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------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/master06.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/preview.wmf Content-Transfer-Encoding: base64 Content-Type: image/x-wmf AQAJAAADYAEAAAUALQAAAAAABAAAAAMBCAAFAAAACwIAAAAABQAAAAwCeQChAAMAAAAeAAcAAAD8 AgAA////AAAABAAAAC0BAAAIAAAA+gIFAAAAAAD///8ABAAAAC0BAQAOAAAAJAMFAP///////3gA oAB4AKAA////////CAAAAPoCAAAAAAAAAAAAAAQAAAAtAQIABAAAAC0BAAAEAAAAJwH//xwAAAD7 Avn/AAAAAAAAkAEAAAAAAEAAIkFyaWFsAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABAAAAC0B AwAEAAAALgEYAAQAAAACAQEABQAAAAkCAAAAAg0AAAAyCgkAIgAEAAAAU2lnbgUAAgAEAAQABAAA AC4BAAAcAAAA+wIQAAcAAAAAALwCAAAAAAECAiJTeXN0ZW0AAAAAAAAAAAAAGAAAAAEAAABQ1B8A 5AQAAAQAAAAtAQQABAAAAPABAwAcAAAA+wL5/wAAAAAAAJABAAAAAABAACJBcmlhbAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAQAAAAtAQMABAAAAC4BGAAEAAAAAgEBAAUAAAAJAgAAAAIJAAAA MgoJADEAAQAAAC0mAgAEAAAALgEAAAQAAAAtAQQABAAAAPABAwAcAAAA+wL5/wAAAAAAAJABAAAA AABAACJBcmlhbAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAQAAAAtAQMABAAAAC4BGAAEAAAA AgEBAAUAAAAJAgAAAAItAAAAMgoJADMAGQAAAENoYXJ0cyBhbmQgSW5lcXVhbGl0aWVzIEkABQAE AAQAAgACAAQAAgAEAAQABAACAAIABAAEAAQABAAEAAEAAgACAAIAAwAEAAIAAQAEAAAALgEAAAQA AAAtAQQABAAAAPABAwADAAAAAAA= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0123.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0129.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
We use the factored polynomials or rational expressions to determi= ne the signs of the outputs. 
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0130.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
We use the factored polynomials or rational expressions to determi= ne the signs of the outputs. 
That is, given a formu= la f and a value for x, factor f to determine if the outpu= t is positive (f > 0) or negative (f < 0) when f is evaluated with x.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0131.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
We use the factored polynomials or rational expressions to determi= ne the signs of the outputs. 
Example A: Determ= ine the outcome is + or  –<= span style=3D'mso-spacerun:yes'>  for
x2 – 2x – 3 if x =3D -3/2= , -1/2.
That is, given a formu= la f and a value for x, factor f to determine if the outpu= t is positive (f > 0) or negative (f < 0) when f is evaluated with x.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0132.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
We use the factored polynomials or rational expressions to determi= ne the signs of the outputs. 
Example A: Determ= ine the outcome is + or  –<= span style=3D'mso-spacerun:yes'>  for
x2 – 2x – 3 if x =3D -3/2= , -1/2.
In factored form x – 2x – 3 =3D (x –= ; 3)(x + 1)
That is, given a formu= la f and a value for x, factor f to determine if the outpu= t is positive (f > 0) or negative (f < 0) when f is evaluated with x.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0133.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
We use the factored polynomials or rational expressions to determi= ne the signs of the outputs. 
Example A: Determ= ine the outcome is + or  –<= span style=3D'mso-spacerun:yes'>  for
x2 – 2x – 3 if x =3D -3/2= , -1/2.
In factored form x – 2x – 3 =3D (x –= ; 3)(x + 1)
Hence, f= or x =3D -3/2:
(-3/2 &#= 8211; 3)(-3/2 + 1)
That is, given a formu= la f and a value for x, factor f to determine if the outpu= t is positive (f > 0) or negative (f < 0) when f is evaluated with x.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0134.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
We use the factored polynomials or rational expressions to determi= ne the signs of the outputs. 
