NeXT Education Software Sampler 1992 Fall

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LaTeX Document | 1992-06-23 | 10.8 KB
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|00000be0| 26 6c 74 3b 2f 49 26 67 | 74 3b 20 26 61 6d 70 3b |&lt;/I&g|t; &amp;|
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|00000c00| 67 74 3b 61 26 6c 74 3b | 2f 49 26 67 74 3b 7c 20 |gt;a&lt;|/I&gt;| |
|00000c10| 3d 20 26 6c 74 3b 49 26 | 67 74 3b 61 26 6c 74 3b |= &lt;I&|gt;a&lt;|
|00000c20| 2f 49 26 67 74 3b 2e 20 | 20 4d 6f 72 65 6f 76 65 |/I&gt;. | Moreove|
|00000c30| 72 20 20 7c 20 2d 20 26 | 6c 74 3b 49 26 67 74 3b |r  | - &|lt;I&gt;|
|00000c40| 61 26 6c 74 3b 2f 49 26 | 67 74 3b 7c 20 3d 20 26 |a&lt;/I&|gt;| = &|
|00000c50| 6c 74 3b 49 26 67 74 3b | 61 26 6c 74 3b 2f 49 26 |lt;I&gt;|a&lt;/I&|
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|00000c80| 20 4d 41 54 48 0a 20 3c | 21 2d 2d 20 4d 41 54 48 | MATH. <|!-- MATH|
|00000c90| 0a 20 24 7c 2d 61 7c 20 | 3d 20 7c 61 7c 24 0a 20 |. $|-a| |= |a|$. |
|00000ca0| 2d 2d 3e 0a 7c 20 2d 20 | 3c 49 3e 61 3c 2f 49 3e |-->.| - |<I>a</I>|
|00000cb0| 7c 20 3d 20 7c 20 3c 49 | 3e 61 3c 2f 49 3e 7c 0a || = | <I|>a</I>|.|
|00000cc0| 20 26 6e 64 61 73 68 3b | 26 67 74 3b 0a 7c 20 2d | &ndash;|&gt;.| -|
|00000cd0| 20 26 6c 74 3b 49 26 67 | 74 3b 61 26 6c 74 3b 2f | &lt;I&g|t;a&lt;/|
|00000ce0| 49 26 67 74 3b 7c 20 3d | 20 7c 20 26 6c 74 3b 49 |I&gt;| =| | &lt;I|
|00000cf0| 26 67 74 3b 61 26 6c 74 | 3b 2f 49 26 67 74 3b 7c |&gt;a&lt|;/I&gt;||
|00000d00| 20 61 6c 6c 20 74 68 65 | 20 74 69 6d 65 21 0a 0a | all the| time!..|
|00000d10| 3c 50 3e 0a 26 6c 74 3b | 50 26 67 74 3b 0a 26 6c |<P>.&lt;|P&gt;.&l|
|00000d20| 74 3b 2f 4c 49 26 67 74 | 3b 0a 26 6c 74 3b 4c 49 |t;/LI&gt|;.&lt;LI|
|00000d30| 26 67 74 3b 46 6f 72 20 | 26 6c 74 3b 49 26 67 74 |&gt;For |&lt;I&gt|
|00000d40| 3b 61 26 6c 74 3b 2f 49 | 26 67 74 3b 20 26 61 6d |;a&lt;/I|&gt; &am|
|00000d50| 70 3b 6c 74 3b 20 30 2c | 20 20 7c 20 26 6c 74 3b |p;lt; 0,|  | &lt;|
|00000d60| 49 26 67 74 3b 61 26 6c | 74 3b 2f 49 26 67 74 3b |I&gt;a&l|t;/I&gt;|
|00000d70| 7c 20 3d 20 2d 20 26 6c | 74 3b 49 26 67 74 3b 61 || = - &l|t;I&gt;a|
