1. **State the problem:** We are given a line segment EC with length expressed in two ways: as a sum of lengths $23 + 40$ and as an algebraic expression $2x + 10 + 2x + 11$. We need to find the value of $x$. 2. **Write the equation:** Since both expressions represent the same length, set them equal: $$23 + 40 = 2x + 10 + 2x + 11$$ 3. **Simplify both sides:** $$63 = 4x + 21$$ 4. **Isolate the variable $x$:** Subtract 21 from both sides: $$63 - 21 = 4x + \cancel{21} - \cancel{21}$$ $$42 = 4x$$ 5. **Solve for $x$ by dividing both sides by 4:** $$\frac{42}{\cancel{4}} = \frac{4x}{\cancel{4}}$$ $$x = \frac{42}{4} = 10.5$$ 6. **Final answer:** $$\boxed{10.5}$$ This means the value of $x$ that makes the segment lengths equal is 10.5.