1. **State the problem:** Solve the equation $$\frac{4}{x} + 1 = \frac{5}{3}$$ for $x$. 2. **Isolate the fraction:** Subtract 1 from both sides: $$\frac{4}{x} = \frac{5}{3} - 1$$ 3. **Simplify the right side:** $$\frac{5}{3} - 1 = \frac{5}{3} - \frac{3}{3} = \frac{2}{3}$$ 4. **Rewrite the equation:** $$\frac{4}{x} = \frac{2}{3}$$ 5. **Cross-multiply to solve for $x$:** $$4 \times 3 = 2 \times x$$ $$12 = 2x$$ 6. **Divide both sides by 2:** $$\frac{\cancel{12}}{\cancel{2}} = \frac{2x}{2}$$ $$6 = x$$ **Final answer:** $$x = 6$$