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Negative RHS Transformation

Definition

The negative right-hand-side transformation multiplies an entire constraint by −1-1 when its right-hand side is negative. An inequality reverses direction; an equality remains an equality.

Interpretation

Multiplying a whole constraint by −1-1 rewrites it without changing which points satisfy it. The elementary simplex starting construction needs nonnegative right-hand sides, and this transformation provides them while keeping the same feasible decisions.

A shaded half-plane described both by -x1 + x2 at most -3 and, after multiplying by -1, by x1 - x2 at least 3, which has a nonnegative right-hand side.

The shaded half-plane is described by both forms: a point satisfies −x1+x2≤−3-x_1+x_2\le-3 exactly when it satisfies x1−x2≥3x_1-x_2\ge3; for example, (5,1)(5,1) gives −4≤−3-4\le-3 and 4≥34\ge3.

Multiplying both sides of an inequality by a negative number reverses its direction, so ≤\le becomes ≥\ge and every coefficient changes sign. Nonnegative right-hand sides matter because the starting solution sets the original variables to zero, so each slack starts at its right-hand side; with x1−x2+s=−1x_1-x_2+s=-1 this would give s=−1<0s=-1<0. After the transformation, −x1+x2≥1-x_1+x_2\ge1 receives a surplus and an artificial variable instead.

Changing only the sign of the right-hand side turns −x1+x2≤−3-x_1+x_2\le-3 into −x1+x2≤3-x_1+x_2\le3, a different half-plane: the origin satisfies the new constraint but not the original one.

Key Results

  • Change every coefficient and the right-hand side.
  • A ≤\le constraint can become a ≥\ge constraint and consequently require a surplus and an artificial variable.
  • Changing only the right-hand side would change the problem.

Examples

  • x1−x2≤−1x_1-x_2\le-1 becomes −x1+x2≥1-x_1+x_2\ge1.
  • −x1+x2=−3-x_1+x_2=-3 becomes x1−x2=3x_1-x_2=3.

Prerequisites

Sources

Last updated: 20 October 2026 03:02