Download Chapter 3.1

Solving Inequalities
a<b
a<b
0
a < b means a is less than b
0
a < b means a is less than b
2
0
a < b means a is less than b
2
0
a < b means a is less than b
2 
0
a < b means a is less than b
2  3
0
a < b means a is less than b
2  3
0
a < b means a is less than b
1
0
a < b means a is less than b
1
0
a < b means a is less than b
1 
0
a < b means a is less than b
1  3
0
a < b means a is less than b
1  3
0
a < b means a is less than b
0
0
a < b means a is less than b
0
0
a < b means a is less than b
0 
0
a < b means a is less than b
0  3
0
a < b means a is less than b
0  3
0
a < b means a is less than b
3
0
a < b means a is less than b
3
0
a < b means a is less than b
3 
0
a < b means a is less than b
3  3
0
a < b means a is less than b
3  3
0
a < b means a is less than b
3
0
a < b means a is less than b
3
0
a < b means a is less than b
3 
0
a < b means a is less than b
3  0
0
a < b means a is less than b
3  0
0
a < b means a is less than b
3
0
a < b means a is less than b
3
0
a < b means a is less than b
3 
0
a < b means a is less than b
3  2
0
a < b means a is less than b
3  2
0
a < b means a is less than b
This is true when a is to the left of b
on the number line.
0
a < b means a is less than b
This is true when a is to the left of b
on the number line.
0
a>b
0
a > b means a is greater than b
0
a > b means a is greater than b
3
0
a > b means a is greater than b
3
0
a > b means a is greater than b
3 
0
a > b means a is greater than b
3 2
0
a > b means a is greater than b
3 2
0
a > b means a is greater than b
3
0
a > b means a is greater than b
3
0
a > b means a is greater than b
3 
0
a > b means a is greater than b
3  3
0
a > b means a is greater than b
3  3
0
a > b means a is greater than b
2
0
a > b means a is greater than b
2
0
a > b means a is greater than b
2 
0
a > b means a is greater than b
 2  3
0
a > b means a is greater than b
 2  3
0
a > b means a is greater than b
This is true when a is to the right of
b on the number line.
0
a > b means a is greater than b
This is true when a is to the right of
b on the number line.
0
A replacement that makes an
inequality true is called a solution.
A replacement that makes an
inequality true is called a solution.
The set of all solutions is called the
solution set.
A replacement that makes an
inequality true is called a solution.
The set of all solutions is called the
solution set.
When we have found the set of all
solutions of an inequality,
A replacement that makes an
inequality true is called a solution.
The set of all solutions is called the
solution set.
When we have found the set of all
solutions of an inequality, we say
we have solved the inequality.
Determine whether the number is a
solution of x < 2.
0
Determine whether the number is a
solution of x < 2.
0
Determine whether the number is a
solution of x < 2.
 2.7
0
Determine whether the number is a
solution of x < 2.
 2.7
0
Determine whether the number is a
solution of x < 2.
 2.7
yes
0
Determine whether the number is a
solution of x < 2.
0
Determine whether the number is a
solution of x < 2.
2
0
Determine whether the number is a
solution of x < 2.
2
no
0
Determine whether the number is a
solution of x  6.
0
Determine whether the number is a
solution of x  6.
0
Determine whether the number is a
solution of x  6.
6
0
Determine whether the number is a
solution of x  6.
6
yes
0
Determine whether the number is a
solution of x  6.
0
Determine whether the number is a
solution of x  6.
4

3
0
Determine whether the number is a
solution of x  6.
4

3
0
Determine whether the number is a
solution of x  6.
4

3
no
0
Some solutions of x  2 are
0
2
Some solutions of x  2 are
0
2
Some solutions of x  2 are
0,
0
2
Some solutions of x  2 are
0,
0
2
Some solutions of x  2 are
0,  3,
0
2
Some solutions of x  2 are
0,  3,
3
0
2
Some solutions of x  2 are
0,  3, 1,
3
0
2
Some solutions of x  2 are
0,  3, 1,
3
0
1
2
Some solutions of x  2 are
0,  3, 1, 0.45,
3
0
1
2
Some solutions of x  2 are
0,  3, 1, 0.45,
3
0
1
0.45
2
Some solutions of x  2 are
0,  3, 1, 0.45,   ,
3
0
1
0.45
2
Some solutions of x  2 are
0,  3, 1, 0.45,   ,

