1. **State the problem:** We are given two angles at point B formed by line segments BA, BC, BD, and BE. The angle between BA and BC is $2x+7$ degrees, and the angle between BD and BE is $4x-14$ degrees. Since these two angles are adjacent and form a straight line at B, their sum is $180^\circ$. 2. **Write the equation:** $$2x + 7 + 4x - 14 = 180$$ 3. **Simplify the equation:** $$2x + 4x + 7 - 14 = 180$$ $$6x - 7 = 180$$ 4. **Isolate $x$:** $$6x = 180 + 7$$ $$6x = 187$$ 5. **Divide both sides by 6:** $$x = \frac{187}{6}$$ 6. **Final answer:** $$x = 31.1667$$ (approximately) Thus, the value of $x$ is approximately 31.17 degrees.