|00000d80| 26 6c 74 3b 2f 49 26 67 | 74 3b 3b 20 20 7c 20 2d |&lt;/I&g|t;;  | -|
|00000d90| 20 26 6c 74 3b 49 26 67 | 74 3b 61 26 6c 74 3b 2f | &lt;I&g|t;a&lt;/|
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|00000db0| 26 67 74 3b 61 26 6c 74 | 3b 2f 49 26 67 74 3b 2e |&gt;a&lt|;/I&gt;.|
|00000dc0| 0a 0a 3c 50 3e 0a 26 6c | 74 3b 50 26 67 74 3b 0a |..<P>.&l|t;P&gt;.|
|00000dd0| 26 6c 74 3b 2f 4c 49 26 | 67 74 3b 0a 26 6c 74 3b |&lt;/LI&|gt;.&lt;|
|00000de0| 4c 49 26 67 74 3b 46 6f | 72 20 26 6c 74 3b 49 26 |LI&gt;Fo|r &lt;I&|
|00000df0| 67 74 3b 78 26 6c 74 3b | 2f 49 26 67 74 3b 26 61 |gt;x&lt;|/I&gt;&a|
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|00000e10| 74 3b 61 26 6c 74 3b 2f | 49 26 67 74 3b 2c 20 74 |t;a&lt;/|I&gt;, t|
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|00000e40| 20 4d 41 54 48 0a 20 3c | 21 2d 2d 20 4d 41 54 48 | MATH. <|!-- MATH|
|00000e50| 0a 20 24 62 20 5c 6c 65 | 71 20 78 20 2d 20 61 20 |. $b \le|q x - a |
|00000e60| 5c 6c 65 71 20 63 24 0a | 20 2d 2d 3e 0a 3c 49 3e |\leq c$.| -->.<I>|
|00000e70| 62 3c 2f 49 3e 26 23 38 | 38 30 34 3b 3c 49 3e 78 |b</I>&#8|804;<I>x|
|00000e80| 3c 2f 49 3e 20 2d 20 3c | 49 3e 61 3c 2f 49 3e 26 |</I> - <|I>a</I>&|
|00000e90| 23 38 38 30 34 3b 3c 49 | 3e 63 3c 2f 49 3e 0a 20 |#8804;<I|>c</I>. |
|00000ea0| 26 6e 64 61 73 68 3b 26 | 67 74 3b 0a 26 6c 74 3b |&ndash;&|gt;.&lt;|
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|00000ed0| 26 67 74 3b 78 26 6c 74 | 3b 2f 49 26 67 74 3b 20 |&gt;x&lt|;/I&gt; |
|00000ee0| 2d 20 26 6c 74 3b 49 26 | 67 74 3b 61 26 6c 74 3b |- &lt;I&|gt;a&lt;|
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|00000f00| 3b 26 6c 74 3b 49 26 67 | 74 3b 63 26 6c 74 3b 2f |;&lt;I&g|t;c&lt;/|
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|00000f20| 6c 74 3b 49 26 67 74 3b | 61 26 6c 74 3b 2f 49 26 |lt;I&gt;|a&lt;/I&|
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|00000f50| 3b 20 4d 41 54 48 0a 20 | 3c 21 2d 2d 20 4d 41 54 |; MATH. |<!-- MAT|
|00000f60| 48 0a 20 24 61 20 2b 20 | 62 20 5c 6c 65 71 20 78 |H. $a + |b \leq x|
|00000f70| 20 5c 6c 65 71 20 61 20 | 2b 20 63 24 0a 20 2d 2d | \leq a |+ c$. --|
|00000f80| 3e 0a 3c 49 3e 61 3c 2f | 49 3e 20 2b 20 3c 49 3e |>.<I>a</|I> + <I>|