3
0
1
0.45
2
Some solutions of x  2 are
0,  3, 1, 0.45,   ,

3
0
1
0.45
2
5
8
Some solutions of x  2 are
0,  3, 1, 0.45,   ,
5
8

3
0
1
0.45
2
5
8
Some solutions of x  2 are
0,  3, 1, 0.45,   ,
and so on.
5
8

3
0
1
0.45
2
5
8
Some solutions of x  2 are
0,  3, 1, 0.45,   ,
and so on.
5
8

3
0
1
0.45
2
5
8
In fact, there are infinitely many
real numbers that are solutions.
In fact, there are infinitely many
real numbers that are solutions.
Because we cannot list them all
individually,
In fact, there are infinitely many
real numbers that are solutions.
Because we cannot list them all
individually, it is helpful to
make a drawing that represents
all the solutions.
A graph of an inequality is a
drawing that represents its
solutions.
A graph of an inequality is a
drawing that represents its
solutions. An inequality in one
variable can be graphed on a
number line.
A graph of an inequality is a
drawing that represents its
solutions. An inequality in one
variable can be graphed on a
number line. An inequality if
two variables can be graphed on
a coordinate plane;
A graph of an inequality is a
drawing that represents its
solutions. An inequality in one
variable can be graphed on a
number line. An inequality if
two variables can be graphed on
a coordinate plane; we will
study such graphs in Chapter 11.
Graph x  2
0
Graph x  2
0
Graph x  2
0
Graph x  2
0
Graph x  - 3
0
Graph x  - 3
0
Graph x  - 3
0
Graph x  - 3
0
Graph - 3  x  2
Graph - 3  x  2
-3  x
Graph - 3  x  2
- 3  x and
Graph - 3  x  2
- 3  x and x  2
Graph - 3  x  2
- 3  x and x  2
0
Graph - 3  x  2
- 3  x and x  2
0
Graph - 3  x  2
- 3  x and x  2
0
Graph - 3  x  2
- 3  x and x  2
0
Graph - 3  x  2
- 3  x and x  2
0
Graph - 3  x  2
- 3  x and x  2
0
For any real numbers a, b, and c,
For any real numbers a, b, and c,

For any real numbers a, b, and c,

ab
For any real numbers a, b, and c,

a  b is equivalent to
For any real numbers a, b, and c,

a  b is equivalent to a  c  b  c
For any real numbers a, b, and c,

a  b is equivalent to a  c  b  c
ab
For any real numbers a, b, and c,

a  b is equivalent to a  c  b  c
a  b is equivalent to
For any real numbers a, b, and c,

a  b is equivalent to a  c  b  c
a  b is equivalent to a  c  b  c
For any real numbers a, b, and c,
For any real numbers a, b, and c,

For any real numbers a, b, and c,
ab

For any real numbers a, b, and c,

a  b is equivalent to
For any real numbers a, b, and c,

a  b is equivalent to a  c  b  c
For any real numbers a, b, and c,

a  b is equivalent to a  c  b  c
ab
For any real numbers a, b, and c,

a  b is equivalent to a  c  b  c
a  b is equivalent to
For any real numbers a, b, and c,

a  b is equivalent to a  c  b  c
a  b is equivalent to a  c  b  c
Solve x  2  8
Use set builder notation.
Then graph.
Solve
x  2  8
Solve
x  2  8
Solve
x  2  8
2
Solve
x  2  8
2 2
Solve
x  2  8
2 2
x
Solve
x  2  8
2 2
x

Solve
x  2  8
2 2
x
 6
Solve x  2  8
Use set builder notation.
Then graph.
0
Solve x  2  8
Use set builder notation.
Then graph.
x6
0
Solve x  2  8
Use set builder notation.
Then graph.
x6
0
Solve x  2  8
Use set builder notation.
Then graph.
x6
0
Solve x  2  8
Use set builder notation.
Then graph.
x6
0
Solve x  2  8
Use set builder notation.
Then graph.
x6
0
Solve x  2  8
Use set builder notation.
Then graph.
x6
0


