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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

Anti reflexive Property

For all real numbers x ,

  • x x and x x
Anti symmetry Property

For all real numbers x and y ,

  • if x < y , then y x .
  • if x > y , then y x .
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
  • { x z < y z , if z > 0. x z > y z , if z < 0. x z = y z , if z = 0.