1. **State the problem:** Solve the equation $$\frac{4x - 2}{6} = \frac{3}{2x + 7}$$ for $x$. 2. **Use the cross-multiplication rule:** When two fractions are equal, their cross products are equal: $$ (4x - 2)(2x + 7) = 6 \times 3 $$ 3. **Expand the left side:** $$ (4x)(2x) + (4x)(7) - 2(2x) - 2(7) = 18 $$ $$ 8x^2 + 28x - 4x - 14 = 18 $$ 4. **Simplify the left side:** $$ 8x^2 + 24x - 14 = 18 $$ 5. **Bring all terms to one side to set the equation to zero:** $$ 8x^2 + 24x - 14 - 18 = 0 $$ $$ 8x^2 + 24x - 32 = 0 $$ 6. **Divide the entire equation by 8 to simplify:** $$ \frac{\cancel{8}x^2}{\cancel{8}} + \frac{24x}{8} - \frac{32}{8} = 0 $$ $$ x^2 + 3x - 4 = 0 $$ 7. **Factor the quadratic:** $$ (x + 4)(x - 1) = 0 $$ 8. **Set each factor equal to zero and solve for $x$:** $$ x + 4 = 0 \Rightarrow x = -4 $$ $$ x - 1 = 0 \Rightarrow x = 1 $$ 9. **Check for restrictions:** The original denominators are $6$ and $2x + 7$. The denominator $2x + 7$ cannot be zero: $$ 2x + 7 \neq 0 \Rightarrow x \neq -\frac{7}{2} $$ Neither $-4$ nor $1$ equals $-\frac{7}{2}$, so both solutions are valid. **Final answer:** $$x = -4 \text{ or } x = 1$$