Math 7 inequalities and intervals

Let’s Start!

Inequalities & Intervals

A Mathematics 7 Lecture

What’s an inequality?

• It is a range of

values, rather than

ONE set number

• It is an algebraic

relation showing that

a quantity is greater

than or less than

another quantity.

Inequalities and Intervals

Less than

Greater than

Less than OR EQUAL TO

Greater than OR EQUAL TO

Inequality Symbols


True or false?

Inequality Symbols

5 4

3 2

5 4

6.5 6.4

3 3

3 2

5 6

3 3

True

False

False

True

False

True

False

True


INEQUALITIES AND KEYWORDS

< >

•less than

•fewer

than

•greater

than

•more

than

•exceeds

•less than

or equal to

•no more

than

•at most

•greater

than or

equal to

•no less

than

•at least

Keywords


Keywords

Examples Write as inequalities.

1. A number x is more than 5 x > 5

2. A number x increased by 3 is fewer

than 4x + 3 < 4

3. A number x is at least 10 x 10

4. Three less than twice a number x is at

most 7

2x 3 7


Recall: order on the number line

On a number line, the number on the

right is greater than the number on

the left.

If a and b are numbers on the number line so that

the point representing a lies to the left of the

point representing b, then

a < b or b > a.

Graphs of Inequalities


Given a real number a and any real

number x:


all values of x to the

LEFT of a

x > ax < a

a


RIGHT of a

The point is

has a hole

because a is

excluded



number x:



LEFT of a,

INCLUDING a

x ax a

a


RIGHT of a,

INCLUDING a

The point is

shaded because

a is included



number x:


a

darken the part of

the number line

that is to the

RIGHT of the

constant

Place a point

with a HOLE at

x = a

x a


Place a point

with a HOLE at

x = a


number x:


a

darken the part of

the number line

that is to the LEFT

of the constant

x a



number x:


a

darken the part of

the number line

that is to the

RIGHT of the

constant

Place a point

with a SHADE

at x = a

x a


Place a point

with a SHADE

at x = a


number x:


a

darken the part of

the number line

that is to the LEFT

of the constant

x a



number x:


a

darken the part of

the number line

that is to the

RIGHT of the

constant

Place a point

with a SHADE

at x = a

x a



Let’s

compare

graphs!


Examples


x < 0

x > 2


Examples


Linear Inequalities

x 5

x 3

Check your understanding

Sketch the graph of the following

inequalities on a number line.

1. x > 6

2. x 7

3. x 1

4. x < 8


State the inequality represented by

the given graph.

1.

2.

3.

4.

• These are also called double

inequalities.

• These inequalities represent

“betweeness” of values; i.e., values

between two real numbers

Compound Inequalities



Linear Inequalities

a x b x is between a and b

x is greater than a and

less than b

a x b x is between a and b

inclusive

x is greater than or equal

to a and less than or

equal to b


We can also have:

a x b x is greater than a and

less than or equal to b

a x b x is greater than or equal

to a and less than b



Graphs

a x b

a x b

a x b

a x b


Example:


This inequality means that x is

BETWEEN 2 and 3

2 3x

This also means that x is GREATER

than 2 and LESS THAN 3


Example:


Linear Inequalities

2 3x

Example:


2 3x

This inequality means that x is

BETWEEN 2 and 3 INCLUSIVE


Example:


2 3x

2 3x



Sketch the graph of the following

inequalities on a number line.

1. 5 < x < 5

2. 4 x 7

3. 3 < x 1

4. 2 x < 8

Interval Notation

• The set of all numbers between two

endpoints is called an interval.

• An interval may be described either by an

inequality, by interval notation, or by a

straight line graph.

• An interval may be:

– Bounded:

• Open - does not include the endpoints

• Closed - does include the endpoints

• Half-Open - includes one endpoint

– Unbounded: one or both endpoints are infinity


Notations

• A parenthesis ( ) shows an open (not

included) endpoint

• A bracket [ ] shows a closed [included]

endpoint

• The infinity symbol () is used to describe

very large or very small numbers

+ or - all numbers GREATER than another

- all numbers GREATER than another

Note that “” is NOT A NUMBER!

Interval Notation


Interval Notation

INEQUALITY SET NOTATIONINTERVAL

NOTATION

x > a { x | x > a } (a, +)

x < a { x | x < a } (-, a)

x a { x | x a } [a, +)

x a { x | x a } (-, a]


Unbounded Intervals

Interval Notation

INEQUALITY SET NOTATIONINTERVAL

NOTATION

a < x < b { x | a < x < b } (a, b)

a x b { x | a x b } [a, b]

a < x b { x | a < x b } (a, b]

a x < b { x | a x < b } [a, b)

Bounded Intervals


Interval Notation

Example:

This represents all numbers

GREATER THAN OR EQUAL TO 1

1,

In inequality form, this is

x 1


Interval Notation

Example:

The symbol before the –1 is a square bracket

which means “is greater than or equal to."

The symbol after the infinity sign is a parenthesis

because the interval goes on forever (unbounded)

and since infinity is not a number, it doesn't

equal the endpoint (there is no endpoint).


1,

Interval Notation

Example:

Write the following inequalities using

interval notation

2x 2,

2x ,2

2x 2,

2x ,2


Interval Notation

Example:


interval notation

0 2x 0,2

0 2x 0,2

0 2x 0,2

0 2x 0,2


Interval Notation

Example:


interval notation

0 2x 0,2

0 2x 0,2

0 2x 0,2

0 2x 0,2


Interval Notation

Example:

Graph the following intervals:

(, 0)

[3, +)


Interval Notation

Linear Inequalities

Example:

Graph the following intervals:

2,3

1,6


Write the following using interval

notation, then sketch the graph.

1. 1 < x < 1

2. 4 x < 7

3. x 2

4. x > 6

Thank you!

Date post:	18-Jul-2015
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Math 7 inequalities and intervals

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