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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