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Multiplying both sides of an inequality by a nonzero real number z produces an equivalent inequality. The following conditions about z need to be considered when applying this property.
| Positive z | If z is positive, the inequality sign remains the same. |
|---|---|
| Negative z | If z is negative, the inequality sign needs to be reversed to produce an equivalent inequality. |
For example, let x, y, and z be real numbers such that x< y and z≠0. Then, the equivalent inequalities can be written depending on the sign of z.
This property holds for the other types of inequalities.
Using these properties, the following conditional statements can be proven.
Each conditional statement will be analyzed separately.
As it is given that x< y, then by the first property, it is known that y-x is greater than 0.
x< y ⇔ y-x>0
Furthermore, because z> 0, from the second property, it can be stated that the product of z and y-x is also greater than 0.
y-x>0 &and z>0 &⇓ z(y-&x)>0
Now the second part of this conditional statement can be rewritten using the Distributive Property. z(y-x)>0 ⇔ zy-zx>0
From the first property, it can be said that zy-zx>0 if and only if zx
The following statement is valid because x
Distribute (- z)
(- a)b = - ab
a-(- b)=a+b
LHS+zy>RHS+zy
Finally, the property has been proven because x
If x