Absolute Value 4A986E

1. **State the problem:** We need to solve the absolute value equation $$|4x - 5| = 7$$ for $x$. 2. **Recall the rule for absolute value equations:** If $$|A| = B$$ where $B > 0$, then $$A = B$$ or $$A = -B$$. 3. **Apply the rule:** Set up two equations: $$4x - 5 = 7$$ and $$4x - 5 = -7$$ 4. **Solve the first equation:** $$4x - 5 = 7$$ Add $5$ to both sides: $$4x - 5 + 5 = 7 + 5$$ $$4x = 12$$ Divide both sides by $\cancel{4}$ to isolate $x$: $$\frac{4x}{\cancel{4}} = \frac{12}{\cancel{4}}$$ $$x = 3$$ 5. **Solve the second equation:** $$4x - 5 = -7$$ Add $5$ to both sides: $$4x - 5 + 5 = -7 + 5$$ $$4x = -2$$ Divide both sides by $\cancel{4}$: $$\frac{4x}{\cancel{4}} = \frac{-2}{\cancel{4}}$$ $$x = -0.5$$ 6. **Final answer:** The solutions to the equation are $$x = 3$$ and $$x = -0.5$$. 7. **Match with given options:** The correct choice is d.) $x = -0.5$ and $x = 3$.