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Absolute Value Inequalities
------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0091.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Absolute Value Inequalities
Since |x| means “the distance between x and 0”, so the = expressio= n |x| < c means “the distance between x and 0 is less than c”. 
------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0092.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example B. Draw the inequality |x| < 7.
Absolute Value Inequalities
Since |x| means “the distance between x and 0”, so the = expressio= n |x| < c means “the distance between x and 0 is less than c”. 
------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0093.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example B. Draw the inequality |x| < 7.
Absolute Value Inequalities
Since |x| means “the distance between x and 0”, so the = expressio= n |x| < c means “the distance between x and 0 is less than c”. 
We are to draw all numbers which are within 7 units from the number 0.
------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0094.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example B. Draw the inequality |x| < 7.
-7
-7
7
0
x
Absolute Value Inequalities
Since |x| means “the distance between x and 0”, so the = expressio= n |x| < c means “the distance between x and 0 is less than c”. 
We are to draw all numbers which are within 7 units from the number 0.
x
------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0094_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqgEKAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAwCm AQUAgAAAAAAAAAJIhI+py+0Po5y02ouz3rz7Dz5BEJbmiabqyrbuq4wjINf2jef6zvf+DwwKh8Si 8YhMKpfMJhEGjUqn1Kr1CpFht9yu9wu2kg4FADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0094_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAB AA4AgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0094_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAO AA4AgAAAAAAAAAIfhB9xa6ng2GMQ0aXmrXqnjXXXBzYiRYLPaTVj58RLAQA7 ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0094_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAO AA4AgAAAAAAAAAIhhBFxecatlGtrImqRxPvxtVFdCIFYFp3d9bGa6IKRp8YFADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0094_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh7AAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAwDn AA8AgQAAAAAAAP8AAAECAwKOhI8JIuoPo5y02ouz3olxxHjfSJbmiZJh8wTSKqbyTNc2BLOJ++Tx DQwKhxOfzsDrGJfMpvMJjUqn1Kr1Kj0kDdiu9wsOi8dWwJaMTqvXbPSW247L53Q0smck6vd82xLx pgTTR1homOGjEKgweOj4CLmw0lIREnmJqWcJsYiT+QlKc0QZWmoa2XmqugpUAAA7 ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0095.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example B. Draw the inequality |x| < 7.
-7 < x < 7
-7
-7
7
0
x
Absolute Value Inequalities
Since |x| means “the distance between x and 0”, so the = expressio= n |x| < c means “the distance between x and 0 is less than c”. 
We are to draw all numbers which are within 7 units from the number 0.
x
------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0095_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqgEKAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAwCm AQUAgAAAAAAAAAJIhI+py+0Po5y02ouz3rz7Dz5BEJbmiabqyrbuq4wjINf2jef6zvf+DwwKh8Si 8YhMKpfMJhEGjUqn1Kr1CpFht9yu9wu2kg4FADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0095_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAB AA4AgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0095_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAO AA4AgAAAAAAAAAIfhB9xa6ng2GMQ0aXmrXqnjXXXBzYiRYLPaTVj58RLAQA7 ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0095_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAO AA4AgAAAAAAAAAIhhBFxecatlGtrImqRxPvxtVFdCIFYFp3d9bGa6IKRp8YFADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0095_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh7AAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAwDn AA8AgQAAAAAAAP8AAAECAwKOhI8JIuoPo5y02ouz3olxxHjfSJbmiZJh8wTSKqbyTNc2BLOJ++Tx DQwKhxOfzsDrGJfMpvMJjUqn1Kr1Kj0kDdiu9wsOi8dWwJaMTqvXbPSW247L53Q0smck6vd82xLx pgTTR1homOGjEKgweOj4CLmw0lIREnmJqWcJsYiT+QlKc0QZWmoa2XmqugpUAAA7 ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0096.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example B. Draw the inequality |x| < 7.
-7 < x < 7
-7
-7
7
0
x
Absolute Value Inequalities
Since |x| means “the distance between x and 0”, so the = expressio= n |x| < c means “the distance between x and 0 is less than c”. 
We are to draw all numbers which are within 7 units from the number 0.
x
The open circles means the end points are not included in the solut= ion.
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I. (<= /span>one= piece |   |–inequalities= )
Example B. Draw the inequality |x| < 7.
-7 < x < 7
-7
-7
7
0
x
Absolute Value Inequalities
Since |x| means “the distance between x and 0”, so the = expressio= n |x| < c means “the distance between x and 0 is less than c”. 
We are to draw all numbers which are within 7 units from the number 0.
x
The open circles means the end points are not included in the solut= ion.
