reviews the basic ideas you need to start calculus. The topics includes Cartesian coordinates in the plane, straight lines.

Lecture no.1 : reviews the basic ideas you need to start calculus. The topics includes Cartesian coordinates in the plane, straight lines.

General concepts :
Real number : are numbers that can be expressed as decimals, such as :
,
The real number can be represented geometrically as points on a number line called the real line.



-3 -2 -1 0 +1 +2 +3


The real number properties:
1. algebric : can be added, subtracted, multiplied and divided.
2. order properties :







Where c= constant value



Inequalities :
Rule of inequalities , if a, b and c are real numbers then :
1. a < b, a+c < b+c
2. a < b, a-c < b-c
3. a < b and c > 0 , ac < bc
4. a < b and c < 0 , ac > bc
Special case a <b , -b <-a
5. a >0 , 1/a > 0
6. if a and b are both positive or negative then a < b , 1/b <1/a

Notice : multiplying by a negative number reverses the inequality: 2<5 but -2>-5.

Intervals :
A subset of the real line is called an interval if it contains at least an interval if it contains at least two numbers and contains all the real numbers lying between any two of its elements.
Example : x> b , -2< x <5.

Types of intervals:
Finite : (a,b){x/a< x< b} open

Closed

Half open

Infinite :


Example: solve the following inequalities and show their solution sets on the real line.
a. 2x-1<x+3 b. –x/3<2x+1 c. 6/(x-1)>5
solution:
a. 2x-1<x+3
2x-x<4
X<4 interval(- ,4) open

b. –x<6x+3
7x>-3
x>-3/7 (-3/7, )

c. 6>5x-5
11>5x
X<5/11 (1,11/5] half open.

Absolute value :

The absolute value of a number X denoted by ; is defined by the formula :





3

Geometrically the absolute value of x is the distance from x to zero on the real number line.
the distance between x and y since the distance are always positive or 0.

Absolute value properties :

1.
2.
3.
4.



Example : solve
2x-3= 7
2x= - 7+3 or 2x-3=7
2x= - 4 2x=10
X= -2 x=5

Note : if D is any positive number then




Example : solve

-8< <8 or 8> x-5
-8 < x – 5 x < 13
-3 < x

Homework : solve

Note: reciprocal and multiply by negative number reverse the inequality

Example : solve the inequality and show the solution on real line :



Solution :








1 2

Home works: solve



4. prove that