Math Properties . . . Transitive Property of Inequality . . .

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Math Properties
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Transitive Property of Inequality

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Math Properties - Transitive - Properties of Inequality : In mathematics, inequalities compare numbers and expressions that are not equal to one another.

Generally, the following symbols are used for comparison:

$Math Symbol - Greater Than$ , $Math Symbol - Less Than$ , $Math Symbol - Greater Than or Equal To$ , $Math Symbol - Less Than or Equal To$ .
Each of these symbols compares the relative size of two numbers to show which number is bigger, and which number is smaller.

Inequalities are important because they are common to everyday life experiences.

People intuitively use the concept of inequality to compare options and make choices based on relative value.

For example, a person may ask themselves: "Can I buy a new car with $10,000 if a new car costs $11,000?" The comparison of these choices is an inequality. One number is smaller than the other number.

Automatic control systems rely on this concept as well. A thermostat uses an inequality (comparing temperatures) to decide whether to turn on a heater or air conditioner. For example, the thermostat may turn on an air conditioner when the temperature rises above 85 degrees (or turn off the air conditioner when the temperature falls below 80 degrees).

The Properties of Inequality of Real Numbers
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$Math Properties - Math Inequalities - Trichotomy$

Trichotomy
Property of Inequality

$Math Properties - Math Inequalities - Transitive$

Transitive
Property of Inequality

$Math Properties - Math Inequalities - Reversal$

Reversal
Property of Inequality

$Math Properties - Math Inequalities - Addition & Subtraction$

Additive
Property of Inequality

$Math Properties - Math Inequalities - Multiplication & Division$

Multiplicative
Property of Inequality

$Math Properties - Math Inequalities - Exponents & Roots$

Exponents and Roots
Properties of Inequality

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The Transitive Property of Inequality is similar to the Transitive Property of Equality described under "Equivalence" Relationships. However, there are a large number of possible variations when applying the Transitive Property to inequalities.

This being said, the logic can be illustrated as follows: if a number is less than or equal to a second number, and the second number is less than or equal to a third number, then the first number is also less than or equal to the third number.

Examples follow. However, the examples shown below are not inclusive. There are many more possibilities which could be shown.

a = any real number, b = any real number, c = any real number

If a $Math Symbol - Greater Than$ b and b $Math Symbol - Greater Than$ c, then a $Math Symbol - Greater Than$ c

If a $Math Symbol - Less Than$ b and b $Math Symbol - Less Than$ c, then a $Math Symbol - Less Than$ c

If a $Math Symbol - Greater Than or Equal To$ b and b $Math Symbol - Greater Than or Equal To$ c, then a $Math Symbol - Greater Than or Equal To$ c

If a $Math Symbol - Less Than or Equal To$ b and b $Math Symbol - Less Than or Equal To$ c, then a $Math Symbol - Less Than or Equal To$ c

If a $Math Symbol - Greater Than or Equal To$ b and b $Math Symbol - Greater Than$ c, then a $Math Symbol - Greater Than$ c

If a $Math Symbol - Less Than$ b and b $Math Symbol - Less Than or Equal To$ c, then a $Math Symbol - Less Than$ c

If a $Math Symbol - Greater Than$ b and b $Math Symbol - Greater Than or Equal To$ c, then a $Math Symbol - Greater Than$ c

If a $Math Symbol - Less Than or Equal To$ b and b $Math Symbol - Less Than$ c, then a $Math Symbol - Less Than$ c

If 5 $Math Symbol - Greater Than$ 1 and 1 $Math Symbol - Greater Than$ 0, then 5 $Math Symbol - Greater Than$ 0

If 15 $Math Symbol - Less Than or Equal To$ 17 and 17 $Math Symbol - Less Than or Equal To$ 19, then 15 $Math Symbol - Less Than or Equal To$ 19

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Math Properties
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Transitive Property of Inequality