25.3 Multiplicative Inequality Transformations

Multiplicative Inequality Transformations are the operations of multiplying or dividing both sides of an inequality by a fixed quantity, extending the multiplication and division properties of equality to the setting of an inequality, but requiring a critical distinction based on the sign of that quantity that has no counterpart in equation solving.

Multiplication by a Positive Quantity is the action of multiplying both sides of an inequality by a value known to be positive, producing a new inequality that retains the same comparison symbol as the original, shown in the general principle below.

where c is positive. Multiplying both quantities by the same positive scale factor preserves their relative order, since scaling by a positive number never reverses which quantity is larger.

Division by a Positive Quantity is the corresponding action of dividing both sides of an inequality by a positive value, likewise producing a new inequality that retains the same comparison symbol as the original, for the identical reason that division by a positive number preserves relative order.

Positive Scaling without Relation Reversal is the summary property shared by Multiplication by a Positive Quantity and Division by a Positive Quantity: whenever the quantity used to multiply or divide both sides is confirmed to be positive, the direction of the inequality symbol remains completely unchanged, behaving exactly as the multiplication and division properties of equality do for ordinary equations.

Multiplication by a Negative Quantity is the action of multiplying both sides of an inequality by a value known to be negative, which produces a new inequality with the comparison symbol reversed relative to the original, shown in the general principle below.

where c is negative. Multiplying by a negative number flips the sign of each quantity, which inverts their relative order on the number line, so the quantity that was originally larger becomes, after scaling by a negative factor, the smaller of the two resulting values.

Division by a Negative Quantity is the corresponding action of dividing both sides of an inequality by a negative value, which likewise reverses the direction of the comparison symbol, for the same underlying reason that dividing by a negative number inverts relative order.

Relation Reversal after Negative Scaling is the critical rule uniting Multiplication by a Negative Quantity and Division by a Negative Quantity: whenever the quantity used to multiply or divide both sides is confirmed to be negative, the inequality symbol must be flipped, meaning a greater-than symbol becomes a less-than symbol and a less-than symbol becomes a greater-than symbol, with the corresponding inclusive symbols reversing in the same manner. Overlooking this reversal is among the most consequential errors possible in inequality solving, since it produces a solution set describing the wrong side of the boundary entirely.

Zero Scaling Exclusion is the necessary restriction accompanying every multiplicative inequality transformation: the quantity used to multiply or divide both sides must be nonzero, since multiplying an inequality by zero destroys the comparison entirely by making both sides equal to zero, and dividing by zero is undefined altogether. This restriction mirrors the Nonzero Symbolic Factor Condition required for literal equations, but carries the additional consequence, unique to inequalities, that the sign of the nonzero quantity, not merely its nonzero status, determines whether Positive Scaling without Relation Reversal or Relation Reversal after Negative Scaling applies.