Let's solve the inequalities step by step and understand how to mark the points that satisfy all three conditions.

Given inequalities:

1. $y \geq 2$
2. $x + y \leq 4$
3. $y \leq 2x + 2$

Step-by-step approach:

1. $y \geq 2$:

   • This inequality states that the $y$-coordinate of any solution must be greater than or equal to 2. On the graph, this is the region above or on the horizontal line $y = 2$.

2. $x + y \leq 4$:

   • Rearranging this gives $y \leq 4 - x$.
   • This is the region below or on the line $y = 4 - x$, which slopes downward from $(0, 4)$ to $(4, 0)$.

3. $y \leq 2x + 2$:

   • This represents the region below or on the line $y = 2x + 2$, which has a steeper slope. It starts at $(0, 2)$ and rises with a slope of 2.

Plotting:

• From the graph in the image, we see three red lines:

  • Horizontal line $y = 2$.
  • Line $x + y = 4$ or $y = 4 - x$.
  • Line $y = 2x + 2$.

  We are interested in the region where all three inequalities overlap, considering $x$ and $y$ must be integers.

Intersection of Regions:

• You need to identify where the shaded regions corresponding to the inequalities intersect.
  • The region above $y = 2$ but below both $y = 4 - x$ and $y = 2x + 2$.

Integer points:

From the graph, the points of intersection that are integers and satisfy all three inequalities are:

• $(0, 2)$
• $(1, 2)$
• $(0, 3)$

These are the points that should be marked on the graph.

Would you like more details or have any questions? Here are some related questions for deeper understanding:

1. How do you verify if a point satisfies multiple inequalities?
2. What happens if the inequalities were strict, such as $y > 2$?
3. How can you graphically find the intersection of multiple inequalities?
4. What role do the integer constraints play in solving inequalities?
5. How would the solution change if one inequality had a different slope?

Tip: When solving graphically, always consider the region where all conditions overlap, and mark integer solutions accordingly.