2.
Inequalities
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
3.
Inequalities
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
–
-3
-2
-1
0
1
2
3
+
4.
Inequalities
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
–
-3
-2
-1
0
1
2/3
2
3
+
5.
Inequalities
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
–
-3
-2
-1
0
1
2/3
2
3
2½
+
6.
Inequalities
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
–
-3
–π
-2
–3.14..
-1
0
1
2/3
2
3
2½ π
+
3.14..
7.
Inequalities
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
–
-3
–π
-2
–3.14..
-1
0
1
2/3
2
+
3
2½ π
3.14..
This line with each position addressed by a real number is
called the real (number) line.
8.
Inequalities
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
–
-3
–π
-2
–3.14..
-1
0
1
2/3
2
3
2½ π
+
3.14..
This line with each position addressed by a real number is
called the real (number) line.
Given two numbers corresponding to two points on the real
line, we define the number to the right to be greater than the
number to the left.
9.
Inequalities
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
–
-3
–π
-2
-1
–3.14..
0
1
2
3
2½ π
2/3
+
3.14..
This line with each position addressed by a real number is
called the real (number) line.
Given two numbers corresponding to two points on the real
line, we define the number to the right to be greater than the
number to the left.
–
L
R
+
10.
Inequalities
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
–
-3
–π
-2
-1
0
–3.14..
1
2
3
2½ π
2/3
+
3.14..
This line with each position addressed by a real number is
called the real (number) line.
Given two numbers corresponding to two points on the real
line, we define the number to the right to be greater than the
number to the left.
L
<
R
+
–
We write this as L < R and called this the natural form because
it corresponds to their respective positions on the real line.
11.
Inequalities
We associate each real number with a position on a line,
positive numbers to the right and negative numbers to the left.
–
-3
–π
-2
-1
0
–3.14..
1
2
3
2½ π
2/3
+
3.14..
This line with each position addressed by a real number is
called the real (number) line.
Given two numbers corresponding to two points on the real
line, we define the number to the right to be greater than the
number to the left.
L
<
R
+
–
We write this as L < R and called this the natural form because
it corresponds to their respective positions on the real line.
This relation may also be written as R > L (less preferable).
12.
Inequalities
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
13.
Inequalities
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
14.
Inequalities
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x".
15.
Inequalities
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a).
16.
Inequalities
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
a
a<x
+
–
open dot
17.
Inequalities
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
a
a<x
+
–
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x.
18.
Inequalities
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
a
a<x
+
–
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
a
a<x
–
solid dot
+
19.
Inequalities
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
a
a<x
+
–
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
a
a<x
–
+
solid dot
The numbers x fit the description a < x < b where a < b are all
the numbers x between a and b.
20.
Inequalities
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
a
a<x
+
–
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
a
a<x
–
+
solid dot
The numbers x fit the description a < x < b where a < b are all
the numbers x between a and b.
–
a
a<x<b
b
+
21.
Inequalities
Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". In general, we write "a < x" for all
the numbers x greater than a (excluding a). In picture,
a
a<x
+
–
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
a
a<x
–
+
solid dot
The numbers x fit the description a < x < b where a < b are all
the numbers x between a and b. A line segment as such is
called an interval.
a<x<b
b
a
+
–
23.
Example B.
a. Draw –1 < x < 3.
It’s in the natural form.
Inequalities
24.
Inequalities
Example B.
a. Draw –1 < x < 3.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
25.
Inequalities
Example B.
a. Draw –1 < x < 3.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
x
+
–
-1
0
3
26.
Inequalities
Example B.
a. Draw –1 < x < 3.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
x
+
–
-1
b. Draw 0 > x > –3
0
3
27.
Inequalities
Example B.
a. Draw –1 < x < 3.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
x
+
–
-1
0
b. Draw 0 > x > –3
Put it in the natural form –3 < x < 0.
3
28.
Inequalities
Example B.
a. Draw –1 < x < 3.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
x
+
–
-1
0
b. Draw 0 > x > –3
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
3
29.
Inequalities
Example B.
a. Draw –1 < x < 3.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
x
+
–
-1
0
b. Draw 0 > x > –3
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
x
–
-3
0
3
+
30.
Inequalities
Example B.
a. Draw –1 < x < 3.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
x
+
–
-1
0
3
b. Draw 0 > x > –3
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
x
–
-3
0
Expressions such as 2 < x > 3 or 2 < x < –3 do not have
any solution.
+
31.
Inequalities
Example B.
a. Draw –1 < x < 3.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
x
+
–
-1
0
3
b. Draw 0 > x > –3
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
x
–
-3
0
Expressions such as 2 < x > 3 or 2 < x < –3 do not have
any solution.
Adding or subtracting the same quantity to both retains the
inequality sign,
+
32.