|00000f90| 62 3c 2f 49 3e 26 23 38 | 38 30 34 3b 3c 49 3e 78 |b</I>&#8|804;<I>x|
|00000fa0| 3c 2f 49 3e 26 23 38 38 | 30 34 3b 3c 49 3e 61 3c |</I>&#88|04;<I>a<|
|00000fb0| 2f 49 3e 20 2b 20 3c 49 | 3e 63 3c 2f 49 3e 0a 20 |/I> + <I|>c</I>. |
|00000fc0| 26 6e 64 61 73 68 3b 26 | 67 74 3b 0a 26 6c 74 3b |&ndash;&|gt;.&lt;|
|00000fd0| 49 26 67 74 3b 61 26 6c | 74 3b 2f 49 26 67 74 3b |I&gt;a&l|t;/I&gt;|
|00000fe0| 20 2b 20 26 6c 74 3b 49 | 26 67 74 3b 62 26 6c 74 | + &lt;I|&gt;b&lt|
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|00001020| 3b 26 6c 74 3b 49 26 67 | 74 3b 61 26 6c 74 3b 2f |;&lt;I&g|t;a&lt;/|
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|00001070| 38 30 35 3b 26 6c 74 3b | 49 26 67 74 3b 62 26 6c |805;&lt;|I&gt;b&l|
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|000010d0| 3b 62 26 6c 74 3b 2f 49 | 26 67 74 3b 26 61 6d 70 |;b&lt;/I|&gt;&amp|
|000010e0| 3b 23 38 38 30 35 3b 20 | 30 20 74 68 65 20 6c 6f |;#8805; |0 the lo|
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|00001140| 26 67 74 3b 61 26 6c 74 | 3b 2f 49 26 67 74 3b 2e |&gt;a&lt|;/I&gt;.|
|00001150| 20 20 54 68 75 73 20 70 | 61 72 74 20 6f 66 20 74 |  Thus p|art of t|
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|00001170| 69 73 20 74 68 65 20 63 | 6c 6f 73 65 64 20 69 6e |is the c|losed in|
|00001180| 74 65 72 76 61 6c 2c 20 | 20 26 6c 74 3b 21 26 6e |terval, | &lt;!&n|
|00001190| 64 61 73 68 3b 20 4d 41 | 54 48 0a 20 3c 21 2d 2d |dash; MA|TH. <!--|
|000011a0| 20 4d 41 54 48 0a 20 24 | 5b 61 20 2b 20 62 2c 20 | MATH. $|[a + b, |
|000011b0| 61 20 2b 20 63 5d 24 0a | 20 2d 2d 3e 0a 5b 3c 49 |a + c]$.| -->.[<I|
|000011c0| 3e 61 3c 2f 49 3e 20 2b | 20 3c 49 3e 62 3c 2f 49 |>a</I> +| <I>b</I|
|000011d0| 3e 2c 20 3c 49 3e 61 3c | 2f 49 3e 20 2b 20 3c 49 |>, <I>a<|/I> + <I|
|000011e0| 3e 63 3c 2f 49 3e 5d 0a | 20 26 6e 64 61 73 68 3b |>c</I>].| &ndash;|
|000011f0| 26 67 74 3b 0a 5b 26 6c | 74 3b 49 26 67 74 3b 61 |&gt;.[&l|t;I&gt;a|
|00001200| 26 6c 74 3b 2f 49 26 67 | 74 3b 20 2b 20 26 6c 74 |&lt;/I&g|t; + &lt|
|00001210| 3b 49 26 67 74 3b 62 26 | 6c 74 3b 2f 49 26 67 74 |;I&gt;b&|lt;/I&gt|
|00001220| 3b 2c 20 26 6c 74 3b 49 | 26 67 74 3b 61 26 6c 74 |;, &lt;I|&gt;a&lt|