Solve x  2  8
Use set builder notation.
Then graph.
x6
0
x

Solve x  2  8
Use set builder notation.
Then graph.
x6
0
x
|

Solve x  2  8
Use set builder notation.
Then graph.
x6
0
x
| x

Solve x  2  8
Use set builder notation.
Then graph.
x6
0
x
| x 

Solve x  2  8
Use set builder notation.
Then graph.
x6
0
x
| x  6

Solve 3x  1  2x - 3
Use set builder notation.
Then graph.
0
Solve
3x  1  2x  3
Solve
3x  1  2x  3
Solve
3x  1  2x  3
 2x
Solve
3x  1  2x  3
 2x
 2x
Solve
3x  1  2x  3
 2x
 2x
x
Solve
3x  1  2x  3
 2x
 2x
x 1
Solve
3x  1  2x  3
 2x
 2x
x 1 
Solve
3x  1  2x  3
 2x
 2x
x 1 
 3
Solve
3x  1  2x  3
 2x
 2x
x 1 
 3
1
1
Solve
3x  1  2x  3
 2x
 2x
x 1 
 3
1
1
x
Solve
3x  1
 2x
x 1
1
x
 2x  3
 2x
 3

1

Solve
3x  1
 2x
x 1
1
x
 2x  3
 2x
 3

1

 4
Solve
3x  1  2x
 2x
 2x
x 1 
1
x

x  -4
 3
 3
1
 4
Solve 3x  1  2x - 3
Use set builder notation.
Then graph.
0
Solve 3x  1  2x - 3
Use set builder notation.
Then graph. x  -4
0
Solve 3x  1  2x - 3
Use set builder notation.
Then graph. x  -4
0
Solve 3x  1  2x - 3
Use set builder notation.
Then graph. x  -4
0
Solve 3x  1  2x - 3
Use set builder notation.
Then graph. x  -4
0
Solve 3x  1  2x - 3
Use set builder notation.
Then graph. x  -4
0
Solve 3x  1  2x - 3
Use set builder notation.
Then graph. x  -4
0


Solve 3x  1  2x - 3
Use set builder notation.
Then graph. x  -4
0
x

Solve 3x  1  2x - 3
Use set builder notation.
Then graph. x  -4
0
x
|

Solve 3x  1  2x - 3
Use set builder notation.
Then graph. x  -4
0
x
| x

Solve 3x  1  2x - 3
Use set builder notation.
Then graph. x  -4
0
x
| x 

Solve 3x  1  2x - 3
Use set builder notation.
Then graph. x  -4
0
x
| x   4
1 5
Solve x  
3 4
Use set builder notation.
Then graph.
0
Solve
1 5
x 
3 4
Solve
1 5
x 
3 4
x
1

3
5
4
Solve
1 5
x 
3 4
12  x 
1

3
5
4
Solve
1 5
x 
3 4
1
12  x  12  
3
5
4
Solve
1 5
x 
3 4
1
5
12  x  12   12 
3
4
1 5
x 
3 4
Solve
12
1
1
5
 x  12   12 
3
4
1 5
x 
3 4
Solve
12
1
 x 
12
1
1
5
  12 
3
4
1 5
x 
3 4
Solve
12
1
 x 
12
1
1
 
3
12
1
5

4
1 5
x 
3 4
Solve
12
1
 x 
12
1

1
3

12
1
5

4
1 5
x 
3 4
Solve
12
1
 x 
12
1

1
3

12
1

5
4
1 5
x 
3 4
Solve
12
1
 x 
12
1

1
3

12
1

5
4
1 5
x 
3 4
Solve
12
1
 x 
12
1

1
3

12
1

5
4
1 5
x 
3 4
Solve
12
1
 x 
4
12
1

1
3

12
1

5
4
1 5
x 
3 4
Solve
12
1
 x 
4
12
1

1
3
1

12
1

5
4
1 5
x 
3 4
Solve
12
1
 x 
4
12
1

1
3
1

12
1

5
4
1 5
x 
3 4
Solve
12
1
 x 
4
12
1

1
3
1

12
1

5
4
1 5
x 
3 4
Solve
12
1
 x 
4
12
1

1
3
1

3
12
1

5
4
1 5
x 
3 4
Solve
12
1
 x 
4
12
1

1
3
1

3
12
1

5
4
1
1 5
x 
3 4
Solve
12
1
 x 
12x
4
12
1

1
3
1

3
12
1

5
4
1
1 5
x 
3 4
Solve
12
1
 x 
4
12
1

12x  4
1
3
1

3
12
1

5
4
1
1 5
x 
3 4
Solve
12
1
 x 
4
12
1

1
3
1
12x  4 

3
12
1

5
4
1
1 5
x 
3 4
Solve
12
1
 x 
4
12
1

1
3
1

3
12
1
12x  4  15

5
4
1
Solve
12x  4  15
Solve
12x  4  15
4 4
Solve
12x  4  15
4 4
12x
Solve
12x  4  15
4 4

