Solve a linear programming problem with inequality constraints.

Question: Solve a linear programming problem with inequality constraints.

Answer: Linear programming is a method used to optimize a linear objective function subject to linear inequality constraints.

Linear programming is a mathematical technique used to find the best possible solution for a given problem. It is used to optimize a linear objective function subject to linear inequality constraints. The objective function is the function that needs to be maximized or minimized, while the inequality constraints are the conditions that need to be satisfied.

To solve a linear programming problem, we first need to identify the objective function and the constraints. Let's consider an example:

Maximize Z = 3x + 4y
Subject to:
2x + y ≤ 10
x + 2y ≤ 8
x, y ≥ 0

Here, the objective function is Z = 3x + 4y, which needs to be maximized. The inequality constraints are 2x + y ≤ 10 and x + 2y ≤ 8. The last constraint, x, y ≥ 0, is a non-negativity constraint.

To solve this problem, we can use the graphical method. We first plot the two inequality constraints on a graph and shade the feasible region, which is the region that satisfies both constraints.

The feasible region is the shaded area in the graph. We then find the corner points of the feasible region, which are the points where the two lines intersect.

The corner points are (0, 0), (0, 5), and (4, 2). We then substitute these corner points into the objective function to find the maximum value of Z.

Z(0, 0) = 3(0) + 4(0) = 0
Z(0, 5) = 3(0) + 4(5) = 20
Z(4, 2) = 3(4) + 4(2) = 16 + 8 = 24

Therefore, the maximum value of Z is 24, which occurs at the point (4, 2).