Equivalent Inequalities

Two inequalities are said to be equivalent inequalities when they have exactly the same solution set.

For example, consider the pair of inequalities:

$$ x + 1 > 0 $$

$$ x + 2 > 1 $$

These two statements are equivalent because they yield the identical solution set:

$$ S = \{\, x \in \mathbb{R} \mid x > -1 \,\} $$

This example also provides a clear illustration of the first principle of equivalence, which asserts that adding or subtracting the same quantity on both sides of an inequality preserves its set of solutions.

Note. The principles of equivalence for inequalities form the basis of the standard algebraic rules used when transforming inequalities. These include reversing the direction of an inequality when multiplying both sides by a negative number, and changing the sign of a term when moving it from one side of the inequality to the other.

And so on.