This document provides an introduction to inequalities and intervals. It defines inequalities as ranges of values rather than single numbers, showing that a quantity is greater than or less than another. Symbols for inequalities like <, >, ≤, ≥ are introduced. Compound inequalities representing values between two numbers are explained. Interval notation is defined using brackets and parentheses to indicate whether endpoints are included or not. Examples of writing inequalities as intervals and graphing intervals on number lines are provided.
4. 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
5.
Less than
Greater than
Less than OR EQUAL TO
Greater than OR EQUAL TO
Inequality Symbols
Inequalities and Intervals
7. 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
8. 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
Inequalities and Intervals
9. 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
10. 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
11. 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
12. 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
13. 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
14. 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
15. 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
16. 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
22. • These are also called double
inequalities.
• These inequalities represent
“betweeness” of values; i.e., values
between two real numbers
Compound Inequalities
Inequalities and Intervals
23. 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
24. 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
30. 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
31. 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
32. 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
33. 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]
Inequalities and Intervals
Unbounded Intervals
34. 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
Inequalities and Intervals
36. 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,
37. Interval Notation
Example:
Write the following inequalities using
interval notation
2x 2,
2x ,2
2x 2,
2x ,2
Inequalities and Intervals
38. 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
39. 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