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Chapter 1
Numbers and Inequalities
This chapter reviews real numbers, inequalities, intervals, and absolute values
...
1 Real Numbers
The real numbers can be represented geometrically as points on a number line called
the real line
...

We distinguish four special subsets of real numbers
...
};
2) The integers ℤ = {
...
};
3) The rational numbers ℚ, namely the numbers that can be expressed in the form
of a fraction , where
1
,
2

are integers and  0
...
For
examples
, √2, √5,

log 3
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2 Inequalities
If

and

are real numbers and if

greater than or equal to
≥

or

or

−

is a nonnegative number, we say that

is less than or equal to , and write, respectively,

≤
...

1

>

Geometrically,

if the point on the real axis corresponding to

lies to the right

of the point corresponding to
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Rules of inequalities
If , , and

are any given real numbers, then:

1) Either

< ,

2) < ⟹

=
±

<

> ;
± ;

3)

<

>0⟹

<

;

4)

<

<0⟹

>



5)

>0⟹



: <

⟹− >− ;

> 0;

6) If a and b are both positive or negative, then

< ⟹

>
...
3 Intervals
A subset of the real line is called an interval if it contains at least two numbers and
contains all the real numbers lying between any two of its elements
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Types of intervals
Notation
( , )
Finite:

Infinite:

Set description
{ : < < }

Type
Open

[ , ]

{ :

≤

≤ }

Closed

[ , )

{ :

≤

< }

Half-open

( , ]

{ :

<

≤ }

Half-open

( , ∞)

{ :

> }

Open

[ , ∞)

{ :

≥ }

Closed

Mathematical Analysis

Picture

2

(−∞, )

{ :

< }

Open

(−∞, ]

{ :

≤ }

Closed

(−∞, ∞)

ℝ
Real number

Open
Closed

1
...

Example 1
...

6
) 2 − 1 < + 3
) − < 2 + 1
)
≥5
3
−1
Solution
2 −1<

a)

+3

Add 1 to both sides, we have
2 <
Subtract

+ 4

from both sides, we get
<4

The solution set is the open interval (−∞, 4) (see Figure 1
...
1

b)

−

3

<2 +1

Multiply both sides by 3, we obtain
− < 6 +3
Add

to both sides, we have

Mathematical Analysis

3

0 <7 +3
Subtract 3 from both sides, we get
−3 < 7
Divide by 7, we have
−

3
<
7

The solution set is the open interval − , ∞ (see Figure 1
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2

c)

The inequality

≥ 5 can hold only if

> 1, because otherwise

is

undefined or negative
...

5
The solution set is the half-open interval 1,

(see Figure 1
...
3
Mathematical Analysis

4

Example 1
...


=

or If 2 − 7 < 0 ⟹

>

and

<

and
3 +2
< 0
2 −7

3 +2
< 0
2 −7

Multiply both sides by 2 − 7

Multiply both sides by 2 − 7

3 +2< 0⟹

<−

2
3

3 +2> 0⟹

Mathematical Analysis

2
3

2 7
= − ,
3 2

=Φ
The solution set is the interval

>−

=

∪

= − ,


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5 Absolute Value
The absolute value of a number , denoted by | |, is defined by the formula
| |=

,
− ,

≥ 0;
< 0
...

to 0 on the real number

line
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2) | − | = the distance between x and y on the real line
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The absolute value has the following properties
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The following statements are all consequences of the definition of absolute value and
are often helpful when solving equations or inequalities involving absolute values
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Example 1
...
4)
...
4
|2 − 3| ≥ 1

b)
2 −3≥1

2 − 3 ≤ −1

or

2 ≥4

2 ≤2

≥2

≤1

Add 3 to both sides
Divide by 2

The solution set is the interval (−∞, 1]⋃[2, ∞) (see Figure 1
...


Figure 1
...
5
...


b) We have
+
If

>2⟹

+ 2 > 0
⟹

+

− 2 > 0 ⟹ ( + 2)( − 1) > 0

−1> 0

> −2

or If

>1

⟹

= (1, ∞)
The solution set is the interval

+ 2 < 0
< −2

−1<0
<1

= (−∞, −2)
=

∪

= (1, ∞) ∪ (−∞, −2)
...


−

− 6 = ( − 3)( + 2) ≥ 0,
8

it means that
( − 3) ≥ 0
≥ 3
So

( + 2) ≥ 0
≥ −2
= [3, ∞)

Then the solution set is

Mathematical Analysis

⋃

or

( − 3) ≤ 0
≤ 3
So

( + 2) ≤ 0
≤ −2

= (−∞, −2]

= [3, ∞)⋃(−∞, −2]

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