Final Answer:
The given linear program is inconsistent; there is no solution that simultaneously satisfies both the objective function (3x₁+4x₂+3x₃+6x₄) and the constraint (2x₁+x₂-x₃+x₄ ≥ 12).
Step-by-step explanation:
The inconsistency arises from the constraint 2x₁+x₂-x₃+x₄ ≥ 12, which implies a lower bound on the combination of variables x₁, x₂, x₃, and x₄. However, the objective function 3x₁+4x₂+3x₃+6x₄ is seeking to maximize a linear combination of these variables. If the constraint is satisfied at its minimum (when each variable is at its lowest value to satisfy the inequality), the objective function cannot achieve a value greater than or equal to 12, resulting in an inconsistency.
To further illustrate, let's consider the extreme case where x₁, x₂, x₃, and x₄ are all set to their lowest possible values to satisfy the constraint. In this scenario, the objective function evaluates to a value less than 12, making it impossible to find a solution that simultaneously satisfies the given constraint and maximizes the objective function.
In summary, the linear program is inconsistent because the constraint imposes a lower bound that prevents the objective function from reaching its maximum value. This misalignment renders the problem unsolvable, and no combination of variables exists that satisfies both the constraint and the objective function simultaneously.