1. Appendix A: Numbers, Inequalities, and Absolute values
There are many number systems
Real number system (1-dimension system)
Complex number system (2-dimension system)
Hamilton Number system (3-dimension system) ……
Calculus is based on real number system

2. There is a 1-1 correspondence between real # and points on the real line
Comparing two real numbers requires symbols that indicates their order on the real line
Symbol
Meaning
a < b
a is less than b
a > b
a is greater than b
a  b
a is less than or equal to b
a  b
a is greater than or equal to b

3. A set is a collection of objects, and these objects are called the elements of the set.
Commonly used sets: R = the set of all real numbers, Z = the set of all integers
Symbol
Meaning
a S
a is an element of the set S
a  S
a is not an element of the set S

4. Two ways to define a set:
Roster Method: Listing the elements
Ex: A = {a, b, c, d}— elements are listed within
set braces.
Set Builder method:
-- Use a variable, braces {}, and a vertical bar |.
-- {} is read as “set of” and | is read as “such that”
Ex:
B is the set of all real numbers such that
hold.

5. Like numbers, we have some operations on sets
Like numbers, we have some operations on sets. In this course, we only use the following two operations:
Note:
The set that contains no elements is called empty set, denoted by  or { }.
S  T means that S is a subset of T, that is, each element in S is also in T.
Symbol
Meaning
S  T
Set of elements that are in either S or T or both
S  T
Set of elements that are in both S and T

6. A set that consists of all the real numbers between two points on the real line is called an interval. The commonly used interval notions:

7. Inequality Properties
(i) If a < b, then a + c < b + c for any cR.
(ii) If a < b and c > 0, then ac < bc and a/c < b/c.
(iii) If a < b and c < 0, then ac > bc and a/c > b/c.
Note:
(a) They are still true if “<“ is replaced by “” and “>” by “”.
(b) If multiplying or dividing a negative
number into both sides of a inequality,
the inequality must be reversed.

8. In this section, we only review:
(1) linear inequalities
(2) quadratic inequalities

9. (1) Linear Inequality
ax + b is called a linear expression, where a and b
are constants with a  0 and x is a variable.
A linear inequality is an inequality that only
involves linear expressions.
Ex. 3 – 2x  5 + 3x is a linear inequality.
|x – 3|< 4 and 2x2 – 3x + 1 > 0 are not linear.

10. Ex: Solve 3 – 2x  5 + 3x
Note: To solve an inequality means to determine the set of all real numbers for which the given inequality holds.
Solution: Idea is to get variable x to one side and numbers on the other side.
3 – 2x  5 + 3x  3 – 5  3x + 2x
 – 2  5x  – 2/5  x or x  – 2/5
We can write the solution in terms of interval
notation: solution set = [– 2/5, ).
Another solution:
3 – 2x  5 + 3x  – 2x – 3x  5 – 3
 – 5x   x  – 2/5
Solution set = [– 2/5, ).

11. (2) Quadratic Inequality
ax2 + bx + c is called a quadratic expression, where a, b, and c are constants with a  0 and x is a variable.
A quadratic inequality is an inequality that only involves quadratic expressions.
Ex: 2x2 – 3x > – 1 is a quadratic inequality.

14. Absolute values Definition: The absolute value of a, denoted by |a|,
is defined to be the distance from a to the origin (0).
Ex: |– 3| = 3. |4.3| = 4.3. |0| = 0.
Definition: |a – b| = the distance between a and b.
Comment: |a – b| = |b – a|, |a| = |a – 0|
Ex: |3 – | = |  – 3| =  – 3.