12x
Solve
12x  4  15
4 4
 11
12x
Solve
12x  4  15
4 4
 11
12x
12x

11
Solve
12x  4  15
4 4
 11
12x
12x
12

11
12
Solve
12x  4  15
4 4
 11
12x
12x
12
x

11
12
Solve
12x  4  15
4 4
 11
12x
12x
12

x 
11
12
Solve
12x  4  15
4 4
 11
12x

11
12
x 
11
12
12x
12
1 5
Solve x  
3 4
Use set builder notation.
Then graph.
0
1 5
Solve x  
3 4
Use set builder notation.
11
Then graph.
x  12
0
1 5
Solve x  
3 4
Use set builder notation.
11
Then graph.
x  12
0
1 5
Solve x  
3 4
Use set builder notation.
11
Then graph.
x  12
0
1 5
Solve x  
3 4
Use set builder notation.
11
Then graph.
x  12
0
1 5
Solve x  
3 4
Use set builder notation.
11
Then graph.
x  12
0
1 5
Solve x  
3 4
Use set builder notation.
11
Then graph.
x  12
0


1 5
Solve x  
3 4
Use set builder notation.
11
Then graph.
x  12
0
x

1 5
Solve x  
3 4
Use set builder notation.
11
Then graph.
x  12
0
x
|

1 5
Solve x  
3 4
Use set builder notation.
11
Then graph.
x  12
0
x
| x

1 5
Solve x  
3 4
Use set builder notation.
11
Then graph.
x  12
0
x
| x 

1 5
Solve x  
3 4
Use set builder notation.
11
Then graph.
x  12
0
x
| x 
11
12

0
0
3
0
3
3
0
3 
3
0
3  4
3
0
3  4
3
4
0
3  4
0
3
4
0
3  4
0 
3
4
0
3  4
0  4
3
4
0
3  4
0  4
4
3
4
4
3  4
0  4
4
0
3
4
4
3  4
0  4
4 
0
3
4
4
3  4
0  4
4  4
0
3
4
4
3  4
0  4
4  4
4
0
3
4
4
3  4
0  4
4  4
4 
0
3
4
4
3  4
0  4
4  4
 4  3
0
3
4
4 3
3  4
0  4
4  4
 4  3
0
3
4
4 3
3  4
0  4
4  4
 4  3
0
3
4
4
4 3
3  4
0  4
4  4
 4  3
0
3
4 
4
4 3
3  4
0  4
4  4
 4  3
0
3
4  3
4
4 3
0
3
3  4
4  3
0  4
4
4  4
 4  3
4
3
4 3
3  4
0  4
4  4
 4  3
0
3
4
4  3
1  4
3
4 3
3  4
0  4
4  4
 4  3
0
3
4
4  3
1  4
1  3
4 3
3  4
0
3
4  3
0  4
1  4
4  4
4
 4  3
4
1  3
4 3
3  4
0
3
4  3
0  4
1  4
4  4
4
 4  3
4
1  3
3
4 3
3  4
0  4
4  4
 4  3
0
3
4
4  3
1  4
1  3
 4  3
4 3
0
3
4
3  4
4  3
0  4
1  4  1  3
4  4
 4  3
 4  3
4 3
0
3
4
4  3
1  4  1  3
 4  3
4 3
0
3
4  3
4
4 3
0
3
4
4  3
1  4
1  3
4 3
0
3
4
4  3
1  4  1  3
For any real numbers a, b,
For any real numbers a, b,
and for any positive number c,
For any real numbers a, b,
and for any positive number c,
ab
For any real numbers a, b,
and for any positive number c,
a  b is equivalent to
For any real numbers a, b,
and for any positive number c,
a  b is equivalent to
ac  bc
For any real numbers a, b,
and for any positive number c,
a  b is equivalent to
ab
ac  bc
For any real numbers a, b,
and for any positive number c,
a  b is equivalent to
a  b is equivalent to
ac  bc
For any real numbers a, b,
and for any positive number c,
a  b is equivalent to
ac  bc
a  b is equivalent to
ac  bc
For any real numbers a, b,
For any real numbers a, b,
and for any negative number c,
For any real numbers a, b,
and for any negative number c,
ab
For any real numbers a, b,
and for any negative number c,
a  b is equivalent to
For any real numbers a, b,
and for any negative number c,
a  b is equivalent to
ac  bc
For any real numbers a, b,
and for any negative number c,
a  b is equivalent to
ab
ac  bc
For any real numbers a, b,
and for any negative number c,
a  b is equivalent to
a  b is equivalent to
ac  bc
For any real numbers a, b,
and for any negative number c,
a  b is equivalent to
ac  bc
a  b is equivalent to
ac  bc
Solve 4 x  28
Use set builder notation.
Then graph.
0
4x  28
4x  28
4x
28
4x  28
4x
4
28
4x  28
4x
4
28
4
4x  28
4x
4

