1. **Problem:** Solve the equation $$|4x - 1| = |x - 3|$$. 2. **Formula and rules:** The absolute value equation $$|A| = |B|$$ implies either $$A = B$$ or $$A = -B$$. 3. **Step 1:** Set up the two cases: Case 1: $$4x - 1 = x - 3$$ Case 2: $$4x - 1 = -(x - 3)$$ 4. **Step 2:** Solve Case 1: $$4x - 1 = x - 3$$ Subtract $$x$$ from both sides: $$4x - x - 1 = -3$$ $$3x - 1 = -3$$ Add 1 to both sides: $$3x = -3 + 1$$ $$3x = -2$$ Divide both sides by 3: $$\frac{\cancel{3}x}{\cancel{3}} = \frac{-2}{3}$$ $$x = -\frac{2}{3}$$ 5. **Step 3:** Solve Case 2: $$4x - 1 = -(x - 3) = -x + 3$$ Add $$x$$ to both sides: $$4x + x - 1 = 3$$ $$5x - 1 = 3$$ Add 1 to both sides: $$5x = 4$$ Divide both sides by 5: $$\frac{\cancel{5}x}{\cancel{5}} = \frac{4}{5}$$ $$x = \frac{4}{5}$$ 6. **Step 4:** Final solutions for part (i): $$x = -\frac{2}{3}$$ or $$x = \frac{4}{5}$$ --- 7. **Problem:** Hence solve the equation $$|4^{x+1} - 1| = |4^x - 3|$$ correct to 3 significant figures. 8. **Step 1:** Let $$y = 4^x$$, then $$4^{x+1} = 4 \cdot 4^x = 4y$$. Rewrite the equation as: $$|4y - 1| = |y - 3|$$ 9. **Step 2:** Using the same logic as part (i), set up two cases: Case 1: $$4y - 1 = y - 3$$ Case 2: $$4y - 1 = -(y - 3) = -y + 3$$ 10. **Step 3:** Solve Case 1: $$4y - 1 = y - 3$$ Subtract $$y$$ from both sides: $$3y - 1 = -3$$ Add 1 to both sides: $$3y = -2$$ Divide both sides by 3: $$\frac{\cancel{3}y}{\cancel{3}} = \frac{-2}{3}$$ $$y = -\frac{2}{3}$$ Since $$y = 4^x > 0$$, discard this solution. 11. **Step 4:** Solve Case 2: $$4y - 1 = -y + 3$$ Add $$y$$ to both sides: $$5y - 1 = 3$$ Add 1 to both sides: $$5y = 4$$ Divide both sides by 5: $$\frac{\cancel{5}y}{\cancel{5}} = \frac{4}{5}$$ $$y = \frac{4}{5}$$ 12. **Step 5:** Recall $$y = 4^x$$, so: $$4^x = \frac{4}{5}$$ Take logarithm base 4 of both sides: $$x = \log_4 \left(\frac{4}{5}\right)$$ Rewrite using natural logs: $$x = \frac{\ln \left(\frac{4}{5}\right)}{\ln 4}$$ 13. **Step 6:** Calculate numerically: $$\ln \left(\frac{4}{5}\right) = \ln 0.8 \approx -0.223143$$ $$\ln 4 \approx 1.386294$$ So: $$x \approx \frac{-0.223143}{1.386294} \approx -0.161$$ Rounded to 3 significant figures: $$x \approx -0.161$$ --- **Final answers:** (i) $$x = -\frac{2}{3}$$ or $$x = \frac{4}{5}$$ (ii) $$x \approx -0.161$$ (to 3 significant figures)