------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0097_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqgEKAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAwCm AQUAgAAAAAAAAAJIhI+py+0Po5y02ouz3rz7Dz5BEJbmiabqyrbuq4wjINf2jef6zvf+DwwKh8Si 8YhMKpfMJhEGjUqn1Kr1CpFht9yu9wu2kg4FADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0097_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAB AA4AgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0097_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAO AA4AgAAAAAAAAAIfhB9xa6ng2GMQ0aXmrXqnjXXXBzYiRYLPaTVj58RLAQA7 ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0097_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAO AA4AgAAAAAAAAAIhhBFxecatlGtrImqRxPvxtVFdCIFYFp3d9bGa6IKRp8YFADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0097_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh7AAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAwDn AA8AgQAAAAAAAP8AAAECAwKOhI8JIuoPo5y02ouz3olxxHjfSJbmiZJh8wTSKqbyTNc2BLOJ++Tx DQwKhxOfzsDrGJfMpvMJjUqn1Kr1Kj0kDdiu9wsOi8dWwJaMTqvXbPSW247L53Q0smck6vd82xLx pgTTR1homOGjEKgweOj4CLmw0lIREnmJqWcJsYiT+QlKc0QZWmoa2XmqugpUAAA7 ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0098.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
I. (one= piece |   |–inequalities= )
If |x| < c then –c < x = < c.
Example B. Draw the inequality |x| < 7.
-7 < x < 7
-7
-7
7
0
x
Absolute Value Inequalities
Since |x| means “the distance between x and 0”, so the = expressio= n |x| < c means “the distance between x and 0 is less than c”. 
We are to draw all numbers which are within 7 units from the number 0.
x
The open circles means the end points are not included in the solut= ion.
------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0098_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqgEKAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAwCm AQUAgAAAAAAAAAJIhI+py+0Po5y02ouz3rz7Dz5BEJbmiabqyrbuq4wjINf2jef6zvf+DwwKh8Si 8YhMKpfMJhEGjUqn1Kr1CpFht9yu9wu2kg4FADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0098_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAB AA4AgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0098_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAO AA4AgAAAAAAAAAIfhB9xa6ng2GMQ0aXmrXqnjXXXBzYiRYLPaTVj58RLAQA7 ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0098_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAO AA4AgAAAAAAAAAIhhBFxecatlGtrImqRxPvxtVFdCIFYFp3d9bGa6IKRp8YFADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0098_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh7AAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAwDn AA8AgQAAAAAAAP8AAAECAwKOhI8JIuoPo5y02ouz3olxxHjfSJbmiZJh8wTSKqbyTNc2BLOJ++Tx DQwKhxOfzsDrGJfMpvMJjUqn1Kr1Kj0kDdiu9wsOi8dWwJaMTqvXbPSW247L53Q0smck6vd82xLx pgTTR1homOGjEKgweOj4CLmw0lIREnmJqWcJsYiT+QlKc0QZWmoa2XmqugpUAAA7 ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0099.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
I. (one= piece |   |–inequalities= )
If |x| < c then –c < x = < c.
Example B. Draw the inequality |x| < 7.
-7 < x < 7
-7
-7
7
0
x
Absolute Value Inequalities
Since |x| means “the distance between x and 0”, so the = expressio= n |x| < c means “the distance between x and 0 is less than c”. 
We are to draw all numbers which are within 7 units from the number 0.
x
In general, if we have=
        = ;            &n= bsp;       |expression| < c
The open circles means the end points are not included in the solut= ion.
------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0099_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqgEKAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAwCm AQUAgAAAAAAAAAJIhI+py+0Po5y02ouz3rz7Dz5BEJbmiabqyrbuq4wjINf2jef6zvf+DwwKh8Si 8YhMKpfMJhEGjUqn1Kr1CpFht9yu9wu2kg4FADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0099_image027.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAB AA4AgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0099_image028.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAO AA4AgAAAAAAAAAIfhB9xa6ng2GMQ0aXmrXqnjXXXBzYiRYLPaTVj58RLAQA7 ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0099_image029.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAO AA4AgAAAAAAAAAIhhBFxecatlGtrImqRxPvxtVFdCIFYFp3d9bGa6IKRp8YFADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0099_image030.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh7AAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAwDn AA8AgQAAAAAAAP8AAAECAwKOhI8JIuoPo5y02ouz3olxxHjfSJbmiZJh8wTSKqbyTNc2BLOJ++Tx DQwKhxOfzsDrGJfMpvMJjUqn1Kr1Kj0kDdiu9wsOi8dWwJaMTqvXbPSW247L53Q0smck6vd82xLx pgTTR1homOGjEKgweOj4CLmw0lIREnmJqWcJsYiT+QlKc0QZWmoa2XmqugpUAAA7 ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0100.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
I. (one= piece |   |–inequalities= )
If |x| < c then –c < x = < c.
Example B. Draw the inequality |x| < 7.
-7 < x < 7
-7
-7
7
0
x
Absolute Value Inequalities
Since |x| means “the distance between x and 0”, so the = expressio= n |x| < c means “the distance between x and 0 is less than c”. 
We are to draw all numbers which are within 7 units from the number 0.
x
In general, if we have=
        = ;            &n= bsp;       |expression| < c
we rewrite it without = the "|   |" as = ;
        = ;            &n= bsp; – c < expression < c. 
The open circles means the end points are not included in the solut= ion.
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Example C. Rewrite by dropping the "|     |“, don&#= 8217;t solve.
a.  | x + y | < 2
Absolute Value Inequalities
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Example C. Rewrite by dropping the "|     |“, don&#= 8217;t solve.
a.  | x + y | < 2
 