28
4
4x  28
4x
4
x

28
4
4x  28
4x
4

x 
28
4
4x  28
4x
4

28
4
x  7
Solve 4 x  28
Use set builder notation.
Then graph.
x  7
0
Solve 4 x  28
Use set builder notation.
Then graph.
x  7
0
Solve 4 x  28
Use set builder notation.
Then graph.
x  7
0
Solve 4 x  28
Use set builder notation.
Then graph.
x  7
0


Solve 4 x  28
Use set builder notation.
Then graph.
x  7
0
x

Solve 4 x  28
Use set builder notation.
Then graph.
x  7
0
x
|

Solve 4 x  28
Use set builder notation.
Then graph.
x  7
0
x
| x

Solve 4 x  28
Use set builder notation.
Then graph.
x  7
0
x
| x 

Solve 4 x  28
Use set builder notation.
Then graph.
x  7
0
x
| x  7

Solve  2y  18
Use set builder notation.
Then graph.
0
 2y  18
 2y  18
2y
18
 2y  18
2y
2
18
 2y  18
2y
2
18
2
 2y  18
2y
2

18
2
 2y  18
2y
2
y

18
2
 2y  18
2y
2

y
18
2
 2y  18
2y
2

18
2
y  9
Solve 6  5y  7
Use set builder notation.
Then graph.
Solve 6  5y  7
Use set builder notation.
Then graph.
y  9
0
Solve 6  5y  7
Use set builder notation.
Then graph.
y  9
5
0
Solve 6  5y  7
Use set builder notation.
Then graph.
y  9
9
5
0
Solve 6  5y  7
Use set builder notation.
Then graph.
y  9
9
5
0
Solve 6  5y  7
Use set builder notation.
Then graph.
y  9
9
5
0
Solve 6  5y  7
Use set builder notation.
Then graph.
y  9
9
5

0

Solve 6  5y  7
Use set builder notation.
Then graph.
y  9
9
5
y
0

Solve 6  5y  7
Use set builder notation.
Then graph.
y  9
9
5
y
|
0

Solve 6  5y  7
Use set builder notation.
Then graph.
y  9
9
5
y
| y
0

Solve 6  5y  7
Use set builder notation.
Then graph.
y  9
9
5
y
| y 
0

Solve 6  5y  7
Use set builder notation.
Then graph.
y  9
9
5
y
| y 9
0
Solve 17  5 y  8 y  9
Use set builder notation.
Then graph.
0
Solve
17  5y  8y  9
Solve
17  5y  8y  9
 5y
Solve
17  5y  8y  9
 5y  5y
Solve
17  5y  8y  9
 5y  5y
17
Solve
17  5y  8y  9
 5y  5y

17
Solve
17  5y  8y  9
 5y  5y
 13y
17
Solve
17  5y  8y  9
 5y  5y
 13y  9
17
Solve
17  5y  8y  9
 5y  5y
 13y  9
17
Solve
17  5y  8y  9
 5y  5y
 13y  9
17
9
Solve
17  5y  8y  9
 5y  5y
 13y  9
17
9
9
Solve
17  5y  8y  9
 5y  5y
 13y  9
17
9
9
26
Solve
17  5y  8y  9
 5y  5y
 13y  9
17
9
9