Example A: Determ= ine the outcome is + or  –<= span style=3D'mso-spacerun:yes'>  for
x2 – 2x – 3 if x =3D -3/2= , -1/2.
In factored form x – 2x – 3 =3D (x –= ; 3)(x + 1)
Hence, f= or x =3D -3/2:
(-3/2 &#= 8211; 3)(-3/2 + 1) is (–)(–) =3D + .
That is, given a formu= la f and a value for x, factor f to determine if the outpu= t is positive (f > 0) or negative (f < 0) when f is evaluated with x.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0135.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
We use the factored polynomials or rational expressions to determi= ne the signs of the outputs. 
Example A: Determ= ine the outcome is + or  –<= span style=3D'mso-spacerun:yes'>  for
x2 – 2x – 3 if x =3D -3/2= , -1/2.
In factored form x – 2x – 3 =3D (x –= ; 3)(x + 1)
Hence, f= or x =3D -3/2:
(-3/2 &#= 8211; 3)(-3/2 + 1) is (–)(–) =3D + .
And for = x =3D -1/2:
(-1/2 &#= 8211; 3)(-1/2 + 1)
That is, given a formu= la f and a value for x, factor f to determine if the outpu= t is positive (f > 0) or negative (f < 0) when f is evaluated with x.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0136.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
We use the factored polynomials or rational expressions to determi= ne the signs of the outputs. 
Example A: Determ= ine the outcome is + or  –<= span style=3D'mso-spacerun:yes'>  for
x2 – 2x – 3 if x =3D -3/2= , -1/2.
In factored form x – 2x – 3 =3D (x –= ; 3)(x + 1)
Hence, f= or x =3D -3/2:
(-3/2 &#= 8211; 3)(-3/2 + 1) is (–)(–) =3D + .
And for = x =3D -1/2:
(-1/2 &#= 8211; 3)(-1/2 + 1) is (–)(+) =3D – .
That is, given a formu= la f and a value for x, factor f to determine if the outpu= t is positive (f > 0) or negative (f < 0) when f is evaluated with x.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0124.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Example B: Determ= ine the outcome is + or  –<= span style=3D'mso-spacerun:yes'>  for
        = ;            if x =3D -3/2, -1/2.
x2 – 2x – 3
x2 + x – 2
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0124_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0137.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Example B: Determ= ine the outcome is + or  –<= span style=3D'mso-spacerun:yes'>  for
        = ;            if x =3D -3/2, -1/2.
x2 – 2x – 3
x2 + x – 2
x2 – 2x – 3
x2 + x – 2
In factored form
=3D<= /div>
(x – 3)(x + 1)
(x – 1)(x + 2)
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0137_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0137_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBAECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAD AQEAgAAAAAAAAAITjI+py+0Po1yg2ouz1rP7D4ZLAQA7 ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0138.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Example B: Determ= ine the outcome is + or  –<= span style=3D'mso-spacerun:yes'>  for
        = ;            if x =3D -3/2, -1/2.
x2 – 2x – 3
x2 + x – 2
x2 – 2x – 3
x2 + x – 2
In factored form
=3D<= /div>
(x – 3)(x + 1)
(x – 1)(x + 2)
Hence, for x =3D -3/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=3D<= /div>
(–)(–)
(–)(+)
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0138_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0138_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBAECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAD AQEAgAAAAAAAAAITjI+py+0Po1yg2ouz1rP7D4ZLAQA7 ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0138_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0138_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0139.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Example B: Determ= ine the outcome is + or  –<= span style=3D'mso-spacerun:yes'>  for
        = ;            if x =3D -3/2, -1/2.
x2 – 2x – 3
x2 + x – 2
x2 – 2x – 3
x2 + x – 2
In factored form
=3D<= /div>
(x – 3)(x + 1)
(x – 1)(x + 2)
Hence, for x =3D -3/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=3D<= /div>
(–)(–)
(–)(+)
< 0
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0139_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0139_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBAECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAD AQEAgAAAAAAAAAITjI+py+0Po1yg2ouz1rP7D4ZLAQA7 ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0139_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0139_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0140.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Example B: Determ= ine the outcome is + or  –<= span style=3D'mso-spacerun:yes'>  for
        = ;            if x =3D -3/2, -1/2.
