Chapter 3
INEQUALITIES
Inequality
• A statement that involves
less than ( < ) or greater
than ( > )
• Most often used to answer
questions involving the
idea of “at least” or “no
more than”.
•
•
•
•
> … more than
> … at least
a minimum of
< … smaller than
< … no more than
at most
Graphing inequalities
In general …
• Put a circle (< or >) or a
dot (< or >) at the
endpoint.
• Make an arrow in the
direction of the inequality.
Solving inequalities
• In most cases, you just
solve like an equation.
2x – 19 < 53
+ 19 +19
2x
< 72
2
2
x
< 36
•
There is one important
exception: If you divide by
a negative number,
change the symbol in the
answer.
• > becomes <
• < becomes >
• > becomes <
• < becomes >
-5x + 73 > 128
-73 -73
-5x
> 55
-5
-5
x
< -11
Note the answer is <, not >.
(Technically you also switch
the sign if you ever multiply by
a negative, but dividing is
what usually comes up.)
•
You only flip the sign when
the number by “x” is
negative.
3x + 11 > 2
- 11 -11
3x
> -9
3
3
x
> -3
We don’t flip the symbol,
because the number we
divided by (the number by “x”)
isn’t negative.
Compound Inequalities
AND
 Both parts must be true at
the same time.
 Graph is most often a line
segment
x > 0 and x < 3
Graph x > 1 and x < 6
Graph x > 1 and x < 6
Graph x < 9 and x > 5
Graph x < 9 and x > 5
Graph x < 2 and x > 3
Graph x < 2 and x > 3
NO SOLUTION (don’t overlap)
On an AND problem, you’re
always looking for where the
graphs of the two parts
overlap.
OR
 Either part can be true
 Graph is most often two
rays (arrows) in opposite
directions
Graph x < -3 or x > 2
Graph x < -3 or x > 2
Graph x > 6 or x > 10
Graph x > 6 or x > 10
These go the same direction.
The final answer is just the
larger of the two graphs.
Absolute Value
The distance a number is from
zero.
|5|=5
| -2 | = 2
|0|=0
There are normally 2 solutions
to absolute value problems
that involve variables.
For example, if | x | = 3
then x = 3 or x = -3
Solve
| x | = 13
|x|=9
| x | = -2
Solve
| x | = 13
x = 13 or -13
|x|=9
x=+9
| x | = -2
NO SOLUTION
Solve | 3x – 7 | = 22
Solve | 3x – 7 | = 22
Make it into 2 problems.
3x – 7 = 22
or 3x – 7 = -22
3x = 29
3x = -15
29
x = /3
or
x = -5
Solve | x | < 4
We want numbers closer to 0
than 4
So, x < 4 and x > -4
In general < problems with
absolute value are AND
problems.
The solution is typically
between 2 numbers.
Solve | 2x + 11 | < 19
Solve | 2x + 11 | < 19
2x + 11 < 19 and 2x + 11 > -19
2x
< 8
2x > -30
x
< 4 and
x
> -15
Solve | x | > 5
We want numbers at least as
far from 0 as 5
So, x > 5 or x < -5
In general > problems with
absolute value are OR
problems.
The solution is rays that
extend out from 2 numbers.
Solve | 5x + 9 | > 14
Solve | 5x + 9 | > 14
5x + 9 > 14 or
5x > 5
x> 1
5x + 9 < -14
5x < -23
-23
x < /5
You always get the second
problem by flipping the
direction of the inequality and
making the constant at the
end negative.
| 2x – 7 | > 5
2x – 7 > 5 or 2x – 7 < -5
Set  a collection of things
where you can tell exactly
what is and isn’t part of the
collection.
We usually use capital
letters to stand for sets.
Absolute Value
The distance a number is from
zero.
|5|=5
| -2 | = 2
|0|=0
Element (or member)  one
of the things that is part of
a set.
UNION
 Symbol 
 Combining EVERYTHING
in 2 or more sets
 elements in A or B (or both)
Example:
P = { 1, 2, 3, 4, 5 }
Q = { 5, 6, 7 }
P  Q = { 1, 2, 3, 4, 5, 6, 7 }
Example:
P = { 1, 2, 3, 4, 5 }
Q = { 5, 6, 7 }
PQ=
Example:
P = { 1, 2, 3, 4, 5 }
Q = { 5, 6, 7 }
P  Q = { 1, 2, 3, 4, 5, 6, 7 }
Intersection
 Symbol 
 Elements that are in 2 or
more sets AT THE SAME
TIME.
 What overlaps between
different sets
Example:
P = { 1, 2, 3, 4, 5 }
Q = { 5, 6, 7 }
PQ=
Example:
P = { 1, 2, 3, 4, 5 }
Q = { 5, 6, 7 }
PQ={5}
Example:
E = { 1, 2, 3, 4, 5 }
F = { 6, 7, 8, 9, 10 }
EF=
Example:
E = { 1, 2, 3, 4, 5 }
F = { 6, 7, 8, 9, 10 }
E  F = { } … the empty set
( E and F are called disjoint
sets.)
Who is in the intersection
of band and chorus?
What about the union?
Intersection = { Derek }
Union =
{ Lorrie, Sam, Raul, Derek,
Kyesha, Robin }