Math 11

Ordering of the set of real numbers

The symbols < and >

<

>

Less than

Greater than

The symbols < and >

• If a and b are real numbers

1.a < b if and only if b – a is positive

a = 10 and b = 15

The symbols < and >• If a and b are real numbers

2.a > b if and only if a-b is positive

a = 22 and b = 8

The symbols ≤ and ≥

≤

≥

Less than or equal to

Greater than or equal to

<, >, ≤ and ≥

<>≤≥

Strict inequalities

Non Strict inequalities

Continued Inequality

• If a < x and x < b then

a < x < b

a = 5, x = 8 and b = 10

Inequalities and Set Notation

The set of all x such that x is greater than -9 and less than 8

{x | 8 > x > -9 }

Inequalities and Set Notation

{x | 2x+ 4 ≥ 0}

The set of all x such that 2x+4 is nonnegative.

Inequalities and Set Notation

The set of all a such that a-2 is greater than -5 and less than or

equal to 7

{a | 7 ≥ a – 2 > -5 }

Inequalities and Set Notation

{z | 15 ≥ 2z+5 ≥ -1}

The set of all z such that 2z+5 is between and including

-1 and 15.

Geometric Interpretation

Origin = 0

REAL NUMBER

LINE

0

Geometric Interpretation{ 6 4}x x− < ≤

(-6,4]

0-6 4

( ]

Interval notation

Geometric Interpretation

{ 1 9}x x≤ ≤

[1,9]

0 1 9

[ ]

Geometric Interpretation

(a,b) [a,b]

(a,b] [a,b)

Geometric Interpretation

(a,+∞) [a,+∞)

(- ∞,b] (- ∞,b)

How do we represent it in a real number line?

Geometric Interpretation

(a,+∞)

a

(

Geometric Interpretation

[a,+∞)

a

[

Geometric Interpretation

(- ∞,b]

b

]

Geometric Interpretation

(- ∞,b)

b

)

Inequalities and Set Operations

{ | 2} { | 12}x x x x> ∪ <

What if we combine set operations and inequalities?

Inequalities and Set Operations

{ | 2} { | 12}x x x x> ∩ <

{ | 3} { | 10}x x x x> ∪ <

inequalities and set operations

{ | 8} { | 0}x x x x> − ∩ ≤