The image you uploaded contains a mathematical problem involving a system of linear inequalities and a shaded region that represents the solution set. Here’s a breakdown of the tasks associated with the problem:

Problem Statement:

1. Given System of Inequalities: &3x + 2y \leq 12, \\ &2x + 5y \leq 20, \\ &y \geq 1. \end{aligned}$$
2. Graph Representation:
   • The inequalities are represented on the graph with the lines corresponding to each equation, and the shaded region represents the solution set.

Specific Tasks:

a. Find the coordinates of the vertices of the region.

b. Determine the values of $x$ and $y$ for which the solution set is bounded by $y = 1$.

c. Find the points where the solution satisfies $x = 4$.

d. Determine the maximum area of the complete bounded region.

Solution Steps:

a. Find the Coordinates of the Vertices:

• Solve the system of linear equations where the lines intersect to find the vertices.

b. Determine the Values for $y = 1$:

• This involves finding the intersection points of the line $y = 1$ with the boundary lines of the region.

c. Find Points Where $x = 4$:

• Substitute $x = 4$ into the system to determine where it intersects with the region.

d. Maximum Area of the Region:

• Calculate the area of the polygon formed by the vertices using the formula for the area of a polygon.

Would you like to go through the detailed solution for each part?