|00001230| 3b 2f 49 26 67 74 3b 20 | 2b 20 26 6c 74 3b 49 26 |;/I&gt; |+ &lt;I&|
|00001240| 67 74 3b 63 26 6c 74 3b | 2f 49 26 67 74 3b 5d 2e |gt;c&lt;|/I&gt;].|
|00001250| 20 20 53 69 6d 69 6c 61 | 72 20 61 6e 61 6c 79 73 |  Simila|r analys|
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|00001270| 67 74 3b 78 26 6c 74 3b | 2f 49 26 67 74 3b 20 26 |gt;x&lt;|/I&gt; &|
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|00001290| 3b 61 26 6c 74 3b 2f 49 | 26 67 74 3b 20 72 61 6e |;a&lt;/I|&gt; ran|
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|000012e0| 2f 4c 49 26 67 74 3b 0a | 26 6c 74 3b 4c 49 26 67 |/LI&gt;.|&lt;LI&g|
|000012f0| 74 3b 26 6c 74 3b 4f 4c | 26 67 74 3b 0a 26 6c 74 |t;&lt;OL|&gt;.&lt|
|00001300| 3b 4c 49 26 67 74 3b 54 | 68 65 20 73 6f 6c 75 74 |;LI&gt;T|he solut|
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|00001320| 2e 0a 26 6c 74 3b 2f 4c | 49 26 67 74 3b 0a 26 6c |..&lt;/L|I&gt;.&l|
|00001330| 74 3b 4c 49 26 67 74 3b | 54 68 65 20 73 6f 6c 75 |t;LI&gt;|The solu|
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|00001350| 73 61 6d 65 20 61 73 20 | 66 6f 72 20 26 6c 74 3b |same as |for &lt;|
|00001360| 49 26 67 74 3b 62 26 6c | 74 3b 2f 49 26 67 74 3b |I&gt;b&l|t;/I&gt;|
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|00001390| 74 3b 0a 0a 3c 50 3e 0a | 26 6c 74 3b 50 26 67 74 |t;..<P>.|&lt;P&gt|
|000013a0| 3b 0a 26 6c 74 3b 2f 4c | 49 26 67 74 3b 0a 26 6c |;.&lt;/L|I&gt;.&l|
|000013b0| 74 3b 4c 49 26 67 74 3b | 26 6c 74 3b 4f 4c 26 67 |t;LI&gt;|&lt;OL&g|
|000013c0| 74 3b 0a 26 6c 74 3b 4c | 49 26 67 74 3b 54 68 65 |t;.&lt;L|I&gt;The|
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|00001410| 20 63 24 0a 20 2d 2d 3e | 0a 3c 49 3e 62 3c 2f 49 | c$. -->|.<I>b</I|
|00001420| 3e 26 23 38 38 30 34 3b | 7c 20 3c 49 3e 78 3c 2f |>&#8804;|| <I>x</|
|00001430| 49 3e 20 2d 20 3c 49 3e | 61 3c 2f 49 3e 7c 20 26 |I> - <I>|a</I>| &|
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|00001450| 64 61 73 68 3b 26 67 74 | 3b 0a 26 6c 74 3b 49 26 |dash;&gt|;.&lt;I&|
|00001460| 67 74 3b 62 26 6c 74 3b | 2f 49 26 67 74 3b 26 61 |gt;b&lt;|/I&gt;&a|