Inequalities
Example B.
a. Draw –1 < x < 3.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
x
+
–
-1
0
3
b. Draw 0 > x > –3
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
x
–
-3
0
Expressions such as 2 < x > 3 or 2 < x < –3 do not have
any solution.
Adding or subtracting the same quantity to both retains the
inequality sign, i.e. if a < b, then a ± c < b ± c.
+
33.
Inequalities
Example B.
a. Draw –1 < x < 3.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
x
+
–
-1
0
3
b. Draw 0 > x > –3
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
x
–
-3
0
Expressions such as 2 < x > 3 or 2 < x < –3 do not have
any solution.
Adding or subtracting the same quantity to both retains the
inequality sign, i.e. if a < b, then a ± c < b ± c.
For example 6 < 12, then 6 + 3 < 12 + 3.
+
34.
Inequalities
Example B.
a. Draw –1 < x < 3.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
x
+
–
-1
0
3
b. Draw 0 > x > –3
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
x
–
-3
0
Expressions such as 2 < x > 3 or 2 < x < –3 do not have
any solution.
Adding or subtracting the same quantity to both retains the
inequality sign, i.e. if a < b, then a ± c < b ± c.
For example 6 < 12, then 6 + 3 < 12 + 3.
We use the this fact to solve inequalities.
+
35.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
36.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
37.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
38.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
x
–
0
15
+
39.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
x
–
0
A number c is positive means that 0 < c.
15
+
40.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
x
–
0
15
+
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign,
41.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
x
–
0
15
+
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b
42.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
x
–
0
15
+
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
43.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
x
–
0
15
+
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
For example 6 < 12 is true,
44.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
x
–
0
15
+
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12
45.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
x
–
0
15
+
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12 or 18 < 36 is also true.
46.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
x
–
0
15
+
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12 or 18 < 36 is also true.
Example D. Solve 3x > 12 and draw the solution.
47.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
x
–
0
15
+
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12 or 18 < 36 is also true.
Example D. Solve 3x > 12 and draw the solution.
3x > 12
48.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
x
–
0
15
+
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12 or 18 < 36 is also true.
Example D. Solve 3x > 12 and draw the solution.
3x > 12
divide by 3 and keep the inequality sign
3x/3 > 12/3
49.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
x
–
0
15
+
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12 or 18 < 36 is also true.
Example D. Solve 3x > 12 and draw the solution.
3x > 12
divide by 3 and keep the inequality sign
3x/3 > 12/3
x > 4 or 4 < x
50.
Inequalities
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12
add 3 to both sides
x – 3 + 3 < 12 + 3
x < 15
x
–
0
15
+
A number c is positive means that 0 < c. We may multiply or
divide a positive number to the inequality and keep the same
inequality sign, i.e. if 0 < c and a < b, then ac < bc.
For example 6 < 12 is true, then multiplying by 3
3*6 < 3*12 or 18 < 36 is also true.
Example D. Solve 3x > 12 and draw the solution.
3x > 12
divide by 3 and keep the inequality sign
3x/3 > 12/3
x > 4 or 4 < x
x
–
0
4
+
51.
Inequalities
A number c is negative means c < 0.
52.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
53.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
54.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
For example 6 < 12 is true.
55.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
56.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
57.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
–
0
6
<
12
+
58.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
–
–6
0
6
<
12
+
59.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
–
–12
<
–6
0
6
<
12
+
60.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
–
0
<
–6
< 12
6
Example E. Solve –x + 2 < 5 and draw the solution.
–12
+
61.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
–
0
<
–6
< 12
6
Example E. Solve –x + 2 < 5 and draw the solution.
–12
–x + 2 < 5
+
62.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
–
0
<
–6
< 12
6
Example E. Solve –x + 2 < 5 and draw the solution.
–12
–x + 2 < 5
–x < 3
subtract 2 from both sides
+
63.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
–
0
<
–6
< 12
6
Example E. Solve –x + 2 < 5 and draw the solution.
–12
–x + 2 < 5
–x < 3
–(–x) > –3
x > –3
+
subtract 2 from both sides
multiply by –1 to get x, reverse the inequality
64.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
–
0
<
–6
< 12
6
Example E. Solve –x + 2 < 5 and draw the solution.
–12
+
–x + 2 < 5
subtract 2 from both sides
–x < 3
multiply by –1 to get x, reverse the inequality
–(–x) > –3
x > –3 or –3 < x
65.
Inequalities
A number c is negative means c < 0. Multiplying or dividing by
an negative number reverses the inequality sign,
i.e. if c < 0 and a < b then
ca > cb .
For example 6 < 12 is true. If we multiply –1 to both sides then
(–1)6 > (–1)12
– 6 > –12 which is true.
Multiplying by –1 switches the left-right positions of the originals.
–
0
<
–6
< 12
6
Example E. Solve –x + 2 < 5 and draw the solution.
