The task is to minimize the function $-4x - 5y$ over the bounded feasible region shown in the graph. Let’s solve this step-by-step:

Step 1: Identify the vertices of the feasible region

The graph represents a polygon-shaped region where constraints are met. From the graph:

• The vertices of the feasible region are points where the constraint lines intersect.
• The coordinates of these vertices are essential because the minimum or maximum of a linear function occurs at one of these points.

From the graph, the vertices seem to be approximately:

1. $(1, 2)$
2. $(4, 6)$
3. $(6, 0)$
4. $(9, 3)$

Step 2: Evaluate the function $-4x - 5y$ at each vertex

Substitute each vertex into the function $-4x - 5y$:

1. At $(1, 2)$:
   $-4(1) - 5(2) = -4 - 10 = -14$

2. At $(4, 6)$:
   $-4(4) - 5(6) = -16 - 30 = -46$

3. At $(6, 0)$:
   $-4(6) - 5(0) = -24 - 0 = -24$

4. At $(9, 3)$:
   $-4(9) - 5(3) = -36 - 15 = -51$

Step 3: Determine the minimum value

The function values at the vertices are:

• At $(1, 2)$: $-14$
• At $(4, 6)$: $-46$
• At $(6, 0)$: $-24$
• At $(9, 3)$: $-51$

The minimum value is $-51$, occurring at $(9, 3)$.

Final Answer:

The minimum value of $-4x - 5y$ is $-51$, achieved at the point $(9, 3)$.

Do you want me to explain any of the steps in more detail?

Here are 5 related questions for practice:

1. How do you determine the feasible region from constraints?
2. Why do the minima and maxima of linear functions occur at vertices of the feasible region?
3. Can you formulate a linear programming problem with similar constraints and solve it?
4. What happens if the feasible region is unbounded?
5. How would you solve this problem if the objective function were $4x + 5y$?

Tip: Always check the feasibility of each vertex by verifying that it satisfies all constraints before evaluating the objective function.