|00001470| 6d 70 3b 23 38 38 30 34 | 3b 7c 20 26 6c 74 3b 49 |mp;#8804|;| &lt;I|
|00001480| 26 67 74 3b 78 26 6c 74 | 3b 2f 49 26 67 74 3b 20 |&gt;x&lt|;/I&gt; |
|00001490| 2d 20 26 6c 74 3b 49 26 | 67 74 3b 61 26 6c 74 3b |- &lt;I&|gt;a&lt;|
|000014a0| 2f 49 26 67 74 3b 7c 20 | 26 61 6d 70 3b 6c 74 3b |/I&gt;| |&amp;lt;|
|000014b0| 20 26 6c 74 3b 49 26 67 | 74 3b 63 26 6c 74 3b 2f | &lt;I&g|t;c&lt;/|
|000014c0| 49 26 67 74 3b 20 69 73 | 20 26 6c 74 3b 21 26 6e |I&gt; is| &lt;!&n|
|000014d0| 64 61 73 68 3b 20 4d 41 | 54 48 0a 20 3c 21 2d 2d |dash; MA|TH. <!--|
|000014e0| 20 4d 41 54 48 0a 20 24 | 28 61 20 2d 20 63 2c 20 | MATH. $|(a - c, |
|000014f0| 61 20 2d 20 62 5d 20 5c | 63 75 70 20 5b 61 20 2b |a - b] \|cup [a +|
|00001500| 20 62 2c 20 61 20 2b 20 | 63 29 24 0a 20 2d 2d 3e | b, a + |c)$. -->|
|00001510| 0a 28 3c 49 3e 61 3c 2f | 49 3e 20 2d 20 3c 49 3e |.(<I>a</|I> - <I>|
|00001520| 63 3c 2f 49 3e 2c 20 3c | 49 3e 61 3c 2f 49 3e 20 |c</I>, <|I>a</I> |
|00001530| 2d 20 3c 49 3e 62 3c 2f | 49 3e 5d 26 23 38 37 34 |- <I>b</|I>]&#874|
|00001540| 36 3b 5b 3c 49 3e 61 3c | 2f 49 3e 20 2b 20 3c 49 |6;[<I>a<|/I> + <I|
|00001550| 3e 62 3c 2f 49 3e 2c 20 | 3c 49 3e 61 3c 2f 49 3e |>b</I>, |<I>a</I>|
|00001560| 20 2b 20 3c 49 3e 63 3c | 2f 49 3e 29 0a 20 26 6e | + <I>c<|/I>). &n|
|00001570| 64 61 73 68 3b 26 67 74 | 3b 0a 28 26 6c 74 3b 49 |dash;&gt|;.(&lt;I|
|00001580| 26 67 74 3b 61 26 6c 74 | 3b 2f 49 26 67 74 3b 20 |&gt;a&lt|;/I&gt; |
|00001590| 2d 20 26 6c 74 3b 49 26 | 67 74 3b 63 26 6c 74 3b |- &lt;I&|gt;c&lt;|
|000015a0| 2f 49 26 67 74 3b 2c 20 | 26 6c 74 3b 49 26 67 74 |/I&gt;, |&lt;I&gt|
|000015b0| 3b 61 26 6c 74 3b 2f 49 | 26 67 74 3b 20 2d 20 26 |;a&lt;/I|&gt; - &|
|000015c0| 6c 74 3b 49 26 67 74 3b | 62 26 6c 74 3b 2f 49 26 |lt;I&gt;|b&lt;/I&|
|000015d0| 67 74 3b 5d 26 61 6d 70 | 3b 23 38 37 34 36 3b 5b |gt;]&amp|;#8746;[|
|000015e0| 26 6c 74 3b 49 26 67 74 | 3b 61 26 6c 74 3b 2f 49 |&lt;I&gt|;a&lt;/I|
|000015f0| 26 67 74 3b 20 2b 20 26 | 6c 74 3b 49 26 67 74 3b |&gt; + &|lt;I&gt;|
|00001600| 62 26 6c 74 3b 2f 49 26 | 67 74 3b 2c 20 26 6c 74 |b&lt;/I&|gt;, &lt|
|00001610| 3b 49 26 67 74 3b 61 26 | 6c 74 3b 2f 49 26 67 74 |;I&gt;a&|lt;/I&gt|
