1. **State the problem:** We are given that segment QN bisects angle PQS, meaning it divides angle PQS into two equal angles. 2. **Given:** - $\angle PQN = (12x + 4)^\circ$ - $\angle NQS = (18x - 14)^\circ$ - Since QN bisects angle PQS, these two angles are equal. 3. **Set up the equation:** $$12x + 4 = 18x - 14$$ 4. **Solve for $x$:** Subtract $12x$ from both sides: $$\cancel{12x} + 4 = 18x - 14 - \cancel{12x}$$ $$4 = 6x - 14$$ Add 14 to both sides: $$4 + 14 = 6x - 14 + 14$$ $$18 = 6x$$ Divide both sides by 6: $$\frac{18}{\cancel{6}} = \frac{6x}{\cancel{6}}$$ $$3 = x$$ 5. **Final answer:** $$\boxed{3}$$ Thus, the value of $x$ is 3.