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3. Inequalities And Trichotomy

Inequality tells us about the relative size of two values.

The 4 Inequalities

Properties of Inequalities:

• Transitivity
• Converse
• Addition and subtraction
• Multiplication and division
• Additive inverse
• Multiplicative inverse
• Applying a function to both sides

Inequality between means:

There are many inequalities between means. For example, for any positive numbers a1, a2, an we have H ≤ G ≤ A ≤ Q, where

Law of Trichotomy:

For any two real numbers a and b, exactly one of the following is true.

i. a < b

ii. a = b

iii. a > b

Rule for Quadratics:

When working with quadratic equations and inequalities in one variable;

• If the equation has two real solutions, the resulting interval will be the solution set for one of the inequalities and the union of the rays will be the solution set for the other inequality.
• If the equation has one real solution the union of the resulting two rays will be the solution set for one of the inequalities and the solution set for the other inequality is the empty set Ø.
• If the equation has no real solution, the solution set for one of the inequalities is the set of real numbers R and the solution set for the other inequality is the empty set Ø.

Law of Trichotomy Applied to Equations and Inequalities in Two Variables:

Linear Equations and Inequalities in Two Variables: A linear equation in two variables is an equation which may be written in the form Ax + By = C where A, B, and C are real numbers and B is not zero. A linear inequality in two variables x and y is an inequality which can be written as

Ax + By < C,

Ax + By > C.