## Math Properties . . . Properties of Multiplication and Division Applied to Inequalities

Math Properties - Inequalities: Multiplication & Division

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:

, , , .
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).

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Property of Multiplication (and Division) Applied to Inequalities

Multiplication and division involving inequalities can change the direction of the inequality sign. Careful attention to detail is important when performing these operations.

Multiplying or Dividing by a "positive number" will not change the direction of the inequality sign.

Multiplying inequalities by a "positive number".

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

c = a "positive" real number not equal to 0

If a b , then a * c b * c

ac bc

If a b , then a * c b * c

ac bc

If a b , then a * c b * c

ac bc

If a b , then a * c b * c

ac bc

If a = 4, b = 2, c = 2

If 4 2 , then 4 * 2 2 * 2

8 4

If a = 4, b = 6, c = 2

If 4 6 , then 4 * 2 6 * 2

8 12

Dividing inequalities by a "positive number".

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

c = a "positive" real number not equal to 0

If a b , then a ÷ c b ÷ c

If a b , then a ÷ c b ÷ c

If a b , then a ÷ c b ÷ c

If a b , then a ÷ c b ÷ c

If a = 4, b = 2, c = 2

If 4 2 , then 4 ÷ 2 2 ÷ 2

2 1

If a = 4, b = 6, c = 2

If 4 6 , then 4 ÷ 2 6 ÷ 2

2 3

Multiplying or Dividing by a "negative number" will change the direction of the inequality sign.

Multiplying inequalities by a "negative number".

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

c = a "positive" real number not equal to 0

If a b , then a * (-c) b * (-c)

-ac -bc

If a b , then a * (-c) b * (-c)

-ac -bc

If a b , then a * (-c) b * (-c)

-ac -bc

If a b , then a * (-c) b * (-c)

-ac -bc

If a = 4, b = 2, c = 2

If 4 2 , then 4 * (-2) 2 * (-2)

-8 -4

If a = 4, b = 6, c = 2

If 4 6 , then 4 * (-2) 6 * (-2)

-8 -12

Dividing inequalities by a "negative number".

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

c = a "positive" real number not equal to 0

If a b , then a ÷ (-c) b ÷ (-c)

If a b , then a ÷ (-c) b ÷ (-c)

If a b , then a ÷ (-c) b ÷ (-c)

If a b , then a ÷ (-c) b ÷ (-c)

If a = 4, b = 2, c = 2

If 4 2 , then 4 ÷ (-2) 2 ÷ (-2)

-2 -1

If a = 4, b = 6, c = 2

If 4 6 , then 4 ÷ (-2) 6 ÷ (-2)

-2 -3

Multiplying and Dividing an inequality by 0 (zero).

Multiplying inequalities by 0 (zero).

a = any real number, b = any real number

If a b , then a * 0 b * 0

0 0

If a = 4, b = 2

If 4 2 , then 4 * 0 2 * 0

0 0

Dividing inequalities by 0 (zero)- undefined (cannot be done).

a = any real number, b = any real number

If a b , then a ÷ 0 (undefined)
b ÷ 0 (undefined)

If a = 4, b = 2

If 4 2 , then 4 ÷ 0 (undefined)
2 ÷ 0 (undefined)