26
Solve
17  5y  8y  9
 5y  5y
 13y  9
17
9
9
 13y
26
Solve
17  5y  8y  9
 5y  5y
 13y  9
17
9
9
 13y
26
26  13y
26  13y
26  13y
13y
26  13y
13y
26
26  13y
13y  26
26  13y
13y  26
13 y
26
26  13y
13y  26
13 y
13
26
13
26  13y
13y  26
13 y
13

26
13
26  13y
13y  26
13 y
13
y

26
13
26  13y
13y  26
13 y
13

y
26
13
26  13y
13y  26
13 y
13

26
13
y 2
Graph y  2
Graph y  2
0
Graph y  2
0
Graph y  2
0
Graph y  2
0


Graph y  2
0
y

Graph y  2
0
y
|

Graph y  2
0
y
| y

Graph y  2
0
y
| y 

Graph y  2
0
y
| y  2
Solve 3(x - 2) - 1  2 - 5(x  6)
Use set builder notation.
Then graph.
0
Solve 3x  2   1  2  5x  6 
Solve 3x  2   1  2  5x  6 
3x
Solve 3x  2   1  2  5x  6 
3x  6
Solve 3x  2   1  2  5x  6 
3x  6  1
Solve 3x  2   1  2  5x  6 
3x  6  1
 2
Solve 3x  2   1  2  5x  6 
3x  6  1
 2  5x
Solve 3x  2   1  2  5x  6 
3x  6  1
 2  5 x  30
Solve 3x  2   1  2  5x  6 
3x  6  1
3x
 2  5 x  30
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7 
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
 5x
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
 5x
 5x
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
 5x
 5x
8x
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
 5x
 5x
8x  7
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
 5x
 5x
8x  7 
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
 5x
 5x
8x  7 
 28
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
 5x
 5x
8x  7 
 28
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
 5x
 5x
8x  7 
 28
7
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
 5x
 5x
8x  7 
 28
7
7
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
 5x
 5x
8x  7 
 28
7
7
8x
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
 5x
 5x
8x  7 
 28
7
7
8x 
Solve 3x  2   1  2  5x  6 
3x  6  1  2  5 x  30
3x  7   5x  28
 5x
 5x
8x  7 
 28
7
7
8x 
 21
8x   21
8x   21
8x
21
8x   21
8x
8
21
8x   21
8x
8
21
8
8x   21
8x
8

21
8
8x   21
8x
8
x

21
8
8x   21
8x
8

x
21
8
8x   21
8x
8

21
8
x 
21
8
21
x


Graph
8
21
x


Graph
8
0
21
x


Graph
8
0
21
x


Graph
8
0
21
x


Graph
8
0


21
x


Graph
8
0
x

21
x


Graph
8
0
x
|

21
x


Graph
8
0
x
| x

21
x


Graph
8
0
x
| x 

21
x


Graph
8
0
x
| x 

21
8
Solve 16.3 - 7.2p  - 8.18
Use set builder notation.
Then graph.
0
Solve
16.3  7.2p   8.18
Solve
16.3  7.2p   8.18
Solve
16.3  7.2p   8.18
 7.2p
Solve
16.3  7.2p   8.18
 7.2p
 7.2p
Solve
16.3  7.2p   8.18
 7.2p
 7.2p
16.3
Solve
16.3  7.2p   8.18
 7.2p
 7.2p
16.3

Solve
16.3  7.2p   8.18
 7.2p
 7.2p
16.3
  8.18
Solve
16.3  7.2p   8.18
 7.2p
 7.2p
16.3
  8.18  7.2p
Solve
16.3  7.2p   8.18
 7.2p
 7.2p
16.3
  8.18  7.2p
Solve
16.3  7.2p   8.18
 7.2p
 7.2p
16.3
  8.18  7.2p
8.18
Solve
16.3  7.2p   8.18
 7.2p
 7.2p
16.3
  8.18  7.2p
8.18
 8.18
Solve
16.3  7.2p   8.18
 7.2p
 7.2p
16.3
  8.18  7.2p
8.18
 8.18
24.48
Solve
16.3  7.2p   8.18
 7.2p
 7.2p
16.3
  8.18  7.2p
8.18
 8.18
24.48