x2 – 2x – 3
x2 + x – 2
x2 – 2x – 3
x2 + x – 2
In factored form
=3D<= /div>
(x – 3)(x + 1)
(x – 1)(x + 2)
Hence, for x =3D -3/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=3D<= /div>
(–)(–)
(–)(+)
< 0
For x =3D -1/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=3D<= /div>
(–)(+)
(–)(+)
Sign-Charts and Inequalities I
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Example B: Determ= ine the outcome is + or  –<= span style=3D'mso-spacerun:yes'>  for
        = ;            if x =3D -3/2, -1/2.
x2 – 2x – 3
x2 + x – 2
x2 – 2x – 3
x2 + x – 2
In factored form
=3D<= /div>
(x – 3)(x + 1)
(x – 1)(x + 2)
Hence, for x =3D -3/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=3D<= /div>
(–)(–)
(–)(+)
< 0
For x =3D -1/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=3D<= /div>
(–)(+)
(–)(+)
> 0
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0141_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0141_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBAECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAD AQEAgAAAAAAAAAITjI+py+0Po1yg2ouz1rP7D4ZLAQA7 ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0141_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0141_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0141_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0141_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0142.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Example B: Determ= ine the outcome is + or  –<= span style=3D'mso-spacerun:yes'>  for
        = ;            if x =3D -3/2, -1/2.
x2 – 2x – 3
x2 + x – 2
x2 – 2x – 3
x2 + x – 2
In factored form
=3D<= /div>
(x – 3)(x + 1)
(x – 1)(x + 2)
Hence, for x =3D -3/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=3D<= /div>
(–)(–)
(–)(+)
< 0
For x =3D -1/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=3D<= /div>
(–)(+)
(–)(+)
> 0
This leads to the sign charts of formulas. The sign- chart of a formula gives the si= gns of the outputs.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0142_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0142_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBAECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAD AQEAgAAAAAAAAAITjI+py+0Po1yg2ouz1rP7D4ZLAQA7 ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0142_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0142_image027.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0142_image028.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0142_image029.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0026.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Here is an example, the sign chart of f =3D x – 1:
1
f =3D 0
 +
 +
–
–
–
–
 x – 1
Sign-Charts and Inequalities I
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Here is an example, the sign chart of f =3D x – 1:
1
f =3D 0
 +
 +
–
–
–
–
 x – 1
The "+" indicates the region where the output is positive = i.e. if 1 < x. 
Sign-Charts and Inequalities I
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Here is an example, the sign chart of f =3D x – 1:
1
f =3D 0
 +
 +
–
–
–
–
 x – 1
The "+" indicates the region where the output is positive = i.e. if 1 < x.  Likewise, the "–" indicates the region where the outpu= t is negative, i.e. x < 1.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0144_image034.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUAECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBO AQEAgAAAAAAAAAIQjI+py+0Po5y02ouz3ry3AgA7 ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0144_image035.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhB13wcIrokwgyNvutDrmnnAYtCnMYxQAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0145.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Constr= uction of the sign-chart of f.
Here is an example, the sign chart of f =3D x – 1:
1
f =3D 0
 +
 +
–
–
–
–
 x – 1
The "+" indicates the region where the output is positive = i.e. if 1 < x.  Likewise, the "–" indicates the region where the outpu= t is negative, i.e. x < 1.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0145_image036.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUAECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBO AQEAgAAAAAAAAAIQjI+py+0Po5y02ouz3ry3AgA7 ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0145_image037.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhB13wcIrokwgyNvutDrmnnAYtCnMYxQAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0146.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Construction of the sign-chart of f.
I. Solve for f =3D 0 (= and denominator =3D 0) if there is any denominator.
Here is an example, the sign chart of f =3D x – 1:
1
f =3D 0
 +
 +
–
–
–
–
 x – 1
The "+" indicates the region where the output is positive = i.e. if 1 < x.  Likewise, the "–" indicates the region where the outpu= t is negative, i.e. x < 1.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0146_image038.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUAECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBO AQEAgAAAAAAAAAIQjI+py+0Po5y02ouz3ry3AgA7 ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0146_image039.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhB13wcIrokwgyNvutDrmnnAYtCnMYxQAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0147.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Construction of the sign-chart of f.
