Math Properties . . . Inequalities: Properties of Exponents and Roots . . .

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Math Properties
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Properties of Exponents and Roots Applied to Inequalities

Math Properties - Inequalities: Exponents and Roots - 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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Exponents and roots involving inequalities will not change the direction of the inequality sign if all the numbers involved are positive real numbers.

Examples follow.

Exponents .

a = any positive real number, b = any positive real number, c = any positive real number (note: if c = a negative real number, the inequality sign will reverse; if c = 0, the inequality relationship will change to 1 = 1)

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

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

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

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

If a = 10, b = 1, c = 2

If 10 $Math Symbol - Greater Than$ 1, then 10² $Math Symbol - Greater Than$ 1²

100 $Math Symbol - Greater Than$ 1

Roots .

a = zero or any positive real number, b = zero or any positive real number, c = any positive real number

If a $Math Symbol - Greater Than$ b , then $Math - root - example$ $Math Symbol - Greater Than$ $Math - root - example$

If a $Math Symbol - Less Than$ b , then $Math - root - example$ $Math Symbol - Less Than$ $Math - root - example$

If a $Math Symbol - Greater Than or Equal To$ b , then $Math - root - example$ $Math Symbol - Greater Than or Equal To$ $Math - root - example$

If a $Math Symbol - Less Than or Equal To$ b , then $Math - root - example$ $Math Symbol - Less Than or Equal To$ $Math - root - example$

If a = 8, b = 64, c = 3

If 8 $Math Symbol - Less Than$ 64 , then $Math - root - example$ $Math Symbol - Less Than$ $Math - root - example$

2 $Math Symbol - Less Than$ 4