PRELIMINARIES

1. Real Numbers, Sets, and Inequalities (A Review)

2. Absolute Value

3. Coordinate Planes; Distance; Circles

4. Slope of a Line

5. Equations of the Straight Lines

REAL NUMBERS, SETS, AND

INEQUALITIES

integers

…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …

rational numbers ; ratios of integers,

denumerator ≠ 0

irrational numbers

2

5

2

5

2

5,

1000

17,

2

6,

9

0,

3

2

03 19cos,,7,3,21

REAL LINE OR COORDINAT LINE

SET

Set is collection of objects

The objects are called elements or

members of the set

WAY OF DESCRIBING A SET

List its elements between braces

example : {1,2,3,4}

By stating a property that common only to its members

example :

the set of all rational numbers

the set of all real numbers x such that 2x2 -4x + 1 = 0 or {x : 2x2 - 4x + 1 = 0}

To indicate that an element a is a member of a set A, we write

a A

which is read "a is an element of A" or "a belongs to A."

To indicate that the element a is not a member of the set A, we write

a A

which is read, "a is not an element of A" or "a does not belong to A."

EMPTY SET OR NULL SET

A set with no members

Denoted by the symbol

Example :

= {x : x2 <0}

INTERVAL :

SET OF REAL NUMBERS

LINE SEGMENT (GEOMETRICALLY)

Closed interval

{x: a ≤ x ≤ b}

Open interval

{x: a < x < b}

Closed and open intervals are usually denoted by the

symbols [a, b] and (a, b), where

[a,b]= {x: a ≤ x ≤ b}

(a,b)={x:a<x<b}

A square bracket [ or ] indicates that the endpoint is

included, while a rounded bracket ( or ) indicates

that the endpoint is excluded.

An interval can include one endpoint and exclude

the other. Such intervals are called half open (or

sometimes half-closed ). For example,

[a,b)={x : a ≤ x < b}

(a, b]= {x: a < x ≤ b}

Two sets A and B are said to be equal if they have the same elements, in which case we write

A = B

Example

{x : x2=1} = {-1,1},

{π,0,3}={3, π,0}

{x: x2 < 9} = {x: -3 < x < 3}

If every member of a set A is also a member

of set B, then we say A is a subset of B and

write

A B

In addition, we will agree that the empty set

is a subset of every set.

Example

{-2,4} {-2, 1,0,4}

{x: x is rational} {x: x is a real number}

A (for every set A)

If A and B are two given sets, then the set of all

elements belonging to both A and B is called the

intersection of A and B

it is denoted by A B

If A and B are two given sets, then the set of all

elements belonging to A or B or both is called the

union of A and B

it is denoted by A B

EXAMPLE

Solve

2x-3< 7

(That is, find all real numbers satisfying the inequality.)

Solution.

If x is any solution, then

2x - 3 < 7 [Given]

2x < 10 [We added 3 to both sides.]

x < 5 [We multiplied both sides by ]

At this point we are tempted to conclude that the solutions of

2x - 3 < 7 consist of all x less than 5, that is, all x in the interval (-∞, 5).

2

1

EXAMPLE

Solve

7 ≤ 2-5x < 9

Solution.

7 ≤ 2- 5x < 9 [Given.]

5 ≤ - 5x < 7 [We added -2 to each member.]

- l ≥ x > -7/5 [we multiplied each member by -1/5]

Since the steps are reversible, the set of solutions is

(-7/5, -1].

EXERCISE,

FIND ALL X’S THAT SATISFY THESE INEQUALITIES

1. 3x-2<8

2. 3≤4-2x<7

3. 12

52

x

x

1.2 ABSOLUTE VALUE

Absolute value or magnitude of real number a is denoted by | a | and is defined by

0 if

0 if

aaa

aaa

THEOREM

nnaa

b

a

b

a

baab

aaa

nba

aa a

theninteger,an is and numbers real are and If

,number realany For 2

EXAMPLE

Solve |x-3|=4

Answer :

If x-3 > 0 then x-3 = 4 , we get x = 7

If x-3 < 0 then x-3 = -4 , we get x = -1

So there are two values of x, x = 7 and x = -1, such

that |x-3| = 4

EXERCISE

Solve

1. |6x-2|=7

2. |x+6|<3

3. |5-2x|>4

4. |3x-2|=|5x+4|

3. COORDINAT PLANE, DISTANCE, CIRCLES

Coordinat axes

x-axes(horizontal)

y-axes(vertical)

The name :

Cartesian coordinat system

Rectangular coordinat system

Coordinat plane or xy-plane is a plane in which

rectangular coordinat system has been introduced.

DISTANCE

The distance d

between two points

(x1,y1) and (x2,y2) in

a coordinat plane is

given by :

CIRCLES

If (x0,y0) is a fixed point in the plane, then the circle of radius r centered at (x0,y0) is the set of all points in the plane whose distance from (x0,y0) is r.

Thus a point (x, y) will lie on this circle if and

only if

Or equivalently

4. SLOPE OF A LINE

If P1(x1,y1) and P2(x2,y2) are distinct

points such that x1 ≠ x2, then the

number m given by the formula

is called the slope of the line connecting

P1 and P2

ANGLE OF INCLINATION

For a line L not parallel

to the x-axis, the angle

of inclination is the

smallest angle

measured

counterclockwise from

the direction of the

positive x-axis to L.

For a line parallel the x-

axis, we take = 0.

ANGLE OF INCLINATION

For a line not parallel to the y-axis, the slope and

angle of inclination are related by

m = tan

THEOREM

Two nonvertical lines are parallel if and

only if they have the same slope.

Two nonvertical lines are perpendicular if

and only if the product of their slopes is - l;

equivalently, lines with slopes m1 and m2

are perpendicular if and only if

2

1

1

mm

5. EQUATION OF STRAIGHT LINE

The line passing through P1(x1,y1) and having

slope m is given by the equation

y - y1 = m ( x - x1 )

This is called the point-slope form of the line.

EXAMPLE

Find the point-slope form of the line through (4, -3) with slope 5.

Solution.

Substituting the values x1 = 4, y1 = -3, and m = 5 in y - y1 = m ( x - x1 ) , yields the point-slope form

y + 3 = 5 ( x – 4 )

by simplification : y = 5x -23

THE SLOPE-INTERCEPT FORM

The line with y-intercept b and slope m is given by the equation

y = mx + b

Example :

Comparing y = 4x + 7 to y = mx + b , we have m = 4 and b = 7,

so that the equation represents a line crossing the y-axis at (0 , 7) with slope 4.

THE TWO-POINT FORM

The nonvertical line determined by the

points P1(x1,y1) and P2(x2,y2) can be

represented by the equation :

)( 1

12

121 xx

xx

yyyy

EXAMPLE

Find the slope-intercept form of the line

passing through (3, 4) and (2, -5).

Solution.

Letting (x1,y1) = (3, 4) and (x2,y2) = (2, -5)

and substituting in

)( 1

12

121 xx

xx

yyyy

we obtain the two-point form

which can be written

y - 4 = 9(x - 3)

Solving for y yields the slope-intercept form

y = 9x - 23

)3(32

454

xy