Question

Maximize Z = 3x₁ + 4x₂, if possible,
Subject to the constraints 

\[x_1 - x_2 \leq - 1\]

\[ - x_1 + x_2 \leq 0\]

\[ x_1 , x_2 \geq 0\]

Answer 1

First, we will convert the given inequations into equations, we obtain the following equations:
x₁ − x₂ = −1, −x₁ + x₂ = 0, x₁ = 0 and x₂ = 0

Region represented by x₁ − x₂ ≤ −1:
The line x₁ − x₂ = −1 meets the coordinate axes at A(−1, 0) and B(0, 1) respectively. By joining these points we obtain the line x₁ − x₂ = −1.
Clearly (0,0) does not satisfies the inequation x₁ − x₂ ≤ −1 .So,the region in the plane which does not contain the origin represents the solution set of the inequation x₁ − x₂ ≤ −1.

Region represented by −x₁ + x₂ ≤ 0 or x₁ ≥ x₂:
The line −x₁ + x₂ = 0 or x₁ = x₂ is the line passing through (0, 0).The region to the right of the line x₁ = x₂ will satisfy the given inequation −x₁ + x₂ ≤ 0.
If we take a point (1, 3) to the left of the line x₁ = x₂. Here, 1≤3 which is not satifying the inequation x₁ ≥ x_2. Therefore, region to the right of the line x₁ = x₂ will satisfy the given inequation −x₁ + x₂ ≤ 0.

Region represented by x₁ ≥ 0 and x₂ ≥ 0:
Since, every point in the first quadrant satisfies these inequations. So, the first quadrant is the region represented by the inequations x₁ ≥ 0 and x₂ ≥ 0.

The feasible region determined by the system of constraints, x₁ − x₂ ≤ −1, −x₁ + x₂ ≤ 0, x₁ ≥ 0, and x₂ ≥ 0, are as follows.

We observe that the feasible region of the given LPP does not exist.