1. **Problem Statement:** Given points A, B, and C are collinear with B between A and C, and segment lengths AB = $2x - 5$, BC = $3x + 6$, and total length AC = 46 units, find the value of $x$ and the lengths of AB and BC. 2. **Formula Used:** Segment Addition Postulate states that if B is between A and C, then: $$AB + BC = AC$$ 3. **Set up the equation:** $$2x - 5 + 3x + 6 = 46$$ 4. **Simplify the left side:** $$5x + 1 = 46$$ 5. **Isolate $x$:** $$5x = 46 - 1$$ $$5x = 45$$ 6. **Solve for $x$:** $$x = \frac{45}{5}$$ $$x = 9$$ 7. **Find AB:** $$AB = 2(9) - 5 = 18 - 5 = 13$$ 8. **Find BC:** $$BC = 3(9) + 6 = 27 + 6 = 33$$ **Final answer:** $x = 9$, $AB = 13$, and $BC = 33$ units.