1. **State the problem:** We are given a right triangle with legs 24 and 18. A segment inside the triangle divides the right vertical side (length 18) into two parts labeled $2x+1$ and $3x-1$. We need to find the value of $x$. 2. **Understand the problem:** The right vertical side is split into two segments, so their sum must equal the total length of that side. 3. **Write the equation:** $$2x + 1 + 3x - 1 = 18$$ 4. **Simplify the equation:** $$2x + 3x + 1 - 1 = 18$$ $$5x = 18$$ 5. **Solve for $x$:** $$x = \frac{18}{5}$$ 6. **Check the solution:** Calculate each segment: $$2x + 1 = 2 \times \frac{18}{5} + 1 = \frac{36}{5} + 1 = \frac{36}{5} + \frac{5}{5} = \frac{41}{5} = 8.2$$ $$3x - 1 = 3 \times \frac{18}{5} - 1 = \frac{54}{5} - 1 = \frac{54}{5} - \frac{5}{5} = \frac{49}{5} = 9.8$$ Sum: $$8.2 + 9.8 = 18$$ This matches the total length, so $x = \frac{18}{5}$ is correct. **Final answer:** $$x = \frac{18}{5}$$