Properties of Inequality

The following are the properties of inequality for real numbers. They are closely related to the properties of equality, but there are important differences.

Note especially that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality.

PROPERTIES OF INEQUALITY

Antireflexive Property

For all real numbers x,

Antisymmetry Property

For all real numbers x and y,

Transitive Property

For all real numbers x, y, and z,

  • if x < y and y < z, then x < z.
  • if x > y and y > z, then x > z.
Addition Property

For all real numbers x, y, and z,

  • if x < y, then x + z < y + z.
Subtraction Property

For all real numbers x, y, and z,

  • if x < y, then x – z < y – z.
Multiplication Property

For all real numbers x, y, and z,

  • if x < y, then

  • if x > y, then
Division Property

For all real numbers x, y, and z, with z ≠ 0,

  • if x < y, then

  • if x > y, then