|00001620| 3b 20 2b 20 26 6c 74 3b | 49 26 67 74 3b 63 26 6c |; + &lt;|I&gt;c&l|
|00001630| 74 3b 2f 49 26 67 74 3b | 29 2e 0a 26 6c 74 3b 2f |t;/I&gt;|)..&lt;/|
|00001640| 4c 49 26 67 74 3b 0a 26 | 6c 74 3b 4c 49 26 67 74 |LI&gt;.&|lt;LI&gt|
|00001650| 3b 54 68 65 20 73 6f 6c | 75 74 69 6f 6e 20 73 65 |;The sol|ution se|
|00001660| 74 20 6f 66 20 26 6c 74 | 3b 21 26 6e 64 61 73 68 |t of &lt|;!&ndash|
|00001670| 3b 20 4d 41 54 48 0a 20 | 3c 21 2d 2d 20 4d 41 54 |; MATH. |<!-- MAT|
|00001680| 48 0a 20 24 62 20 3c 20 | 7c 78 20 2d 20 61 7c 20 |H. $b < ||x - a| |
|00001690| 5c 6c 65 71 20 63 24 0a | 20 2d 2d 3e 0a 3c 49 3e |\leq c$.| -->.<I>|
|000016a0| 62 3c 2f 49 3e 20 26 6c | 74 3b 20 7c 20 3c 49 3e |b</I> &l|t; | <I>|
|000016b0| 78 3c 2f 49 3e 20 2d 20 | 3c 49 3e 61 3c 2f 49 3e |x</I> - |<I>a</I>|
|000016c0| 7c 26 23 38 38 30 34 3b | 3c 49 3e 63 3c 2f 49 3e ||&#8804;|<I>c</I>|
|000016d0| 0a 20 26 6e 64 61 73 68 | 3b 26 67 74 3b 0a 26 6c |. &ndash|;&gt;.&l|
|000016e0| 74 3b 49 26 67 74 3b 62 | 26 6c 74 3b 2f 49 26 67 |t;I&gt;b|&lt;/I&g|
|000016f0| 74 3b 20 26 61 6d 70 3b | 6c 74 3b 20 7c 20 26 6c |t; &amp;|lt; | &l|
|00001700| 74 3b 49 26 67 74 3b 78 | 26 6c 74 3b 2f 49 26 67 |t;I&gt;x|&lt;/I&g|
|00001710| 74 3b 20 2d 20 26 6c 74 | 3b 49 26 67 74 3b 61 26 |t; - &lt|;I&gt;a&|
|00001720| 6c 74 3b 2f 49 26 67 74 | 3b 7c 26 61 6d 70 3b 23 |lt;/I&gt|;|&amp;#|
|00001730| 38 38 30 34 3b 26 6c 74 | 3b 49 26 67 74 3b 63 26 |8804;&lt|;I&gt;c&|
|00001740| 6c 74 3b 2f 49 26 67 74 | 3b 20 69 73 20 26 6c 74 |lt;/I&gt|; is &lt|
|00001750| 3b 21 26 6e 64 61 73 68 | 3b 20 4d 41 54 48 0a 20 |;!&ndash|; MATH. |
|00001760| 3c 21 2d 2d 20 4d 41 54 | 48 0a 20 24 5b 61 20 2d |<!-- MAT|H. $[a -|
|00001770| 20 63 2c 20 61 20 2d 20 | 62 29 20 5c 63 75 70 20 | c, a - |b) \cup |
|00001780| 28 61 20 2b 20 62 2c 20 | 61 20 2b 20 63 5d 24 0a |(a + b, |a + c]$.|
|00001790| 20 2d 2d 3e 0a 5b 3c 49 | 3e 61 3c 2f 49 3e 20 2d | -->.[<I|>a</I> -|
|000017a0| 20 3c 49 3e 63 3c 2f 49 | 3e 2c 20 3c 49 3e 61 3c | <I>c</I|>, <I>a<|
|000017b0| 2f 49 3e 20 2d 20 3c 49 | 3e 62 3c 2f 49 3e 29 26 |/I> - <I|>b</I>)&|
|000017c0| 23 38 37 34 36 3b 28 3c | 49 3e 61 3c 2f 49 3e 20 |#8746;(<|I>a</I> |