–2 < x + y < 2
Absolute Value Inequalities
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Example C. Rewrite by dropping the "|     |“, don&#= 8217;t solve.
a.  | x + y | < 2
 
–2 < x + y < 2
Absolute Value Inequalities
b.  | x2 – 2x + 1| < 1
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Example C. Rewrite by dropping the "|     |“, don&#= 8217;t solve.
a.  | x + y | < 2
 
–2 < x + y < 2
Absolute Value Inequalities
b.  | x2 – 2x + 1| < 1 =
 
–1 <x2 – 2x + 1 < 1
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Example C. Rewrite by dropping the "|     |“, don&#= 8217;t solve.
a.  | x + y | < 2
 
–2 < x + y < 2
Absolute Value Inequalities
b.  | x2 – 2x + 1| < 1 =
 
–1 <x2 – 2x + 1 < 1
Example D. Solve the inequa= lity |3 – 2x| < 7 and draw the solution.
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Example C. Rewrite by dropping the "|     |“, don&#= 8217;t solve.
a.  | x + y | < 2
 
–2 < x + y < 2
Absolute Value Inequalities
b.  | x2 – 2x + 1| < 1 =
 
–1 <x2 – 2x + 1 < 1
Example D. Solve the inequality |3 – 2x| < 7 and draw the = solution.
Rewrite the inequality without the |  | as
        = ;  –7 < 3 – 2x < 7
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Example C. Rewrite by dropping the "|     |“, don&#= 8217;t solve.
a.  | x + y | < 2
 
–2 < x + y < 2
Absolute Value Inequalities
b.  | x2 – 2x + 1| < 1 =
 
–1 <x2 – 2x + 1 < 1
Example D. Solve the inequality |3 – 2x| < 7 and draw the = solution.
Rewrite the inequality without the |  | as
        = ;  –7 < 3 – 2x < 7        = ;  subtract 3 from each part
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Example C. Rewrite by dropping the "|     |“, don&#= 8217;t solve.
a.  | x + y | < 2
 
–2 < x + y < 2
Absolute Value Inequalities
b.  | x2 – 2x + 1| < 1 =
 
–1 <x2 – 2x + 1 < 1
Example D. Solve the inequality |3 – 2x| < 7 and draw the = solution.
Rewrite the inequality without the |  | as
        = ;  –7 < 3 – 2x < 7        = ;  subtract 3 from each part
        = ;  –10 < –2x < 4
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Example C. Rewrite by dropping the "|     |“, don&#= 8217;t solve.
a.  | x + y | < 2
 
–2 < x + y < 2
Absolute Value Inequalities
b.  | x2 – 2x + 1| < 1 =
 
–1 <x2 – 2x + 1 < 1
Example D. Solve the inequality |3 – 2x| < 7 and draw the = solution.
Rewrite the inequality without the |  | as
        = ;  –7 < 3 – 2x < 7        = ;  subtract 3 from each part
        = ;  –10 < –2x < 4        = ;    divide by -2, need to switch
        = ;            &n= bsp;            = ;           the inequality around
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Example C. Rewrite by dropping the "|     |“, don&#= 8217;t solve.
a.  | x + y | < 2
 