Solve
16.3  7.2p   8.18
 7.2p
 7.2p
16.3
  8.18  7.2p
8.18
 8.18
24.48
7.2p

Solve
16.3  7.2p   8.18
 7.2p
 7.2p
16.3
  8.18  7.2p
8.18
 8.18
24.48
7.2p

24.48  7.2p
24.48  7.2p
24.48  7.2p
7.2p
24.48  7.2p
7.2p 
24.48  7.2p
7.2p  24.48
24.48  7.2p
7.2p  24.48
7.2p
24.48
24.48  7.2p
7.2p  24.48
7.2p
7 .2
24.48
24.48  7.2p
7.2p  24.48
7.2p
7 .2
24.48
7 .2
24.48  7.2p
7.2p  24.48
7.2p
7 .2

24.48
7 .2
24.48  7.2p
7.2p  24.48
7.2p
7 .2
p

24.48
7 .2
24.48  7.2p
7.2p  24.48
7.2p
7 .2

p 
24.48
7 .2
24.48  7.2p
7.2p  24.48
7.2p
7 .2

24.48
7 .2
p  3.4
Graph p  3.4
Graph p  3.4
0
Graph p  3.4
0
Graph p  3.4
0
Graph p  3.4
0


Graph p  3.4
0
p

Graph p  3.4
0
p
|

Graph p  3.4
0
p
| p

Graph p  3.4
0
p
| p 

Graph p  3.4
0
p
| p  3.4
2
1 1
7
Solve x   x   2 x
3
6 2
6
Use set builder notation.
Then graph.
0
Solve
2
3
x   x   2x
1
6
1
2
7
6
Solve
2
3
2
3
x
x   x   2x
1
6
1
6

7
6
1
2
1
2
x
7
6

2x
Solve
6
1
 x
2
3
2
3
x   x   2x
1
6
1
6

7
6
1
2
1
2
x
7
6

2x
Solve
6
1
2
3
x   x   2x
1
6
 x 
2
3
6
1
1
6

7
6
1
2
1
2
x
7
6

2x
Solve
6
1
2
3
x   x   2x
1
6
 x 
2
3
6
1
1
6
7
6
1
2
 
6
1
1
2
x
7
6

2x
Solve
6
1
2
3
x   x   2x
1
6
 x 
2
3
6
1
1
6
7
6
1
2
 
6
1
1
2
x
6
1
 
7
6
2x
Solve
6
1
2
3
x   x   2x
1
6
 x 
2
3
6
1
1
6
7
6
1
2
 
6
1
1
2
x
6
1
   2x
7
6
6
1
Solve
6
1
2
3
x   x   2x
1
6
 x 
2
3
6
1
1
6
7
6
1
2
 
6
1
1
2
x
6
1
   2x
7
6
6
1
Solve
6
1
2
3
x   x   2x
1
6
 x 
2
3
6
1
1
6
7
6
1
2
 
6
1
1
2
x
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
 x 
2
3
6
1
1
6
7
6
1
2
 
6
1
1
2
x
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
 x 
2
3
1
6
1
1
6
7
6
1
2
 
6
1
1
2
x
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
 x 
2
3
1
6
1
1
6
7
6
1
2
 
6
1
1
2
x
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
 x 
2
3
1
6
1
1
6
7
6
1
2
 
6
1
1
2
x
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
 x 
2
3
1
1
6
1
6
7
6
1
2
 
6
1
1
2
x
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
 x 
2
3
1
1
6
1
6
1
7
6
1
2
 
6
1
1
2
x
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
 x 
2
3
1
1
6
1
6
1
7
6
1
2
 
6
1
1
2
x
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
 x 
2
3
1
1
6
1
6
1
7
6
1
2
 
6
1
1
2
x
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
 x 
2
3
1
1
6
1
6
1
7
6
1
2
3
6
1
 
1
2
x
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
 x 
2
3
1
1
6
1
6
1
7
6
1
2
3
6
1
 
1
2
1
x
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
 x 
2
3
1
1
6
1
6
1
7
6
1
2
3
6
1
 
