1. **State the problem:** Solve the equation $$\frac{7}{3x + 1} = \frac{2}{x + 2}$$. 2. **Use the cross-multiplication method:** When two fractions are equal, their cross products are equal. So, $$7(x + 2) = 2(3x + 1)$$ 3. **Expand both sides:** $$7x + 14 = 6x + 2$$ 4. **Isolate the variable $x$:** Subtract $6x$ from both sides: $$7x - \cancel{6x} + 14 = \cancel{6x} + 2 \Rightarrow x + 14 = 2$$ 5. **Subtract 14 from both sides:** $$x + \cancel{14} = 2 - \cancel{14} \Rightarrow x = -12$$ 6. **Check for restrictions:** The denominators cannot be zero. - For $3x + 1 \neq 0$, $x \neq -\frac{1}{3}$ - For $x + 2 \neq 0$, $x \neq -2$ Since $x = -12$ does not violate these restrictions, it is a valid solution. **Final answer:** $$x = -12$$