1. **State the problem:** We are given two similar triangles $\triangle JKL \sim \triangle MKN$ and need to find the value of $x$. The sides given are $KN = 12$, $NL = 20$, $JK = 3x - 2$, and $MK = x + 1$. 2. **Identify corresponding sides:** Since the triangles are similar, corresponding sides are proportional. The vertices order suggests $J \leftrightarrow M$, $K \leftrightarrow K$, and $L \leftrightarrow N$. 3. **Set up the proportion:** Corresponding sides are $JK \leftrightarrow MK$ and $KL \leftrightarrow KN$. We have: $$\frac{JK}{MK} = \frac{KL}{KN}$$ 4. **Substitute known values:** We know $JK = 3x - 2$, $MK = x + 1$, $KN = 12$, and $NL = 20$. Since $KL$ corresponds to $KN$, and $NL$ is given but not directly used here, we assume $KL = NL = 20$ (assuming a typo or that $NL$ is $KL$). So: $$\frac{3x - 2}{x + 1} = \frac{20}{12}$$ 5. **Simplify the right side:** $$\frac{20}{12} = \frac{5}{3}$$ 6. **Cross multiply:** $$3(3x - 2) = 5(x + 1)$$ 7. **Expand both sides:** $$9x - 6 = 5x + 5$$ 8. **Bring like terms together:** $$9x - 5x = 5 + 6$$ 9. **Simplify:** $$4x = 11$$ 10. **Solve for $x$:** $$x = \frac{11}{4}$$ **Final answer:** $x = \frac{11}{4} = 2.75$