |\frac{5}{3}| = \frac{5}{3}$$ $$m + 2 = -\frac{1}{3} + 2 = \frac{5}{3}$$ Both sides equal $$\frac{5}{3}$$, so valid. - For $$m = 3$$: $$|1 - 2(3)| = |1 - 6| = |-5| = 5$$ $$m + 2 = 3 + 2 = 5$$ Both sides equal 5, so valid. **Final answer:** $$m = -\frac{1}{3}$$ or $$m = 3$$