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Question

Constraints

1) A constraint of the type ≤ (≥) can be converted to an equation by adding a slack variable to (subtracting a surplus variable form) the left side of the constraint.

For illustration, in the constraint: X_I + 2X₂ ≤ 6 we add a slack S_I ≥ 0 to the left side to obtain the equation: X_I + 2X₂ + S_I = 6,     S_I ≥ 0.

If the constraint represents the limit on the usage of a resource, S_I will represent the slack or unused amount of the resource.

After that consider the constraint: 3X_I + 2X₂ - 3X₃ ≥ 5, we subtract a surplus variable S₂ ≥ 0 from the left side to obtain the equation.
3X_I + 2X₂ - 3X₃ - S₂ = 5, S₂ ≥ 0

2) The right side of an equation can always be made non-negative by multiplying both sides by -1.

For illustration, 2X_I + 3X₂ - 7X₃ = -5 is mathematically equivalent to -2X_I - 3X₂ + 7X₃ = +5.

3) The direction of an equation is reversed when both sides are multiplied by -1. For illustration, whereas 2 < 4, -2 > -4. Thus the inequality 2X_I - X₂ ≤-5 can be replaced by -2X_I + X₂ ≥ 5.