–2 < x + y < 2
Absolute Value Inequalities
b.  | x2 – 2x + 1| < 1 =
 
–1 <x2 – 2x + 1 < 1
Example D. Solve the inequality |3 – 2x| < 7 and draw the = solution.
Rewrite the inequality without the |  | as
        = ;  –7 < 3 – 2x < 7        = ;  subtract 3 from each part
        = ;  –10 < –2x < 4        = ;    divide by -2, need to switch
        = ;            &n= bsp;            = ;           the inequality around
    –10/–2 > –2x/–2 > 4/–2
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Example C. Rewrite by dropping the "|     |“, don&#= 8217;t solve.
a.  | x + y | < 2
 
–2 < x + y < 2
Absolute Value Inequalities
b.  | x2 – 2x + 1| < 1 =
 
–1 <x2 – 2x + 1 < 1
Example D. Solve the inequality |3 – 2x| < 7 and draw the = solution.
Rewrite the inequality without the |  | as
        = ;  –7 < 3 – 2x < 7        = ;  subtract 3 from each part
        = ;  –10 < –2x < 4        = ;    divide by -2, need to switch
        = ;            &n= bsp;            = ;           the inequality around
    –10/–2 > –2x/–2 > 4/–2 <= /span>
        = ;       5 > x > –2        = ;            &n= bsp; 
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Example C. Rewrite by dropping the "|     |“, don&#= 8217;t solve.
a.  | x + y | < 2
 
