1. **State the problem:** Solve for $x$ in the equation $$\frac{x + 4}{5} + \frac{2x - 1}{3} = 1.$$\n\n2. **Formula and rules:** To solve equations with fractions, find a common denominator to combine terms or clear fractions by multiplying both sides by the least common denominator (LCD). Here, the denominators are 5 and 3, so the LCD is 15.\n\n3. **Multiply both sides by 15 to clear fractions:**\n$$15 \times \left(\frac{x + 4}{5} + \frac{2x - 1}{3}\right) = 15 \times 1$$\n\n4. **Distribute multiplication:**\n$$15 \times \frac{x + 4}{5} + 15 \times \frac{2x - 1}{3} = 15$$\n\n5. **Simplify each term:**\n$$\cancel{15} \times \frac{x + 4}{\cancel{5}} + \cancel{15} \times \frac{2x - 1}{\cancel{3}} = 15$$\n$$3(x + 4) + 5(2x - 1) = 15$$\n\n6. **Expand the parentheses:**\n$$3x + 12 + 10x - 5 = 15$$\n\n7. **Combine like terms:**\n$$13x + 7 = 15$$\n\n8. **Isolate $x$ by subtracting 7 from both sides:**\n$$13x + 7 - 7 = 15 - 7$$\n$$13x = 8$$\n\n9. **Divide both sides by 13 to solve for $x$:**\n$$\frac{13x}{\cancel{13}} = \frac{8}{\cancel{13}}$$\n$$x = \frac{8}{13}$$\n\n**Final answer:** $x = \frac{8}{13}$