Principles of Equivalence for Inequalities

Equivalent inequalities are governed by two fundamental rules known as the principles of equivalence for inequalities.

First Principle of Equivalence
Second Principle of Equivalence

First Principle of Equivalence

Adding or subtracting the same term (or expression) on both sides of an inequality produces an equivalent inequality.

This principle ensures that a term can be moved from one side of an inequality to the other simply by reversing its sign.

Example

Consider the inequality:

$$ x + 1 > 0 $$

Add 1 to both sides:

$$ x + 1 + 1 > 0 + 1 $$

The resulting inequality is equivalent:

$$ x + 2 > 1 $$

Note. This principle underlies the standard algebraic rule that a term may be transferred across an inequality by changing its sign while keeping the inequality’s direction unchanged. For example: $$ x + 1 > 0 $$ $$ x + 1 + (-1) > 0 + (-1) $$ $$ x > -1 $$

Second Principle of Equivalence

if the term is positive, the resulting inequality is equivalent and the direction remains unchanged
if the term is negative, the resulting inequality is equivalent but the direction must be reversed

In particular, multiplying every term of an inequality by a negative number (such as -1) necessitates reversing the inequality sign to preserve equivalence.

Example

Consider the inequality:

$$ x + 1 > 3 $$

Multiply both sides by 2:

$$ (x + 1) \cdot 2 > 3 \cdot 2 $$

$$ 2x + 2 > 6 $$

The result is an equivalent inequality with the same direction (>).

Example 2

Consider again:

$$ x + 1 > 3 $$

Multiply both sides by -2:

$$ (x + 1) \cdot (-2) > 3 \cdot (-2) $$

$$ -2x - 2 < -6 $$

The resulting inequality is equivalent but has the opposite direction (<).

Note. The inequality $$ x + 1 > 3 $$ has solution set $$ x > 2 $$. Multiplying both sides by -1 reverses the sign of every term: $$ (x + 1) \cdot (-1) > 3 \cdot (-1) $$ Keeping the same direction would produce an inequality with solution set $$ x < 2 $$ which is not equivalent to the original. To obtain an equivalent inequality, the direction must also be reversed: $$ -x - 1 < -3 $$ whose solution set $$ x > 2 $$ matches the original.