Unit Pretest Select the correct answer. A carpenter is building a rectangular table. He wants the perimeter of the tabletop to be no more than 28 feet. He also wants the lel be greater than or equal to the square of 2 feet less than its width. Create a system of inequalities to model the situation, where \( x \) represents the width of the tabletop and \( y \) represents the length of tl this system of inequalities to determine the viable solutions. A. Part of the solution region includes a negative length; therefore, not all solutions are viable for the given situation. B. The entire solution region is viable. C. Part of the solution region includes a negative width; therefore, not all solutions are viable for the given situation. D. No part of the solution reaion is viable because the lenath and width cannot be neaative.

To model the carpenter's situation, let's define the variables and set up the inequalities based on the given conditions: - **Variables:** - \( x \): Width of the tabletop (in feet) - \( y \): Length of the tabletop (in feet) - **Conditions:** 1. **Perimeter Constraint:** The perimeter of a rectangle is given by \( 2x + 2y \). The carpenter wants the perimeter to be no more than 28 feet: \[ 2x + 2y \leq 28 \] Simplifying: \[ x + y \leq 14 \] 2. **Length Constraint:** The carpenter wants the length \( y \) to be at least the square of 2 feet less than its width: \[ y \geq (x - 2)^2 \] - **System of Inequalities:** \[ \begin{cases} x + y \leq 14 \\ y \geq (x - 2)^2 \end{cases} \] - **Analyzing the Solution Region:** - **Graphical Interpretation:** - The inequality \( x + y \leq 14 \) represents a region below the line \( y = 14 - x \). - The inequality \( y \geq (x - 2)^2 \) represents a region above the parabola \( y = (x - 2)^2 \). - **Intersection Points:** Solving \( (x - 2)^2 = 14 - x \) gives \( x = 5 \) and \( x = -2 \). However, \( x = -2 \) (a negative width) is not physically meaningful for the tabletop. - **Viable Solutions:** While there are points that satisfy both inequalities, some of these points involve negative values for \( x \), which would imply a negative width. Since negative dimensions are not feasible in this context, not all solutions within the mathematical system are viable for the actual situation. **Conclusion:** **Option C** accurately describes the scenario: **"Part of the solution region includes a negative width; therefore, not all solutions are viable for the given situation."** **Answer:** **C. Part of the solution region includes a negative width; therefore, not all solutions are viable for the given situation.**