I. Solve for f =3D 0 (= and denominator =3D 0) if there is any denominator.
II.   Draw the real line, mark of= f the answers from I.
Here is an example, the sign chart of f =3D x – 1:
1
f =3D 0
 +
 +
–
–
–
–
 x – 1
The "+" indicates the region where the output is positive = i.e. if 1 < x.  Likewise, the "–" indicates the region where the outpu= t is negative, i.e. x < 1.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0147_image040.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUAECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBO AQEAgAAAAAAAAAIQjI+py+0Po5y02ouz3ry3AgA7 ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0147_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhB13wcIrokwgyNvutDrmnnAYtCnMYxQAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0148.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Construction of the sign-chart of f.
I. Solve for f =3D 0 (= and denominator =3D 0) if there is any denominator.
II.   Draw the real line, mark of= f the answers from I.
III.  Sample each segment for signs by testing a point in each segment. 
Here is an example, the sign chart of f =3D x – 1:
1
f =3D 0
 +
 +
–
–
–
–
 x – 1
The "+" indicates the region where the output is positive = i.e. if 1 < x.  Likewise, the "–" indicates the region where the outpu= t is negative, i.e. x < 1.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0148_image042.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUAECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBO AQEAgAAAAAAAAAIQjI+py+0Po5y02ouz3ry3AgA7 ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0148_image043.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhB13wcIrokwgyNvutDrmnnAYtCnMYxQAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0149.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Construction of the sign-chart of f.
I. Solve for f =3D 0 (= and denominator =3D 0) if there is any denominator.
II.   Draw the real line, mark of= f the answers from I.
III.  Sample each segment for signs by testing a point in each segment. 
Here is an example, the sign chart of f =3D x – 1:
1
f =3D 0
 +
 +
–
–
–
–
 x – 1
The "+" indicates the region where the output is positive = i.e. if 1 < x.  Likewise, the "–" indicates the region where the outpu= t is negative, i.e. x < 1.
Fact: The sig= n stays the same for x's in between the values from step I (wh= ere f =3D 0 or f is undefined.) 
Sign-Charts and Inequalities I
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Example = C: Let f =3D x2 – 3x – 4 , use th= e sign-
chart to indicate when is f =3D 0, f > 0, and f < 0.
Sign-Charts and Inequalities I
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Example = C: Let f =3D x2 – 3x – 4 , use th= e sign-
chart to indicate when is f =3D 0, f > 0, and f < 0.
Solve x2 – 3x –= ; 4 =3D 0
         (x – 4)(x + 1) =3D 0 à x =3D 4 , -1
Sign-Charts and Inequalities I
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Example = C: Let f =3D x2 – 3x – 4 , use th= e sign-
chart to indicate when is f =3D 0, f > 0, and f < 0.
Solve x2 – 3x –= ; 4 =3D 0
         (x – 4)(x + 1) =3D 0 à x =3D 4 , -1
Mark off these po= ints on a line:
(x-4)(x+1)<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:#009999;mso-color-index:6;mso-fareast-language: ZH-CN'> =
4
-1
Sign-Charts and Inequalities I
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Example = C: Let f =3D x2 – 3x – 4 , use th= e sign-
chart to indicate when is f =3D 0, f > 0, and f < 0.
Solve x2 – 3x –= ; 4 =3D 0
         (x – 4)(x + 1) =3D 0 à x =3D 4 , -1
Mark off these po= ints on a line:
(x-4)(x+1)<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:#009999;mso-color-index:6;mso-fareast-language: ZH-CN'> =
Select poin= ts to sample in each segment:
4
-1
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0152_image049.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkwECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQCR AQEAgAAAAAAAAAIRjI+py+0Po5y02ouz3rz7vxUAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0152_image050.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHxozmYdztuqwBkdxCSKUQAAOw== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0152_image051.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHwmTNmuy1rZjoDVBDGJYhQAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0153.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Example = C: Let f =3D x2 – 3x – 4 , use th= e sign-
chart to indicate when is f =3D 0, f > 0, and f < 0.
