# 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 $x$ , $x\nless x\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ngtr x$ Anti symmetry Property For all real numbers $x$ and $y$ , $\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x $\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x>y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{then}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\ngtr x.$ Transitive Property For all real numbers $x,y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z$ , if $x and $y , then $x . if $x>y$ and $y>z$ , then $x>z$ . Addition Property For all real numbers $x,y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z$ , if $x then $x+z . Subtraction Property For all real numbers $x,y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z$ , if $x then $x-z . Multiplication Property For all real numbers $x,y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z$ , if $x , then $\left\{\begin{array}{l}xz0.\\ xz>yz,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z<0.\\ xz=yz,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z=0.\end{array}$