–12
+
–x + 2 < 5
subtract 2 from both sides
–x < 3
multiply by –1 to get x, reverse the inequality
–(–x) > –3
x > –3 or –3 < x
+
–
0
-3
66.
Inequalities
To solve inequalities:
1. Simplify both sides of the inequalities
67.
Inequalities
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides
68.
Inequalities
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
69.
Inequalities
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x.
70.
Inequalities
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around.
71.
Inequalities
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
72.
Inequalities
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Example F. Solve 3x + 5 > x + 9
73.
Inequalities
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9
move the x and 5, change side-change sign
74.
Inequalities
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9
3x – x > 9 – 5
move the x and 5, change side-change sign
75.
Inequalities
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9
3x – x > 9 – 5
2x > 4
move the x and 5, change side-change sign
76.
Inequalities
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9
3x – x > 9 – 5
2x > 4
2x 4
2 > 2
move the x and 5, change side-change sign
div. 2
77.
Inequalities
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign
3x – x > 9 – 5
2x > 4
div. 2
2x 4
2 > 2
x > 2 or 2 < x
78.
Inequalities
To solve inequalities:
1. Simplify both sides of the inequalities
2. Gather the x-terms to one side and the number-terms to the
other sides (use the “change side-change sign” rule).
3. Multiply or divide to get x. If we multiply or divide by
negative numbers to both sides, the inequality sign must be
turned around. This rule can be avoided by keeping the
x-term positive.
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign
3x – x > 9 – 5
2x > 4
div. 2
2x 4
2 > 2
x > 2 or 2 < x
+
–
0
2
79.
Inequalities
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
80.
Inequalities
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x
simplify each side
81.
Inequalities
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x
6 – 3x > 2x + 18 – 2x
simplify each side
82.
Inequalities
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x
6 – 3x > 2x + 18 – 2x
6 – 3x > 18
simplify each side
85.
Inequalities
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x
6 – 3x > 2x + 18 – 2x
6 – 3x > 18
6 – 18 > 3x
– 12 > 3x
–12 > 3x
3
3
simplify each side
move 18 and –3x (change sign)
div. by 3 (no need to switch >)
86.
Inequalities
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x
6 – 3x > 2x + 18 – 2x
6 – 3x > 18
6 – 18 > 3x
– 12 > 3x
–12 > 3x
3
3
–4 > x or x < –4
simplify each side
move 18 and –3x (change sign)
div. by 3 (no need to switch >)
87.
Inequalities
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x
6 – 3x > 2x + 18 – 2x
6 – 3x > 18
6 – 18 > 3x
– 12 > 3x
simplify each side
move 18 and –3x (change sign)
div. by 3 (no need to switch >)
–12 > 3x
3
3
–4 > x or x < –4
-4
0
+
88.
Inequalities
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x
6 – 3x > 2x + 18 – 2x
6 – 3x > 18
6 – 18 > 3x
– 12 > 3x
simplify each side
move 18 and –3x (change sign)
div. by 3 (no need to switch >)
–12 > 3x
3
3
–4 > x or x < –4
-4
0
+
We also have inequalities in the form of intervals.
89.
Inequalities
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x
6 – 3x > 2x + 18 – 2x
6 – 3x > 18
6 – 18 > 3x
– 12 > 3x
simplify each side
move 18 and –3x (change sign)
div. by 3 (no need to switch >)
–12 > 3x
3
3
–4 > x or x < –4
-4
0
+
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
90.
Inequalities
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x
6 – 3x > 2x + 18 – 2x
6 – 3x > 18
6 – 18 > 3x
– 12 > 3x
simplify each side
move 18 and –3x (change sign)
div. by 3 (no need to switch >)
–12 > 3x
3
3
–4 > x or x < –4
-4
0
+
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
Again, we + or – remove the number term in the middle first,
91.
Inequalities
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x
6 – 3x > 2x + 18 – 2x
6 – 3x > 18
6 – 18 > 3x
– 12 > 3x
simplify each side
move 18 and –3x (change sign)
div. by 3 (no need to switch >)
–12 > 3x
3
3
–4 > x or x < –4
-4
0
+
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
Again, we + or – remove the number term in the middle first,
then divide or multiply to get x.
92.
Inequalities
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x
6 – 3x > 2x + 18 – 2x
6 – 3x > 18
6 – 18 > 3x
– 12 > 3x
simplify each side
move 18 and –3x (change sign)
div. by 3 (no need to switch >)
–12 > 3x
3
3
–4 > x or x < –4
-4
0
+
We also have inequalities in the form of intervals. We solve
them by +, –, * , / to all three parts of the inequalities.
Again, we + or – remove the number term in the middle first,
then divide or multiply to get x. The answer is an interval of
numbers.
93.
Inequalities
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw
94.
Inequalities
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4
–6
–6 –6
subtract 6
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