Solve x2 – 3x –= ; 4 =3D 0
         (x – 4)(x + 1) =3D 0 à x =3D 4 , -1
Mark off these po= ints on a line:
(x-4)(x+1)<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:#009999;mso-color-index:6;mso-fareast-language: ZH-CN'> =
4
-1
Select poin= ts to sample in each segment:
Test x =3D = - 2,
-2
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0153_image052.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkwECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQCR AQEAgAAAAAAAAAIRjI+py+0Po5y02ouz3rz7vxUAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0153_image053.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHxozmYdztuqwBkdxCSKUQAAOw== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0153_image054.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHwmTNmuy1rZjoDVBDGJYhQAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0153_image055.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAG AAoAgQAAAAAAAJkzAAECAwIORI6icsvh2pKKvggnQgUAOw== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0158.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Example = C: Let f =3D x2 – 3x – 4 , use th= e sign-
chart to indicate when is f =3D 0, f > 0, and f < 0.
Solve x2 – 3x –= ; 4 =3D 0
         (x – 4)(x + 1) =3D 0 à x =3D 4 , -1
Mark off these po= ints on a line:
(x-4)(x+1)<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:#009999;mso-color-index:6;mso-fareast-language: ZH-CN'> =
4
-1
Select poin= ts to sample in each segment:
Test x =3D - 2, &= #13;
get  – *  – =3D + .
Hence the segment
is positive.  Draw +
sign over it. = 3;
-2
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0158_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkwECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQCR AQEAgAAAAAAAAAIRjI+py+0Po5y02ouz3rz7vxUAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0158_image057.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHxozmYdztuqwBkdxCSKUQAAOw== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0158_image058.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHwmTNmuy1rZjoDVBDGJYhQAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0158_image059.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAG AAoAgQAAAAAAAJkzAAECAwIORI6icsvh2pKKvggnQgUAOw== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0154.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Example = C: Let f =3D x2 – 3x – 4 , use th= e sign-
chart to indicate when is f =3D 0, f > 0, and f < 0.
Solve x2 – 3x –= ; 4 =3D 0
         (x – 4)(x + 1) =3D 0 à x =3D 4 , -1
Mark off these po= ints on a line:
(x-4)(x+1)<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:#009999;mso-color-index:6;mso-fareast-language: ZH-CN'> +  +  +  +  +    
4
-1
Select poin= ts to sample in each segment:
Test x =3D - 2, &= #13;
get  – *  – =3D + .
Hence the segment
is positive.  Draw +
sign over it. = 3;
-2
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0154_image060.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkwECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQCR AQEAgAAAAAAAAAIRjI+py+0Po5y02ouz3rz7vxUAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0154_image061.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHxozmYdztuqwBkdxCSKUQAAOw== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0154_image062.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHwmTNmuy1rZjoDVBDGJYhQAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0154_image063.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAG AAoAgQAAAAAAAJkzAAECAwIORI6icsvh2pKKvggnQgUAOw== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0155.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Example = C: Let f =3D x2 – 3x – 4 , use th= e sign-
chart to indicate when is f =3D 0, f > 0, and f < 0.
Solve x2 – 3x –= ; 4 =3D 0
         (x – 4)(x + 1) =3D 0 à x =3D 4 , -1
Mark off these po= ints on a line:
(x-4)(x+1)<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:#009999;mso-color-index:6;mso-fareast-language: ZH-CN'> +  +  +  +  +    
0
4
-1
Select poin= ts to sample in each segment:
Test x =3D - 2, &= #13;
get  – *  – =3D + .
Hence the segment
is positive.  Draw +
sign over it. = 3;
-2
Test x = =3D 0,
get  – * + =3D  –.
Hence th= is segment
is negative. 
