To find the coordinates that define the solution region for the system of inequalities $x \leq 3$ and $x + y > 2$, we will:

1. Identify the boundary lines:

   • For $x \leq 3$: The boundary line is $x = 3$.
   • For $x + y > 2$: The boundary line is $y = -x + 2$.

2. Find the intersection point of the two lines:

   • The boundary line $x = 3$ intersects with the boundary line $y = -x + 2$ at a specific point.
   • To find this point, substitute $x = 3$ into the equation $y = -x + 2$: $y = -(3) + 2 = -3 + 2 = -1$
   • So, the intersection point is $(3, -1)$.

3. Determine the relevant region:

   • The line $x = 3$ defines a vertical line, and the solution requires $x$ values to be less than or equal to 3, meaning the region lies to the left of this line.
   • The inequality $y > -x + 2$ defines the region above the line $y = -x + 2$.

Coordinates Defining the Region:

The solution region is defined by:

• The vertical line $x = 3$, extending upward from $(3, -1)$.
• The line $y = -x + 2$, starting at $(2, 0)$ and moving leftward and upward.

Boundary Coordinates:

• The intersection point: $(3, -1)$.
• Where the line $y = -x + 2$ crosses the y-axis: $(0, 2)$.

Important Points:

• $(3, -1)$: Intersection of the two boundary lines.
• $(0, 2)$: Point where $y = -x + 2$ crosses the y-axis.
• Any point on the line $x = 3$ where $y > -1$.
• Any point on the line $y = -x + 2$ for $x < 3$.

Conclusion:

The region that satisfies both inequalities is bounded by the line $y = -x + 2$ on the lower side and extends up to and including the vertical line $x = 3$, starting from the point $(3, -1)$.

Would you like more details, or do you have any additional questions? Here are some related questions:

1. How do you determine which points are within the solution region?
2. What is the significance of the intersection point in the context of inequalities?
3. How would the solution region change if the inequality were strict (e.g., $x < 3$)?
4. Can the region be defined using inequalities in terms of $y$?
5. How do you check if a specific point lies within the solution region?

Tip: Always test points within and outside the solution region to ensure your understanding of how the inequalities shape the graph.