Properties of Inequalities

1. Non-negative Property

If a number or expression is greater than or equal to zero, it is non-negative. That is, The non-negative property of inequalities refers to the fact that, if a number or expression is greater than or equal to zero, it is non-negative. This property tells us that if a≥0, then The non-negative property of inequalities refers to the fact that, if a number or expression is greater than or equal to zero, it is non-negative. This property tells us that if a≥0, then a is either positive or zero. Also, the square of a number is always positive, that is a ² ≥ 0.

Example:

• 5 ≥ 0 (True)
  This is true because 5 is positive.
• 0 ≥ 0 (True)
  This is true because 0 is equal to 0.
• -2 ≥ 0 (False)
  This is false because -2 is negative.

2. Addition Property

The addition property of inequalities states that if you add the same value to both sides of an inequality, the inequality remains true. In other words,

• if 𝑎 ≤ 𝑏 , then 𝑎 + 𝑐 ≤ 𝑏 + 𝑐 for any real number 𝑐.

That is, adding the same value to both sides of an inequality keeps it valid.

Example:

• Given 3 < 7, adding 2 to both sides gives 3 + 2 < 7 + 2 , which simplifies to 5 < 9, and the inequality remains valid.
• Given x ≥ -1, adding 4 to both sides gives 𝑥 + 4 ≥ − 1 + 4 , or x + 4 ≥ 3.

3. Multiplication Property

The multiplication property of inequalities has two key rules depending on whether you multiply by a positive or negative number:
That is, multiplying both sides by a positive number keeps the sign, but multiplying by a negative number flips the inequality sign.

• Multiplying by a Positive Number: If 𝑎 ≤ 𝑏 , and 𝑐 > 0 , then 𝑎 𝑐 ≤ 𝑏 𝑐 .
• Multiplying by a Negative Number: If 𝑎 ≤ 𝑏 , and 𝑐 < 0 , then 𝑎 𝑐 ≥ 𝑏 𝑐 (the inequality sign flips).

Example:

• Positive multiplication:
  Multiplying 2 < 5 by 3 gives 6 < 15 (True).
  That is, Given 2 < 5 , multiplying both sides by 3 (a positive number) gives 2 × 3 < 5 × 3 , or 6 < 15 , which is true.
• Negative multiplication:
  Multiplying 2 < 5 by -2 gives -4 > -10 (True).
  That is, Given 2 < 5 , multiplying both sides by -2 (a negative number) flips the inequality to 2 × ( − 2 ) > 5 × ( − 2 ) , or − 4 > − 10 , which is also true.

This property is crucial when solving inequalities, as it determines whether you need to reverse the inequality sign.

4. Reciprocal Property

The reciprocal property of inequalities states that when you take the reciprocal (or inverse) of both sides of a positive inequality, the inequality flips. This applies when the terms are positive, and you switch the direction of the inequality when taking reciprocals.

• If 0 < 𝑎 < 𝑏 , then 1/ 𝑏 < 1/ 𝑎.

Example:

• Given 1 < 4, the reciprocals give 1 > 0.25.
  Tha is, given 1 < 4 , taking the reciprocal of both sides gives 1/ 1 > 1/ 4 , or 1 > 0.25 , which is true.
• Given 2 > 0.5, the reciprocals give 0.5 < 2.
  That is, Given 2 > 0.5 , taking the reciprocal of both sides gives 1/ 2 < 2 , which is true.