The solution region shown in the image is a polygon formed by the intersections of the lines:

1. $4x + y = 6$ (green line)
2. $x + 2y = 5$ (red line)
3. $2x + 6y = 12$ (blue line)

To find the vertices (corners) of the solution region, we compute the intersections of these lines.

Step 1: Solve for intersections

Intersection of $4x + y = 6$ and $x + 2y = 5$:

Solve the system: $4x + y = 6 \quad \text{(1)}$ $x + 2y = 5 \quad \text{(2)}$ From equation (2): $y = \frac{5 - x}{2}$.

Substitute into equation (1): $4x + \frac{5 - x}{2} = 6$ $8x + 5 - x = 12 \quad (\text{multiply by 2 to clear the fraction})$ $7x = 7 \quad \Rightarrow \quad x = 1$ Substitute $x = 1$ into $y = \frac{5 - x}{2}$: $y = \frac{5 - 1}{2} = 2$ Intersection: $(1, 2)$.

Intersection of $4x + y = 6$ and $2x + 6y = 12$:

Solve the system: $4x + y = 6 \quad \text{(3)}$ $2x + 6y = 12 \quad \text{(4)}$ From equation (3): $y = 6 - 4x$.

Substitute into equation (4): $2x + 6(6 - 4x) = 12$ $2x + 36 - 24x = 12$ $-22x = -24 \quad \Rightarrow \quad x = \frac{12}{11}$ Substitute $x = \frac{12}{11}$ into $y = 6 - 4x$: $y = 6 - 4\left(\frac{12}{11}\right) = 6 - \frac{48}{11} = \frac{66}{11} - \frac{48}{11} = \frac{18}{11}$ Intersection: $\left(\frac{12}{11}, \frac{18}{11}\right)$.

Intersection of $x + 2y = 5$ and $2x + 6y = 12$:

Solve the system: $x + 2y = 5 \quad \text{(5)}$ $2x + 6y = 12 \quad \text{(6)}$ From equation (5): $x = 5 - 2y$.

Substitute into equation (6): $2(5 - 2y) + 6y = 12$ $10 - 4y + 6y = 12$ $2y = 2 \quad \Rightarrow \quad y = 1$ Substitute $y = 1$ into $x = 5 - 2y$: $x = 5 - 2(1) = 3$ Intersection: $(3, 1)$.

Step 2: List of vertices

The vertices (corners) of the solution region are:

1. $(1, 2)$
2. $\left(\frac{12}{11}, \frac{18}{11}\right)$
3. $(3, 1)$

Would you like further clarifications or visual confirmation of the solution? Here are some additional questions to consider:

1. How can these points be verified geometrically?
2. Would you like a step-by-step explanation of solving one of these intersections?
3. Do you need the inequalities that define the shaded region?
4. Should I plot these points and the polygon of the feasible region?
5. Can I assist in solving for a specific objective function within this region?

Tip: Always double-check intersection points by substituting back into the original equations.