1. **State the problem:** Solve the equation $x + 3 = 2 \left( \frac{1}{3} x - 1 \right) + 4^3$. 2. **Recall the order of operations and distributive property:** First, calculate powers, then distribute multiplication over addition/subtraction, and finally solve for $x$. 3. **Calculate $4^3$:** $$4^3 = 4 \times 4 \times 4 = 64$$ 4. **Rewrite the equation with this value:** $$x + 3 = 2 \left( \frac{1}{3} x - 1 \right) + 64$$ 5. **Distribute the 2 on the right side:** $$2 \times \frac{1}{3} x = \frac{2}{3} x$$ $$2 \times (-1) = -2$$ So, $$x + 3 = \frac{2}{3} x - 2 + 64$$ 6. **Simplify the right side:** $$-2 + 64 = 62$$ So, $$x + 3 = \frac{2}{3} x + 62$$ 7. **Isolate $x$ terms on one side and constants on the other:** Subtract $\frac{2}{3} x$ from both sides: $$x - \frac{2}{3} x + 3 = 62$$ 8. **Combine like terms:** $$x - \frac{2}{3} x = \frac{3}{3} x - \frac{2}{3} x = \frac{1}{3} x$$ So, $$\frac{1}{3} x + 3 = 62$$ 9. **Subtract 3 from both sides:** $$\frac{1}{3} x = 62 - 3 = 59$$ 10. **Multiply both sides by 3 to solve for $x$:** $$x = 59 \times 3 = 177$$ **Final answer:** $$x = 177$$