1
2
1
x
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
 x 
2
3
1
1
6
1
6
1
7
6
1
2
3
6
1
 
1
2
1
x
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
 x 
2
3
1
1
6
1
6
1
7
6
1
2
3
6
1
 
1
2
1
x
1
6
1
   2x
7
6
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
 x 
2
3
1
1
6
1
6
1
7
6
1
2
3
6
1
 
1
2
1
x
1
6
1
   2x
7
6
1
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
 x 
2
3
1
4x
1
6
1
6
1
7
6
1
2
3
6
1
 
1
2
1
x
1
6
1
   2x
7
6
1
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
1
7
6
1
2
3
6
1
 
1
2
1
x
1
6
1
   2x
7
6
1
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
7
6
1
2
3
6
1
 
 1  3x
1
2
1
x
1
6
1
   2x
7
6
1
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
7
6
1
2
3
6
1
 
 1  3x
1
2
1
x

1
6
1
   2x
7
6
1
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
7
6
1
2
3
6
1
 
 1  3x
1
2
1
x

1
6
1
   2x
7
7
6
1
6
1
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
7
6
1
2
3
6
1
 
 1  3x
1
2
1
x

1
6
1
   2x
7
7
6
1
6
1
 12x
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
3
6
1
 
 1  3x
7x
7
6
1
2
1
2
1
x

1
6
1
   2x
7
7
6
1
6
1
 12x
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
7
6
1
2
3
6
1
 
1
2
1
 1  3x
7x
1
x

1
6
1
   2x
7
7
6
1
6
1
 12x
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
7
6
1
2
3
6
1
 
1
2
1
 1  3x
7x
1
x


1
6
1
   2x
7
7
6
1
6
1
 12x
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
7
6
1
2
3
6
1
 
1
2
1
 1  3x
7x
1
x
1
6
1
   2x
7
6
1

7

7
6
1
 12x
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
7
6
1
2
3
6
1
 
1
2
1
 1  3x
7x
1
x
1
6
1
   2x
7
6
1
6
1

7

7  12x
 12x
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
7
6
1
2
3
6
1
 
1
2
1
 1  3x
7x
 7x
1
x
1
6
1
   2x
7
6
1
6
1

7

7  12x
 7x
 12x
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
7
6
1
2
3
6
1
 
1
2
1
 1  3x
7x
 7x
1
1
x
1
6
1
   2x
7
6
1
6
1

7

7  12x
 7x
 12x
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
7
6
1
2
3
6
1
 
1
2
1
   2x
7
6
1
6
1

7
1

1

7  12x
 7x
7
 1  3x
7x
 7x
x
1
6
1
 12x
Solve
2
6
1
2
3
x   x   2x
1
6
1
6
1
 x 
2
3
1
4x
1
6
1
7
6
1
2
3
6
1
 
1
2
1
   2x
7
6
1
6
1

7
1

1

7  12x
 7x
7  5x
 1  3x
7x
 7x
x
1
6
1
 12x
1 
7  5x
 1  7  5x
7
7
 1  7  5x
7
7
8
 1  7  5x
7
7
8 
 1  7  5x
7
7
8 
 5x
 1  7  5x
7
7
8 
 5x
 5x
 1  7  5x
7
7
8 
 5x
 5x
8
 1  7  5x
7
7
8 
 5x
 5x   8
 1  7  5x
7
7
8 
 5x
 5x   8
5x
8

 1  7  5x
7
7
8 
 5x
 5x   8
5x
8
5 
 1  7  5x
7
7
8 
 5x
 5x   8
5x
8
5 
5
 1  7  5x
7
7
8 
 5x
 5x   8
5x
8
5 
5
x
 1  7  5x
7
7
8 
 5x
 5x   8
5x
8
5 
5
x 
 1  7  5x
7
7
8 
 5x
 5x   8
5x
8
5 
5
x 
8
5
Graph x 
8
5
Graph x 
8
5
0
Graph x 
8
5
0
Graph x 
8
5
0
Graph x 
8
5
0


Graph x 
8
5
0
x

Graph x 
8
5
0
x
|

Graph x 
8
5
0
x
| x

Graph x 
8
5
0
x
| x 

Graph x 
8
5
0
x
| x 
8
5

Lesson 2.7 Numbers 1-84