–2 < x + y < 2
Absolute Value Inequalities
b.  | x2 – 2x + 1| < 1 =
 
–1 <x2 – 2x + 1 < 1
Example D. Solve the inequality |3 – 2x| < 7 and draw the = solution.
Rewrite the inequality without the |  | as
        = ;  –7 < 3 – 2x < 7        = ;  subtract 3 from each part
        = ;  –10 < –2x < 4        = ;    divide by -2, need to switch
        = ;            &n= bsp;            = ;           the inequality around
    –10/–2 > –2x/–2 > 4/–2 <= /span>
        = ;       5 > x > –2        = ;            &n= bsp; 
-2
5
0
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Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is = more than c”. 
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Example E. Draw the inequality |x| > 7.
Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is = more than c”. 
------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0114_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAI AAIAgAAAAAAAAAIEjI8ZBQA7 ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0115.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example E. Draw the inequality |x| > 7.
Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is = more than c”. 
We are to draw all x’s which are 7 or more units from the number 0.
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=
Example E. Draw the inequality |x| > 7.
Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is = more than c”. 
We are to draw all x’s which are 7 or more units from the number 0.
x < –7        = ;            &n= bsp; or     &n= bsp;            = ;     7 < x
-7<= /div>
7
0
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=
Example E. Draw the inequality |x| > 7.
Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is = more than c”. 
We are to draw all x’s which are 7 or more units from the number 0. This includes the end points 7 and –7.
x < –7        = ;            &n= bsp; or     &n= bsp;            = ;     7 < x
-7<= /div>
7
0
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=
Example E. Draw the inequality |x| > 7.
Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is = more than c”. 
We are to draw all x’s which are 7 or more units from the number 0. This includes the end points 7 and –7.
x < –7        = ;            &n= bsp; or     &n= bsp;            = ;     7 < x
-7<= /div>
7
0
The solid= circles means the end points are part of the solution.
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=
II. (<= /span>two= –piece |  |–inequalities)<= /b>
Example E. Draw the inequality |x| > 7.
Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is = more than c”. 
We are to draw all x’s which are 7 or more units from the number 0. This includes the end points 7 and –7.
x < –7        = ;            &n= bsp; or     &n= bsp;            = ;     7 < x
-7<= /div>
7
0
The solid= circles means the end points are part of the solution.
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=
Example E. Draw the inequality |x| > 7.
Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is = more than c”. 
We are to draw all x’s which are 7 or more units from the number 0. This includes the end points 7 and –7.
x < –7        = ;            &n= bsp; or     &n= bsp;            = ;     7 < x
-7<= /div>
7
0
The solid= circles means the end points are part of the solution.
II. (two= –piece |  |–inequalities)<= /b> =
If |x| > c then x &= lt; –c or that= c < x.
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=
Example E. Draw the inequality |x| > 7.
Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is = more than c”. 
We are to draw all x’s which are 7 or more units from the number 0. This includes the end points 7 and –7.
x < –7        = ;            &n= bsp; or     &n= bsp;            = ;     7 < x
-7<= /div>
7
0
-c
c
0
 c < x
x< –c =
The solid= circles means the end points are part of the solution.
II. (two= –piece |  |–inequalities)<= /b> =
If |x| > c then x &= lt; –c or that= c < x.
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R0lGODlhDAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAI AAEAgAAAAAAAAAIDjG8FADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0121_image113.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0122.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
=
Example E. Draw the inequality |x| > 7.
Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is = more than c”. 
We are to draw all x’s which are 7 or more units from the number 0. This includes the end points 7 and –7.
In general, if we have= the inequality
        = ;            &n= bsp;       |expression| > c
we drop the |  | and rewrite it as two inequalities
         expression < – c    or     c < expression 
x < –7        = ;            &n= bsp; or     &n= bsp;            = ;     7 < x
-7<= /div>
7
0
-c
c
0
 c < x
x< –c =
The solid= circles means the end points are part of the solution.
II. (two= –piece |  |–inequalities)<= /b> =
If |x| > c then x &= lt; –c or that= c < x.
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R0lGODlhDAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAI AAEAgAAAAAAAAAIDjG8FADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0122_image129.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAEAgAAAAAAAAAIDjH8FADs= ------=_NextPart_01C96AED.074F9820 Content-Location: file:///C:/1ECBB113/7AbsoluteInequalities_files/slide0123.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Absolute Inequalities
Example F. Rewrite without the "|     |“, don&#= 8217;t solve
A.  | x + y | > 2
Absolute Value Inequalities
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Example F. Rewrite without the "|     |“, don&#= 8217;t solve
A.  | x + y | > 2        = ;     
Drop the |  |, write i= t as two inequalities as
x + y < –2=         = ;            or        = ;         2 < x + y
Absolute Value Inequalities
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Example F. Rewrite without the "|     |“, don&#= 8217;t solve
A.  | x + y | > 2        = ;     
Drop the |  |, write i= t as two inequalities as
x + y < –2        = ;            or        = ;         2 < x + y        = ;            &n= bsp;          
B.  | x2 – 2x + 1| > 1
Absolute Value Inequalities
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Example F. Rewrite without the "|     |“, don&#= 8217;t solve
A.  | x + y | > 2        = ;     
Drop the |  |, write i= t as two inequalities as
x + y < –2        = ;            or        = ;         2 < x + y        = ;            &n= bsp;          
B.  | x2 – 2x + 1| > 1 =
Drop the |   |, write it as two inequali= ties as
x2 – 2x + 1 < –1=         = ;   or        = ;        1 < x2 – 2x + 1
Absolute Value Inequalities
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Example F. Rewrite without the "|     |“, don&#= 8217;t solve
A.  | x + y | > 2        = ;     
Drop the |  |, write i= t as two inequalities as
x + y < –2        = ;            or        = ;         2 < x + y        = ;            &n= bsp;          
B.  | x2 – 2x + 1| > 1 =
Drop the |   |, write it as two inequali= ties as
x2 – 2x + 1 < –1        = ;   or        = ;        1 < x2 – 2x + 1
Absolute Value Inequalities
Example G. Solve the inequality |3x – 2| > 4.  Draw.
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Example F. Rewrite without the "|     |“, don&#= 8217;t solve
A.  | x + y | > 2        = ;     
Drop the |  |, write i= t as two inequalities as
x + y < –2        = ;            or        = ;         2 < x + y        = ;            &n= bsp;          
B.  | x2 – 2x + 1| > 1 =
Drop the |   |, write it as two inequali= ties as
x2 – 2x + 1 < –1        = ;   or        = ;        1 < x2 – 2x + 1
Absolute Value Inequalities
Example G. Solve the inequality |3x – 2| > 4.  Draw.
Rewrite the inequality as two inequalities without the |  | as
        = ;    3x – 2 < - 4  &= nbsp; or     = 4 < 3x – 2
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Example F. Rewrite without the "|     |“, don&#= 8217;t solve
A.  | x + y | > 2        = ;     
Drop the |  |, write i= t as two inequalities as
x + y < –2        = ;            or        = ;         2 < x + y        = ;            &n= bsp;          
B.  | x2 – 2x + 1| > 1 =
Drop the |   |, write it as two inequali= ties as
x2 – 2x + 1 < –1        = ;   or        = ;        1 < x2 – 2x + 1
Absolute Value Inequalities
Example G. Solve the inequality |3x – 2| > 4.  Draw.
Rewrite the inequality as two inequalities without the |  | as
        = ;    3x – 2 < - 4  &= nbsp; or     = 4 < 3x – 2
        = ;         3x < -2