1. We start with the problem: Solve the equation $$\frac{1}{2}x - 5 = \frac{3}{4}x + 2$$. 2. The goal is to isolate $x$ on one side. First, subtract $\frac{3}{4}x$ from both sides: $$\frac{1}{2}x - \frac{3}{4}x - 5 = 2$$ 3. Simplify the left side by finding a common denominator for the $x$ terms: $$\frac{2}{4}x - \frac{3}{4}x - 5 = 2$$ 4. Combine like terms: $$\left(\frac{2}{4} - \frac{3}{4}\right)x - 5 = 2$$ $$-\frac{1}{4}x - 5 = 2$$ 5. Add 5 to both sides to isolate the term with $x$: $$-\frac{1}{4}x = 2 + 5$$ $$-\frac{1}{4}x = 7$$ 6. Multiply both sides by $-4$ to solve for $x$: $$\cancel{-\frac{1}{4}}x \times \cancel{-4} = 7 \times (-4)$$ $$x = -28$$ Final answer: $$x = -28$$