Division Property of Inequality

• xif and only if y-x is positive.
• If x and z are positive, then xz is also positive.
• If z is negative, then - z is positive.

The following conditional statements can be proven using these properties.

• If x0, then xz< yz.
• If x yz.

Each case will be analyzed separately.

z>0

As it is given that xgreater than 0. x0 Furthermore, because z>0, from the second property, it can be stated that y-x divided by z is also greater than 0. y-x>0 &and z>0 &⇓ y-x/z&>0 Now the second part of this conditional statement can be rewritten. y-x/z>0 ⇔ y/z-x/z>0 By the first property, it can be said that xz is less than yz. Then because because x

z<0

The following statement is valid because x0 Additionally, since z<0, it follows that - z is positive by the third property. Moreover, the quotient of y-x and - z will be positive. y-x>0 &and - z>0 &⇓ y-x/- z&>0 Now the second part of this statement can be rewritten.

Finally, the property has been obtained because x