\( \left. \begin{array} { l } { x + y \geq 6 } \\ { x \geq 4 } \\ { ( 4 , - 2 ) } \\ { ( - 1,7 ) } \\ { ( 4,2 ) } \end{array} \right. \)

Certainly! Let's analyze the given system of inequalities and determine which of the provided points satisfy all conditions. ## Given System of Inequalities \[ \begin{cases} x + y \geq 6 \\ x \geq 4 \end{cases} \] ### Understanding the Inequalities 1. **\( x + y \geq 6 \)**: - This inequality represents all the points on or above the line \( x + y = 6 \) in the Cartesian plane. - **Boundary Line**: \( x + y = 6 \) - **Shading**: Above the line since it's \( \geq \). 2. **\( x \geq 4 \)**: - This inequality represents all the points to the right of (and including) the vertical line \( x = 4 \). - **Boundary Line**: \( x = 4 \) - **Shading**: To the right of the line since it's \( \geq \). ### Graphical Representation Below is a graphical representation of the system:  - The **red line** represents \( x + y = 6 \), with the region above shaded. - The **blue vertical line** represents \( x = 4 \), with the region to the right shaded. - The **intersection** of these shaded regions is the solution set where both inequalities are satisfied. ### Evaluating the Given Points Let's examine each point to see if it satisfies both inequalities. 1. **Point \( (4, -2) \)** - **Check \( x \geq 4 \)**: - \( x = 4 \) ✔️ - **Check \( x + y \geq 6 \)**: - \( 4 + (-2) = 2 \) - \( 2 \geq 6 \) ❌ **Result**: Does **not** satisfy both inequalities. 2. **Point \( (-1, 7) \)** - **Check \( x \geq 4 \)**: - \( x = -1 \) - \( -1 \geq 4 \) ❌ - (No need to check the second inequality since the first one fails.) **Result**: Does **not** satisfy both inequalities. 3. **Point \( (4, 2) \)** - **Check \( x \geq 4 \)**: - \( x = 4 \) ✔️ - **Check \( x + y \geq 6 \)**: - \( 4 + 2 = 6 \) - \( 6 \geq 6 \) ✔️ **Result**: **Satisfies** both inequalities. ### Summary - **\( (4, -2) \)**: Does **not** satisfy the system. - **\( (-1, 7) \)**: Does **not** satisfy the system. - **\( (4, 2) \)**: **Satisfies** the system. Only the point **\( (4, 2) \)** lies within the solution region defined by the given inequalities.