- Given the relationship \(CE = CD + DE\).
- Substitute the expressions \(CD = 4x - 3\) and \(DE = -x + 8\).
- Combine like terms to get \(CE = 4x - x - 3 + 8\).
- Simplify to obtain the final expression \(\boxed{3x + 5}\), which corresponds to option D.
Explanation
- Problem Analysis
We are given the segment relationship \(CD + DE = CE\), where the measures of segments \(CD\) and \(DE\) are defined in terms of a variable \(x\). Specifically, \(CD = 4x - 3\) and \(DE = -x + 8\). Our goal is to find a simplified expression for the total length \(CE\) by combining these expressions.
- Simplifying the Expression
To find \(CE\), we substitute the given expressions for \(CD\) and \(DE\) into the relationship \(CE = CD + DE\). This gives us:
\[CE = (4x - 3) + (-x + 8)\]
Next, we remove the parentheses and group the like terms (the terms containing \(x\) and the constant terms):
\[CE = 4x - x - 3 + 8\]
Grouping the terms, we get:
\[CE = (4x - x) + (-3 + 8)\]
Simplifying each group:
\[CE = 3x + 5\]
- Conclusion
Comparing our result, \(CE = 3x + 5\), with the given choices:
A \(5x + 5\)
B \(5x + 11\)
C \(3x - 11\)
D \(3x + 5\)
We can see that our result matches option D. Therefore, the simplified expression for \(CE\) is \(3x + 5\). The final answer is \(\boxed{3x + 5}\).
Examples
In geometry and construction, segments often represent physical distances that need to be combined to find total lengths. For instance, if you are measuring a custom shelf layout along a wall where one section is \(4x - 3\) centimeters and the adjoining section is \(-x + 8\) centimeters, the total length of the shelf is the sum of these segments, calculated as \(CE = (4x - 3) + (-x + 8) = 3x + 5\) centimeters. This linear model is essential in fields like architecture and furniture design for planning and material estimation.