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

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

UpStudy AI · Accepted Answer

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:

![Graph of the inequalities](https://i.imgur.com/zZxkBtK.png)

- 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.
The point $$ (4, 2) $$ satisfies both inequalities $$ x + y \geq 6 $$ and $$ x \geq 4 $$.

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

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