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Inequalities
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We associate each real nu= mber to a position on a line, positive numbers to the right a= nd negative numbers to the left.
Inequalities
------=_NextPart_01C8F61F.BB893690 Content-Location: file:///C:/1ECBB113/6Inequalities_files/slide0151.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
We associate each real nu= mber to a position on a line, positive numbers to the right a= nd negative numbers to the left.
-2
2
0
1
3
+
-1
-3
–
Inequalities
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<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
We associate each real nu= mber to a position on a line, positive numbers to the right a= nd negative numbers to the left.
-2
2
0
1
3
+
-1
-3
–
2/3
Inequalities
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<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
We associate each real nu= mber to a position on a line, positive numbers to the right a= nd negative numbers to the left.
-2
2
0
1
3
+
-1
-3
–
2/3
2½
Inequalities
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<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
We associate each real nu= mber to a position on a line, positive numbers to the right a= nd negative numbers to the left.
-2
2
0
1
3
+
-1
-3
–
2/3
2½
π » 3.14..
Inequalities
 –π » –3.14..
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<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
We associate each real nu= mber to a position on a line, positive numbers to the right a= nd negative numbers to the left.
-2
2
0
1
3
+
-1
-3
–
2/3
2½
π » 3.14..
This line with each position addressed by a real number is = called the real (number) line.
Inequalities
–π » –3.14..
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<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
We associate each real nu= mber to a position on a line, positive numbers to the right a= nd negative numbers to the left.
-2
2
0
1
3
+
-1
-3
–
2/3
2½
π » 3.14..
This line with each position addressed by a real number is = called the real (number) line.
Inequalities
–π » –3.14..
Giv= en two numbers corresponding to two points on the real line, we define the numbe= r to the right to be gre= ater than the <= span style=3D'position:absolute;top:64.75%;left:4.3%;width:91.57%;height:5.25%'= >number to the left. 
------=_NextPart_01C8F61F.BB893690 Content-Location: file:///C:/1ECBB113/6Inequalities_files/slide0157.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
We associate each real nu= mber to a position on a line, positive numbers to the right a= nd negative numbers to the left.
-2
2
0
1
3
+
-1
-3
–
=
2/3
2½
π » 3.14..
This line with each position addressed by a real number is = called the real (number) line.
Inequalities
–π » –3.14..
Giv= en two numbers corresponding to two points on the real line, we define the numbe= r to the right to be gre= ater than the <= span style=3D'position:absolute;top:64.75%;left:4.3%;width:105.61%;height:5.25%= '>number to the left.  In the picture below, R is greate= r than L.
+
–
R
L
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<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
We associate each real nu= mber to a position on a line, positive numbers to the right a= nd negative numbers to the left.
-2
2
0
1
3
+
-1
-3
–
=
2/3
2½
π » 3.14..
This line with each position addressed by a real number is = called the real (number) line.
Inequalities
–π » –3.14..
Giv= en two numbers corresponding to two points on the real line, we define the numbe= r to the right to be gre= ater than the <= span style=3D'position:absolute;top:64.75%;left:4.3%;width:105.61%;height:5.25%= '>number to the left.  In the picture below, R is greate= r than L.
+
–
R
We write this relation as= L < R (or R > L)
L
<
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<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
We associate each real nu= mber to a position on a line, positive numbers to the right a= nd negative numbers to the left.
-2
2
0
1
3
+
-1
-3
–
=
2/3
2½
π » 3.14..
This line with each position addressed by a real number is = called the real (number) line.
Inequalities
π » –3.14..
Giv= en two numbers corresponding to two points on the real line, we define the numbe= r to the right to be gre= ater than the <= span style=3D'position:absolute;top:64.75%;left:4.3%;width:105.61%;height:5.25%= '>number to the left.  In the picture below, R is greate= r than L.
+
–
R
We write this relation as L <= ; R (or R > L) and we say that
"R is greater than L"= or that "L is less than R". 
L
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R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAK AAkAgQAAAAAAALvg4wECAwIWhINogOIeznsxzNkutFrU/l3RkShAAQA7 ------=_NextPart_01C8F61F.BB893690 Content-Location: file:///C:/1ECBB113/6Inequalities_files/slide0159_image081.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAkAgQAAAAAAALvg4wECAwITBIJoJ93BnENyhvoupjvOlShAAQA7 ------=_NextPart_01C8F61F.BB893690 Content-Location: file:///C:/1ECBB113/6Inequalities_files/slide0139.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example A.<= span style=3D'mso-spacerun:yes'>  2 < 4,  –3< –2, 0 > <= /span>–1 are true statements
Inequalities
------=_NextPart_01C8F61F.BB893690 Content-Location: file:///C:/1ECBB113/6Inequalities_files/slide0160.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example A.  2 < 4,  –3< –2, 0 > <= /span>–1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
------=_NextPart_01C8F61F.BB893690 Content-Location: file:///C:/1ECBB113/6Inequalities_files/slide0161.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example A.  2 < 4,  –3< –2, 0 > <= /span>–1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If = we want all the numbers greater than 5, we may denote them as "all number x whe= re 5 < x".
------=_NextPart_01C8F61F.BB893690 Content-Location: file:///C:/1ECBB113/6Inequalities_files/slide0162.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example A.  2 < 4,  –3< –2, 0 > <= /span>–1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If = we want all the numbers greater than 5, we may denote them as "all number x whe= re 5 < x". In general, we write "a < x" for all = the numbers x greater than a (excluding a).
------=_NextPart_01C8F61F.BB893690 Content-Location: file:///C:/1ECBB113/6Inequalities_files/slide0163.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example A.  2 < 4,  –3< –2, 0 > <= /span>–1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If = we want all the numbers greater than 5, we may denote them as "all number x whe= re 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
+
–
a
open dot
a < x
------=_NextPart_01C8F61F.BB893690 Content-Location: file:///C:/1ECBB113/6Inequalities_files/slide0164.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example A.  2 < 4,  –3< –2, 0 > <= /span>–1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If = we want all the numbers greater than 5, we may denote them as "all number x whe= re 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
+
–
a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x.
<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
a < x
------=_NextPart_01C8F61F.BB893690 Content-Location: file:///C:/1ECBB113/6Inequalities_files/slide0165.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example A.  2 < 4,  –3< –2, 0 > <= /span>–1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If = we want all the numbers greater than 5, we may denote them as "all number x whe= re 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
+
–
a
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
<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
+
–
a
solid dot
a < x
a < x
------=_NextPart_01C8F61F.BB893690 Content-Location: file:///C:/1ECBB113/6Inequalities_files/slide0166.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example A.  2 < 4,  –3< –2, 0 > <= /span>–1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If = we want all the numbers greater than 5, we may denote them as "all number x whe= re 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
+
–
=
a
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
<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
+
–
a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all the numbers x between a and b. =
<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
-2
2
0
1
3
+
-1
-3
–
a
b
 x
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Absolute Inequalities
Example A.  2 < 4,  –3< –2, 0 > <= /span>–1 are true statements
and –2 < –5 , 5 < 3 are false statements.
Inequalities
If = we want all the numbers greater than 5, we may denote them as "all number x whe= re 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
+
–
=
a
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
<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
+
–
a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all the numbers x between a and b. This is called an interval.
<= span lang=3DZH-CN style=3D'font-family:\5B8B\4F53;mso-ascii-font-family:Tahoma; mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-family:Tahoma;font-size: 200%;text-shadow:auto;mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
-2
2
0
1
3
+
-1
-3
–
a
b
 x
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