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In the world of linear programming, the feasible region is like the playground where all the solutions live. It represents all the possible solutions that satisfy the given inequality constraints. To visualize this, we usually plot the constraints on a coordinate plane. Here, the inequalities are turned into boundaries that form a region which we call the feasible region.

When we plot inequalities, we often shade the area where all these conditions are true. The feasible region lies where all shaded areas overlap. This is a crucial step, as this region essentially contains all the points that could potentially solve our optimization problem.

For our given exercise, the boundary is formed by the constraints \(3s + t \geq 30\), \(s + t \geq 20\), and \(s + 3t \geq 30\), alongside the non-negativity constraints \(s \geq 0\) and \(t \geq 0\). After plotting these, you will notice a certain area where all conditions meet. This is the feasible region. Every point inside this region is a valid solution to the linear programming problem.

The objective function in a linear programming problem is what you are trying to optimize. This could either be maximizing or minimizing a particular value. In this exercise, we are trying to minimize our objective function, given by \(c = 2s + t\).

The essence of solving such a problem lies in evaluating this function within the feasible region. To find the best possible value, we calculate the function's value at each vertex of the feasible region. Why the vertices? Because of the linear nature of our problem, the minimum or maximum value will always lie at the edges of our feasible region, typically at a corner (vertex).

• Evaluate the function at each vertex.
• Compare to find the minimum or maximum value.

For our problem, the objective function is evaluated at the points (10,10), (9,7), and (6,14). By comparing these values, we identify the vertex that gives us the minimum value. Thus, knowing and understanding the objective function is key to solving linear programming problems.

Inequality constraints define the limits or the boundaries of your feasible region. They are the conditions that your solutions must satisfy. In linear programming, these constraints are usually linear inequalities. Each of these inequalities represents a half-space on the coordinate plane.

By plotting each inequality as an equation, you get a line that serves as the boundary. For example, in our exercise, the constraint \(3s + t \geq 30\) becomes the boundary line \(3s + t = 30\). Similarly, the other constraints do the same.

• Each inequality splits the plane into two halves.
• The feasible region is where all these halves intersect.

This ensures that all the conditions are met simultaneously. Without these constraints, the problem would not have a limited number of solutions. In fact, it might not even have any solutions! That's why inequality constraints play a fundamental role in defining the problem's scope and ensuring feasible solutions can be found within the specified conditions.