Algebra I

An inequality is a statement in which the relationships are not equal. Instead of using an equal sign (=) as in an equation, these symbols are used: > (is greater than) and < (is less than) or ≥ (is greater than or equal to) and ≤ (is less than or equal to).

Axioms and properties of inequalities

For all real numbers a, b, and c, the following are some basic rules for using the inequality signs.

• Trichotomy axiom: a > b, a = b, or a < b.

  These are the only possible relationships between two numbers. Either the first number is greater than the second, the numbers are equal, or the first number is less than the second.

• Transitive axiom: If a > b, and b > c, then a > c.

  Therefore, if 3 > 2 and 2 > 1, then 3 > 1.

  If a < b and b < c, then a < c.

  Therefore, if 4 < 5 and 5 < 6, then 4 < 6.

• Addition property:

  • 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.

  Adding or subtracting the same amount from each side of an inequality keeps the direction of the inequality the same.

  Example: If 3 > 2, then 3 + 1 > 2 + 1 (4 > 3)

  If 12 < 15, then 12 – 4 < 15 – 4 (8 < 11)

• Positive multiplication and division property:

  • If a > b, and c > 0, then ac > bc.

  • If a < b, and c > 0, then ac < bc.

  • If a > b, and c > 0, then .

  • If a < b, and c > 0, then .

  Multiplying or dividing each side of an inequality by a positive number keeps the direction of the inequality the same.

  Example: If 5 > 2, then 5(3) > 2(3), therefore, 15 > 6.

  If 3 < 12, then .

• Negative multiplication and division property:

  • If a > b, and c < 0, then ac < bc.

  • If a < b, and c < 0, then ac > bc.

  • If a > b, and c < 0, then .

  • If a < b, and c < 0, then .

  Multiplying or dividing each side of an inequality by a negative number reverses the direction of the inequality.

  Example: If 5 > 2, then 5(– 3) < 2(– 3); therefore, – 15 < – 6.

  If 3 < 12, then .

Solving inequalities