math 7 inequalities and intervals
TRANSCRIPT
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
Inequalities and Intervals
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 Intervals
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
Inequalities and Intervals
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
Inequalities and Intervals
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
Inequalities and Intervals
Given a real number a and any real
number x:
Graphs of Inequalities
all values of x to the
LEFT of a
x > ax < a
a
all values of x to the
RIGHT of a
The point is
has a hole
because a is
excluded
Inequalities and Intervals
Given a real number a and any real
number x:
Graphs of Inequalities
all values of x to the
LEFT of a,
INCLUDING a
x ax a
a
all values of x to the
RIGHT of a,
INCLUDING a
The point is
shaded because
a is included
Inequalities and Intervals
Given a real number a and any real
number x:
Graphs of Inequalities
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
Inequalities and Intervals
Place a point
with a HOLE at
x = a
Given a real number a and any real
number x:
Graphs of Inequalities
a
darken the part of
the number line
that is to the LEFT
of the constant
x a
Inequalities and Intervals
Given a real number a and any real
number x:
Graphs of Inequalities
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
Inequalities and Intervals
Place a point
with a SHADE
at x = a
Given a real number a and any real
number x:
Graphs of Inequalities
a
darken the part of
the number line
that is to the LEFT
of the constant
x a
Inequalities and Intervals
Given a real number a and any real
number x:
Graphs of Inequalities
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
Inequalities and Intervals
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
• These are also called double
inequalities.
• These inequalities represent
“betweeness” of values; i.e., values
between two real numbers
Compound Inequalities
Inequalities and Intervals
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
Compound Inequalities
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
Inequalities and Intervals
Example:
Compound Inequalities
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
Inequalities and Intervals
Example:
Compound Inequalities
2 3x
This inequality means that x is
BETWEEN 2 and 3 INCLUSIVE
Inequalities and Intervals
Check your understanding
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
Inequalities and Intervals
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
Inequalities and Intervals
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]
Inequalities and Intervals
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
Inequalities and Intervals
Interval Notation
Example:
This represents all numbers
GREATER THAN OR EQUAL TO 1
1,
In inequality form, this is
x 1
Inequalities and Intervals
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).
Inequalities and Intervals
1,
Interval Notation
Example:
Write the following inequalities using
interval notation
2x 2,
2x ,2
2x 2,
2x ,2
Inequalities and Intervals
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
Inequalities and Intervals
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
Inequalities and Intervals
Check your understanding
Write the following using interval
notation, then sketch the graph.
1. 1 < x < 1
2. 4 x < 7
3. x 2
4. x > 6