|000017d0| 2b 20 3c 49 3e 62 3c 2f | 49 3e 2c 20 3c 49 3e 61 |+ <I>b</|I>, <I>a|
|000017e0| 3c 2f 49 3e 20 2b 20 3c | 49 3e 63 3c 2f 49 3e 5d |</I> + <|I>c</I>]|
|000017f0| 0a 20 26 6e 64 61 73 68 | 3b 26 67 74 3b 0a 5b 26 |. &ndash|;&gt;.[&|
|00001800| 6c 74 3b 49 26 67 74 3b | 61 26 6c 74 3b 2f 49 26 |lt;I&gt;|a&lt;/I&|
|00001810| 67 74 3b 20 2d 20 26 6c | 74 3b 49 26 67 74 3b 63 |gt; - &l|t;I&gt;c|
|00001820| 26 6c 74 3b 2f 49 26 67 | 74 3b 2c 20 26 6c 74 3b |&lt;/I&g|t;, &lt;|
|00001830| 49 26 67 74 3b 61 26 6c | 74 3b 2f 49 26 67 74 3b |I&gt;a&l|t;/I&gt;|
|00001840| 20 2d 20 26 6c 74 3b 49 | 26 67 74 3b 62 26 6c 74 | - &lt;I|&gt;b&lt|
|00001850| 3b 2f 49 26 67 74 3b 29 | 26 61 6d 70 3b 23 38 37 |;/I&gt;)|&amp;#87|
|00001860| 34 36 3b 28 26 6c 74 3b | 49 26 67 74 3b 61 26 6c |46;(&lt;|I&gt;a&l|
|00001870| 74 3b 2f 49 26 67 74 3b | 20 2b 20 26 6c 74 3b 49 |t;/I&gt;| + &lt;I|
|00001880| 26 67 74 3b 62 26 6c 74 | 3b 2f 49 26 67 74 3b 2c |&gt;b&lt|;/I&gt;,|
|00001890| 20 26 6c 74 3b 49 26 67 | 74 3b 61 26 6c 74 3b 2f | &lt;I&g|t;a&lt;/|
|000018a0| 49 26 67 74 3b 20 2b 20 | 26 6c 74 3b 49 26 67 74 |I&gt; + |&lt;I&gt|
|000018b0| 3b 63 26 6c 74 3b 2f 49 | 26 67 74 3b 5d 2e 0a 26 |;c&lt;/I|&gt;]..&|
|000018c0| 6c 74 3b 2f 4c 49 26 67 | 74 3b 0a 26 6c 74 3b 4c |lt;/LI&g|t;.&lt;L|
|000018d0| 49 26 67 74 3b 54 68 65 | 20 73 6f 6c 75 74 69 6f |I&gt;The| solutio|
|000018e0| 6e 20 73 65 74 20 6f 66 | 20 26 6c 74 3b 21 26 6e |n set of| &lt;!&n|
|000018f0| 64 61 73 68 3b 20 4d 41 | 54 48 0a 20 3c 21 2d 2d |dash; MA|TH. <!--|
|00001900| 20 4d 41 54 48 0a 20 24 | 62 20 3c 7c 78 20 2d 20 | MATH. $|b <|x - |
|00001910| 61 7c 20 5c 20 3c 63 24 | 0a 20 2d 2d 3e 0a 3c 49 |a| \ <c$|. -->.<I|
|00001920| 3e 62 3c 2f 49 3e 20 26 | 6c 74 3b 20 7c 20 3c 49 |>b</I> &|lt; | <I|
|00001930| 3e 78 3c 2f 49 3e 20 2d | 20 3c 49 3e 61 3c 2f 49 |>x</I> -| <I>a</I|
|00001940| 3e 7c 26 6e 62 73 70 3b | 20 26 6c 74 3b 20 3c 49 |>|&nbsp;| &lt; <I|
|00001950| 3e 63 3c 2f 49 3e 0a 20 | 26 6e 64 61 73 68 3b 26 |>c</I>. |&ndash;&|
|00001960| 67 74 3b 0a 26 6c 74 3b | 49 26 67 74 3b 62 26 6c |gt;.&lt;|I&gt;b&l|
|00001970| 74 3b 2f 49 26 67 74 3b | 20 26 61 6d 70 3b 6c 74 |t;/I&gt;| &amp;lt|
