Properties of Inequality

For all real numbers $x$ ,

• $x\nless x\text{and}x\ngtr x$

For all real numbers $x$ and $y$ ,

• $\text{if}x<y,\text{then}y\nless x.$
• $\text{if}x>y,\text{then}y\ngtr x.$

For all real numbers $x,y,\text{and}z$ ,

• if $x<y$ and $y<z$ , then $x<z$ .
• if $x>y$ and $y>z$ , then $x>z$ .

For all real numbers $x,y,\text{and}z$ ,

• if $x<y$ then $x+z<y+z$ .

For all real numbers $x,y,\text{and}z$ ,

• if $x<y$ then $x-z<y-z$ .

For all real numbers $x,y,\text{and}z$ ,

• if $x<y$ , then

$\{\begin{array}{l}xz<yz,\text{if}z>0.\\ xz>yz,\text{if}z<0.\\ xz=yz,\text{if}z=0.\end{array}$