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Rule

Subtraction Property of Inequality

Subtracting the same number from both sides of an inequality produces an equivalent inequality. The solution set and inequality sign of this equivalent inequality does not change. Let x, y, and z be real numbers such that x< y. Then, the following conditional statement holds true.

If x< y, then x-z< y-z.

This property holds for the other types of inequalities.

Proof

Subtraction Property of Inequality

The case when x< y will be proven. Consider the biconditional statement before beginning the proof. x< y ⇔ y-x> 0 This property can be proven using the Additive Inverse of z, which is - z. Now the Identity Property of Addition can be applied to the second part of the statement.

y-x> 0

IdPropAdd

Identity Property of Addition

y-x+0> 0

Rewrite 0 as (- z)-(- z)

y-x+(- z-(- z))> 0

CommutativePropAdd

Commutative Property of Addition

y+(- z-(- z))-x> 0

RemovePar

Remove parentheses

y-z-(- z)-x> 0

- a-b=-(a+b)

y-z-(- z+x)> 0

CommutativePropAdd

Commutative Property of Addition

y-z-(x-z)> 0

AddPar

Add parentheses

(y-z)-(x-z)> 0

The last inequality can be rewritten using the biconditional statement. (y-z)-(x-z)> 0 ⇕ x-z< y-z Finally, because x< y, the property has been proven.

If x< y, then x-z< y-z.

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