Math 7 inequalities and intervals

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 4
x + 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

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 NOTATION
INTERVAL
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 NOTATION
INTERVAL
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:
Write the following inequalities using
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

Math 7 inequalities and intervals

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