Chapter One
Department of IT
Chapter One
Numbers, Inequalities and Absolute value
Numbers:
Definition: A set is a collection of objects “elements” and will be denoted by a capital letters.
Example:
A = {1, 2,3 ,4 ,5}
Important Sets of Real Numbers
1- The set of natural numbers denoted by
and define as
= {1, 2, 3, …}
2- The set of integers numbers denoted by
and define as
= {... , −2, −1, 0, 1, 2, ...}.
3- The set of rational numbers is the set of numbers that can be written as the quotient of two
integers, where the integer in the denominator is not zero, also denoted by
=
Every rational number can be written as a repeating decimal, whereas no irrational number
can be written in this way. For example,
, = 0.272727 . . . , and
are examples of rational numbers written as repeating decimals.
4- An irrational number is a number that cannot be expressed as a ratio a/b,
where a and b are integers and b is non-zero. The set of irrationals is denoted by Irr or
For example, √ √
.
,e
5- Real numbers
Real Numbers are made up of rational numbers and irrational numbers and denoted by
.
.
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6- The Number Line
We may use the number line to represent all the real numbers graphically; each real number
corresponds to exactly one point on the number line.
and - are not real numbers because
there is no point on the number line corresponding to either of them.
7- , denoting the set of all complex numbers:
For example,
.
Definition: The set that contains no elements is called the empty set and is denoted by .
Definition: A set is a subset of another set , written symbolically as
,iff every
element of is an element of . If contains at least one element that is not in , then
is said to be a proper subset of .
Definition: A set and a set are equal if
an element of and vice versa.
Definition: The union of two sets
belong to either or , that is,
and
and , denoted by
. Thus, every element of
, is the set of elements that
A
Definition: The intersection of two sets
that belong to both and . Thus
and , denoted by
is
B
, is the set of elements
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Intervals
Definition A subset of the realline is called an interval if it contains at least two numbers and
also contains all real nubers between any two of its elements.
Types of intervals
Notation
Set description
Type
Finite:
Open
Closed
Half-open
Half-open
Infinite:
Open
Closed
Open
Closed
set of all real
numbers
Relation
Both open
and closed
Picture
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Definition: Let
denoted by
and be any two non-empty sets, then the relation
is a subset from
Example 1.3: Let
and
from
to
. Then each of
and
is a relation from
to .
Note: A Relation can be represented by a set of ordered pairs of the form
Inequalities:
Definition: The expression,
; in words, ( is less than ) means lies to the right of
on the line number. The expression
; in words ( is greater than ) means is to the
right of on the number line.
if either
or
.
if either
or
.
A number,
is positive if
:
Rules for Inequalities
If
and are reals, then
If
and
are positive or negative, then
.
Solving Inequalities
The process of finding the interval or intervals of numbers that satisfy an inequality in x is called
solving the inequality.
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Example 1.4: Solve the inequality
.
Solution: Factoring yields
. In order that the product be positive, the
factors must both be positive or both negative. That is, we must have either
and
or
and
.
In the first case we have
. In the second we must have
Therefore
if
or
.
, and the solution set is given by
.
Example 1.5: Solve the inequalities
Solution: In this problem we are asked to solve two inequalities:
and
.
It is necessary to find the set of numbers for which the inequalities are satisfied. This
set is called the solution set of the inequalities. We start with
or
.
Thus the solution set is the half-open interval
. Note that each of the steps in the
computation served to simplify the term containing x.
Example 1.6: Solve the following inequalities. Express the solution sets in terms of intervals
and graph them:
(b)
Solution:
(c)
.
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(a)
The solution set is the interval
(b)
.
The solution set is the interval
(c)
The fraction
is undefined at
and is 0 at
. Between these numbers it is positive
if the numerator and denominator have the same sign, and negative if they have opposite sign.
It is easiest to organize this sign information in a chart:
1
+
+
+
0
-
-
0
+
+
+
-
undefined
+
0
-
Thus the solution set of the given inequality is the interval (
]
Examples 1.7: Solving a quadratic inequality
Solution:
Method1: we can write
Since
negative.
as form
means two terms
and
and
or
or
.
and
either positive or both
and
and
.
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The solution set in this case is
or
the solution set in this case is
Then solution set is
.
1
2
Method2:
1
+++++++
----------
+++++++
2
0
3
Then the solution set is
Example 1.8: Solving a quadratic inequality
Solution:
Method1: we can write
as form
Since
means two terms
and
.
and
or
and
and
or
The solution set in this case is
and
.
or the solution set in this case is
Then the solution set is
.
-1
Method2:
as follows
4
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-1
+++++++
+++++++
----------
-2
0
4
5
Then the solution set is
Example 1.9: Solve
( x  2)( x  1)
0
( x  3)
Solution: we can’t multiply by ( x  3) because we don’t know the value of (x ) if it’s (-) or
(+) necessary to change the direction of inequality () .
If ( x  2)  0  x  2
If ( x  1)  0  x  1
If ( x  3)  0  x  3
S  [1,2]  (3, )
Example 1.10: Solve
Solution:
1
2

x2 x3
1
2
( x  3)  2 ( x  2)
x7

0
0
0
x2 x3
( x  2) ( x  3)
( x  2) ( x  3)
If ( x  7)  0  x  7
If ( x  2)  0  x  2
If ( x  3)  0  x  3
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 The solution set is S  (,3)  (2,7)
Absolute value:
Definition: The absolute value of a number , denoted by | | which is defined by the
following formula.
| |
{
For example | |
|
|
.
Some properties of the absolute value
Let
and
be any real numbers then:
| | √
| | | || |
|
| | | | |
|
| || | | ||
| |
| |
Example 1.11: Solve x  x  3
Solution: If x  0 then we have x  x  3  0  3 impossible
If x  0 then we have  x  x  3  2 x  3  x 
 The solution set is x 
3
2
3
2
Example 1.12: Solve (a) |
|
(b) |
|
.
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Department of IT
Solution:
|
(a) |
solutions are
(b) |
. Thus, either
or
. The
and
|
. Thus the solutions lie in the interval [
Example 1.13: Solve the inequality |
Solution: |
|
].
|
if and only if
or
or
Then the solution set is
-10
6
Example 1.14: Solve the equation |
Solution: |

|
|
. The solutions are

. Thus, either
and
|
Example 1.15: Solve the inequality|
Solution: |
.
.
.
|
.
Example 1.16: Solve the inequality |
Solution: |
or
| 
or
.
.
The solution set is
Example 1.1: Solve | 2 x  3 || x  2 |
| and show the solution set on the real line.
.
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Solution:  | x  2 | 2 x  3 | x  2 | .
Case I: x  2  0  ( x  2)  2 x  3  x  2  2  3x  3  2 x  2 
Case II: x  2  0  x  2  2 x  3  ( x  2) 
So
1
 x  5, is the solution.
3
1
 x  5.
3
1
 x and x  5 (impossible).
3