DashboardNewsSettingsHelp
Get Premium
Dashboard
Dashboard Sign out
View details

Reference

Properties of Inequalities

Rule

Anti Reflexive Property of Inequality

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

x ≮ x and x ≯ x

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 x and y, if x is less than y, then y cannot be less than x.

If x< y, then y ≠< x.

Alternatively, if x is greater than y, then y cannot be greater than x.

If x> y, then y ≠> x.

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

Transitive Property of Inequality

Let x, y, and z be real numbers. If x is less than y and y is less than z, then x is less than z.

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

This property also applies to other types of inequalities — >, ≤, and ≥.

  • If x> y and y> z, then x > z.
  • If x≤ y and y≤ z, then x ≤ z.
  • If x≥ y and y≥ z, then x ≥ z.
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 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.

The Addition Property of Inequality for All Types of Inequalities

Proof

 Addition Property of Inequality
The case when x< y will be proven. Before starting the proof, consider the following biconditional statement. x< y ⇔ y-x> 0 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-x-z> 0

- a-b=-(a+b)

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 is obtained.

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

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.

The Subtraction Property of Inequality for All 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.

Rule

Multiplication Property of Inequality

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.

  • If x< y and z> 0, then xz< yz.
  • If x< y and z< 0, then xz> yz.

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 x

  • x< y if and only if y-x>0.
  • If x and y are positive, then xy> 0.
  • If z is negative, then - z is positive.

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

  • If x< y and z> 0, then xz< yz.
  • If x< y and z< 0, then xz> yz.

Each conditional statement will be analyzed separately.

When z Is Greater Than 0

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

If x0, then zx

When z Is Less Than 0

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

- z(y-x)>0
â–¼
Simplify
Distr

Distribute (- z)

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

(- a)b = - ab

- zy-(- zx)>0
SubNeg

a-(- b)=a+b

- zy+zx>0
AddIneq

LHS+zy>RHS+zy

zx>zy

Finally, the property has been proven because x

If xzy.

Rule

Division Property of Inequality

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

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

  • If x0, then xz< yz.
  • If x yz.

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 x

  • 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

If x0, then xz< yz.

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.

y-x/- z>0
â–¼
Simplify
MoveNegDenomToNum

Put minus sign in numerator

-(y-x)/z>0
DistrNegSignSwap

-(b-a)=a-b

x-y/z>0
WriteDiffFrac

Write as a difference of fractions

x/z-y/z>0
AddIneq

LHS+y/z>RHS+y/z

x/z>y/z

Finally, the property has been obtained because x

If x yz.

Exercises

Recommended exercises

To get personalized recommended exercises, please login first.
Return to lesson.
Edit Lesson