Put  –  over it.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0155_image064.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkwECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQCR AQEAgAAAAAAAAAIRjI+py+0Po5y02ouz3rz7vxUAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0155_image065.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAB AAkAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0155_image066.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHxozmYdztuqwBkdxCSKUQAAOw== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0155_image067.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHwmTNmuy1rZjoDVBDGJYhQAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0155_image068.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAG AAoAgQAAAAAAAJkzAAECAwIORI6icsvh2pKKvggnQgUAOw== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0155_image069.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAG AAoAgAAAAAAAAAIIRB6Gy+0PYygAOy== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0156.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Example = C: Let f =3D x2 – 3x – 4 , use th= e sign-
chart to indicate when is f =3D 0, f > 0, and f < 0.
Solve x2 – 3x –= ; 4 =3D 0
         (x – 4)(x + 1) =3D 0 à x =3D 4 , -1
Mark off these po= ints on a line:
(x-4)(x+1)<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:#009999;mso-color-index:6;mso-fareast-language: ZH-CN'> +  +  +  +  +     –  –  –  –  –         =
0
4
-1
Select poin= ts to sample in each segment:
Test x =3D - 2, &= #13;
get  – *  – =3D + .
Hence the segment
is positive.  Draw +
sign over it. = 3;
-2
Test x = =3D 0,
get  – * + =3D  –.
Hence th= is segment
is negative. 
Put  –  over it.
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0156_image070.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkwECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQCR AQEAgAAAAAAAAAIRjI+py+0Po5y02ouz3rz7vxUAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0156_image071.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAB AAkAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0156_image072.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHxozmYdztuqwBkdxCSKUQAAOw== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0156_image073.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHwmTNmuy1rZjoDVBDGJYhQAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0156_image074.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAG AAoAgQAAAAAAAJkzAAECAwIORI6icsvh2pKKvggnQgUAOw== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0156_image075.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAG AAoAgAAAAAAAAAIIRB6Gy+0PYygAOy== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0157.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Example = C: Let f =3D x2 – 3x – 4 , use th= e sign-
chart to indicate when is f =3D 0, f > 0, and f < 0.
Solve x2 – 3x –= ; 4 =3D 0
         (x – 4)(x + 1) =3D 0 à x =3D 4 , -1
Mark off these po= ints on a line:
(x-4)(x+1)<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:#009999;mso-color-index:6;mso-fareast-language: ZH-CN'> +  +  +  +  +     –  –  –  –  –     
0
4
-1
Select poin= ts to sample in each segment:
Test x =3D - 2, &= #13;
get  – *  – =3D + .
Hence the segment
is positive.  Draw +
sign over it. = 3;
-2
Test x = =3D 0,
get  – * + =3D  –.
Hence th= is segment
is negative. 
Put  –  over it.
Test x = =3D 5,
get + * = + =3D +.
Hence th= is segment
is positive.
Put + ov= er it.
5
Sign-Charts and Inequalities I
------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0157_image076.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkwECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQCR AQEAgAAAAAAAAAIRjI+py+0Po5y02ouz3rz7vxUAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0157_image077.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAB AAkAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0157_image078.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHxozmYdztuqwBkdxCSKUQAAOw== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0157_image079.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AAoAgQAAAAAAAP8AAAECAwIXhBFxlyLJnHwmTNmuy1rZjoDVBDGJYhQAOx== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0157_image080.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAG AAoAgQAAAAAAAJkzAAECAwIORI6icsvh2pKKvggnQgUAOw== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0157_image081.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAG AAoAgAAAAAAAAAIIRB6Gy+0PYygAOy== ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0157_image082.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAF AAoAgQAAAAAAADMzmQECAwINjC9pKavBXIMv0nlCAQA7 ------=_NextPart_01C82E61.6D76B260 Content-Location: file:///C:/2EC450E9/4SignChartsandInequalitiesI_files/slide0159.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Example = C: Let f =3D x2 – 3x – 4 , use th= e sign-
chart to indicate when is f =3D 0, f > 0, and f < 0.
Solve x2 – 3x –= ; 4 =3D 0
         (x – 4)(x + 1) =3D 0 à x =3D 4 , -1
Mark off these po= ints on a line:
(x-4)(x+1)<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:#009999;mso-color-index:6;mso-fareast-language: ZH-CN'> +  +  +  +  +     –  –  –  –  –      +  +  +  +  +  
0
4
-1
Select poin= ts to sample in each segment:
Test x =3D -