The next few sets of supplemental notes I am going to call preliminary notes. These notes will be a review of topics covered in college algebra and pre-calculus. The material in these notes should by familiar.

First of all, much of calculus is based on real numbers. Recall that real numbers are any number that can be expressed as decimals, and can be represented geometrically as points on a number line. There are specific properties of real numbers. We can add, subtract, multiply, and divide (except by 0) any two real numbers to produce another real number. Real numbers also satisfy order properties. Here are the order properties of real numbers, which you should know as the rules for inequalities.

RULES FOR INEQUALITIES

If a, b, and c are real numbers, then:

1.

a < b Þ a + c < b + c

2.

a < b Þ a - c < b - c

3.

a < b and c > 0 Þ ac < bc

4.

a < b and c < 0 Þ ac > bc

5.

6.

There are four major subsets of the reals, and they are the natural numbers, integers, rational, and irrational numbers. The following diagram illustrates how they are related.

DEFINITION:

A subset of the real line is called an interval if it contains at least two numbers and contains all the real numbers lying between any two of its elements.

There are two types of intervals, and they are finite and infinite. Geometrically, a finite interval is a line segment because it has two endpoints. Now whether or not it contains its endpoints depends on if it is an open, a closed, or a half-open interval. An open interval does not include its endpoints.

EXAMPLE 1:

(a, b)

{x | a < x < b}

A closed interval contains both of its endpoints.

EXAMPLE 2:

[a, b]

{x | a £ x £ b}

A half-open interval only includes one of its endpoints.

EXAMPLE 3:

(a, b]

{x | a < x £ b}

Geometrically, an infinite interval is a ray or a line. It can include one, none, or have no endpoints. When an infinite interval includes one endpoint or has none, it is said to be a half-open interval.

EXAMPLE 4:

(a, ¥ )

{x | x > a}

(-¥ , b]

{x | x £ b}

When an infinite interval has no endpoints, then it is a line and is the interval (-¥ , ¥ ). Sometimes the endpoints are called boundary points, and the points contained in the interval are called interior points.

DEFINITION:

The absolute value of a number x, denoted | x |, is defined by

What this definition is stating is that when x is positive, then the absolute value of x will be that number. But, when x is negative, take a negative one times that number to get the absolute value. Remember that the absolute value is the distance from zero to that number.

ABSOLUTE VALUE PROPERTIES

1.

|-a | = | a |

2.

| ab | = | a || b |

3.

4.

| a + b | £ | a | + | b |

TRIANGLE INEQUALITY

Inequalities involving absolute values are related to intervals in the following way. If D is any positive number, then

EXAMPLE 5:

| x | £ 3 Û -3 £ x £ 3

| x | > 4 Û x < -4 or x > 4