1. **State the problem:** Solve for $x$ in the equation $$\frac{x + 2}{7} = \frac{2x - 4}{6}.$$\n\n2. **Formula and rules:** To solve equations with fractions, cross-multiply to eliminate denominators: $$a/b = c/d \implies ad = bc.$$\n\n3. **Apply cross-multiplication:** Multiply both sides by $7 \times 6$ to get rid of denominators:\n$$6(x + 2) = 7(2x - 4).$$\n\n4. **Expand both sides:**\n$$6x + 12 = 14x - 28.$$\n\n5. **Isolate $x$ terms on one side:** Subtract $6x$ from both sides:\n$$12 = 14x - 6x - 28,$$\nwhich simplifies to\n$$12 = 8x - 28.$$\n\n6. **Isolate constants on the other side:** Add $28$ to both sides:\n$$12 + 28 = 8x,$$\nso\n$$40 = 8x.$$\n\n7. **Solve for $x$:** Divide both sides by $8$:\n$$x = \frac{40}{8} = 5.$$\n\n**Final answer:** $$x = 5.$$