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Reference

Properties of Inequalities

Rule

Anti Reflexive Property of Inequality

A real number can never be less than or greater than itself.

and

This property is an axiom. Therefore, it can be accepted as true without proof.

Rule

Anti Symmetric Property of Inequality

For any two real numbers and if is less than then cannot be less than

If then

Alternatively, if is greater than then cannot be greater than

If then

This property is an axiom. Therefore, it can be accepted as true without proof.

Rule

Transitive Property of Inequality

Let and be real numbers. If is less than and is less than then is less than

If and then

This property also applies to other types of inequalities — and

  • If and then
  • If and then
  • If and then
Since this property is an axiom, it does not need proof to be accepted as true.

Rule

Addition Property of Inequality

Adding the same number to both sides of an inequality generates an equivalent inequality. This equivalent inequality will have the same solution set and the inequality sign remains the same. Let and be real numbers such that Then, the following conditional statement holds true.

If then

This property holds for the other types of inequalities.
The Addition Property of Inequality for All Types of Inequalities

Proof

Addition Property of Inequality
The case when will be proven. The remaining cases can be proven similarly. Before starting the proof, the following biconditional statement needs to be considered.
Now, the Identity Property of Addition can be applied to the second part of the statement.

Rewrite as

Using the biconditional statement, the last inequality can be rewritten.
Finally, because the property is obtained.

If then

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 and be real numbers such that Then, the following conditional statement holds true.

If then

This property holds for the other types of inequalities.
The Subtraction Property of Inequality for All Types of Inequalities

Proof

Subtraction Property of Inequality
The case when will be proven. The other cases can be proven using a similar reasoning. Consider the biconditional statement before beginning the proof.
This property can be proven using the Additive Inverse of which is Now, the Identity Property of Addition can be applied to the second part of the statement.

Rewrite as

From the biconditional statement, the last inequality can be rewritten.
Finally, because the property has been proven.

If then

Rule

Multiplication Property of Inequality

Multiplying both sides of an inequality by a nonzero real number produces an equivalent inequality. The following conditions about need to be considered when applying this property.

Positive If is positive, the inequality sign remains the same.
Negative If is negative, the inequality sign needs to be reversed to produce an equivalent inequality.

For example, let and be real numbers such that and Then, the equivalent inequalities can be written depending on the sign of

  • If and then
  • If and then
This property holds for the other types of inequalities.
The Multiplication Property of Inequality for All Types of Inequalities

Proof

Multiplication Property of Inequality

The case when will be proven. The remaining cases can be proven following a similar reasoning. Before starting the proof, the following properties of real numbers need to be considered.

  • if and only if
  • If and are positive, then
  • If is negative, then is positive.

Using these properties, the following conditional statements can be proven.

  • If and then
  • If and then

Each conditional statement will be analyzed separately.

When Is Greater Than

It is given that then using the first property, it is known that is greater than
Furthermore, because from the second property, it can be stated that the product of and is also greater than
Now, the second part of this conditional statement can be rewritten using the Distributive Property.
From the first property, it can be said that if and only if Additionally, because the conditional statement has been proven.
If and then

When Is Less Than

Again, because the following statement is valid.
Additionally, since from the third property it follows that is positive. Moreover, the product of and will be positive.
Now, can be distributed in the second part of the statement.
Simplify
Finally, because the property has been proven.

If and then

Rule

Division Property of Inequality

Dividing both sides of an inequality by a nonzero real number produces an equivalent inequality. However, the following conditions need to be considered.

Positive If is positive, the inequality sign remains the same.
Negative If is negative, the inequality sign needs to be reversed to produce an equivalent inequality.

For example, let and be real numbers such that and Then, the equivalent inequalities can be written depending on the sign of

  • If and then
  • If and then
This property holds for the other types of inequalities.
The Division Property of Inequality for All the Types of Inequalities

Proof

Division Property of Inequality

The case when will be proven. The remaining cases can be proven following a similar reasoning. Before starting the proof, the following properties of real numbers need to be considered.

  • if and only if is positive.
  • If and are positive, then is also positive.
  • If is negative, then is positive.

Using these properties, the following conditional statements can be proven.

  • If and then
  • If and then

Each case will be analyzed separately.

It is given that then using the first property, it is known that is greater than
Furthermore, because from the second property, it can be stated that divided by is also greater than
Now, the second part of this conditional statement can be rewritten.
By using the first property, it can be said that is less than Additionally, because the property has been proven.

If and then

Again, because the following statement is valid.
Additionally, since from the third property, it follows that is positive. Moreover, the quotient of and will be positive.
Now, the second part of this statement can be rewritten.
Simplify
Finally, because the property has been obtained.

If and then