1. **State the problem:** Solve for $x$ in the equation $$\frac{7}{3}x - 5 = -\frac{7}{4}x - \frac{1}{2}.$$\n\n2. **Write down the equation:** $$\frac{7}{3}x - 5 = -\frac{7}{4}x - \frac{1}{2}.$$\n\n3. **Add $\frac{7}{4}x$ to both sides to get all $x$ terms on one side:**\n$$\frac{7}{3}x + \frac{7}{4}x - 5 = -\frac{1}{2}.$$\n\n4. **Find common denominator for $x$ terms:** The denominators are 3 and 4, so common denominator is 12.\n$$\frac{28}{12}x + \frac{21}{12}x - 5 = -\frac{1}{2}.$$\n\n5. **Combine $x$ terms:**\n$$\left(\frac{28}{12} + \frac{21}{12}\right)x - 5 = -\frac{1}{2}$$\n$$\frac{49}{12}x - 5 = -\frac{1}{2}.$$\n\n6. **Add 5 to both sides:**\n$$\frac{49}{12}x = -\frac{1}{2} + 5.$$\n\n7. **Convert 5 to fraction with denominator 2:**\n$$5 = \frac{10}{2}.$$\n\n8. **Add fractions on right side:**\n$$-\frac{1}{2} + \frac{10}{2} = \frac{9}{2}.$$\n\n9. **Equation is now:**\n$$\frac{49}{12}x = \frac{9}{2}.$$\n\n10. **Solve for $x$ by dividing both sides by $\frac{49}{12}$:**\n$$x = \frac{\frac{9}{2}}{\frac{49}{12}} = \frac{9}{2} \times \frac{12}{49}.$$\n\n11. **Multiply numerators and denominators:**\n$$x = \frac{9 \times 12}{2 \times 49} = \frac{108}{98}.$$\n\n12. **Simplify the fraction $\frac{108}{98}$ by dividing numerator and denominator by 2:**\n$$x = \frac{\cancel{108}54}{\cancel{98}49} = \frac{54}{49}.$$\n\n**Final answer:** $$x = \frac{54}{49}.$$