|00001980| 3b 20 7c 20 26 6c 74 3b | 49 26 67 74 3b 78 26 6c |; | &lt;|I&gt;x&l|
|00001990| 74 3b 2f 49 26 67 74 3b | 20 2d 20 26 6c 74 3b 49 |t;/I&gt;| - &lt;I|
|000019a0| 26 67 74 3b 61 26 6c 74 | 3b 2f 49 26 67 74 3b 7c |&gt;a&lt|;/I&gt;||
|000019b0| 26 61 6d 70 3b 6e 62 73 | 70 3b 20 26 61 6d 70 3b |&amp;nbs|p; &amp;|
|000019c0| 6c 74 3b 20 26 6c 74 3b | 49 26 67 74 3b 63 26 6c |lt; &lt;|I&gt;c&l|
|000019d0| 74 3b 2f 49 26 67 74 3b | 20 69 73 20 26 6c 74 3b |t;/I&gt;| is &lt;|
|000019e0| 21 26 6e 64 61 73 68 3b | 20 4d 41 54 48 0a 20 3c |!&ndash;| MATH. <|
|000019f0| 21 2d 2d 20 4d 41 54 48 | 0a 20 24 28 61 20 2d 20 |!-- MATH|. $(a - |
|00001a00| 63 2c 20 61 20 2d 20 62 | 29 20 5c 63 75 70 20 28 |c, a - b|) \cup (|
|00001a10| 61 20 2b 20 62 2c 20 61 | 20 2b 20 63 29 24 0a 20 |a + b, a| + c)$. |
|00001a20| 2d 2d 3e 0a 28 3c 49 3e | 61 3c 2f 49 3e 20 2d 20 |-->.(<I>|a</I> - |
|00001a30| 3c 49 3e 63 3c 2f 49 3e | 2c 20 3c 49 3e 61 3c 2f |<I>c</I>|, <I>a</|
|00001a40| 49 3e 20 2d 20 3c 49 3e | 62 3c 2f 49 3e 29 26 23 |I> - <I>|b</I>)&#|
|00001a50| 38 37 34 36 3b 28 3c 49 | 3e 61 3c 2f 49 3e 20 2b |8746;(<I|>a</I> +|
|00001a60| 20 3c 49 3e 62 3c 2f 49 | 3e 2c 20 3c 49 3e 61 3c | <I>b</I|>, <I>a<|
|00001a70| 2f 49 3e 20 2b 20 3c 49 | 3e 63 3c 2f 49 3e 29 0a |/I> + <I|>c</I>).|
|00001a80| 20 26 6e 64 61 73 68 3b | 26 67 74 3b 0a 28 26 6c | &ndash;|&gt;.(&l|
|00001a90| 74 3b 49 26 67 74 3b 61 | 26 6c 74 3b 2f 49 26 67 |t;I&gt;a|&lt;/I&g|
|00001aa0| 74 3b 20 2d 20 26 6c 74 | 3b 49 26 67 74 3b 63 26 |t; - &lt|;I&gt;c&|
|00001ab0| 6c 74 3b 2f 49 26 67 74 | 3b 2c 20 26 6c 74 3b 49 |lt;/I&gt|;, &lt;I|
|00001ac0| 26 67 74 3b 61 26 6c 74 | 3b 2f 49 26 67 74 3b 20 |&gt;a&lt|;/I&gt; |
|00001ad0| 2d 20 26 6c 74 3b 49 26 | 67 74 3b 62 26 6c 74 3b |- &lt;I&|gt;b&lt;|
|00001ae0| 2f 49 26 67 74 3b 29 26 | 61 6d 70 3b 23 38 37 34 |/I&gt;)&|amp;#874|
|00001af0| 36 3b 28 26 6c 74 3b 49 | 26 67 74 3b 61 26 6c 74 |6;(&lt;I|&gt;a&lt|
|00001b00| 3b 2f 49 26 67 74 3b 20 | 2b 20 26 6c 74 3b 49 26 |;/I&gt; |+ &lt;I&|
|00001b10| 67 74 3b 62 26 6c 74 3b | 2f 49 26 67 74 3b 2c 20 |gt;b&lt;|/I&gt;, |
|00001b20| 26 6c 74 3b 49 26 67 74 | 3b 61 26 6c 74 3b 2f 49 |&lt;I&gt|;a&lt;/I|
|00001b30| 26 67 74 3b 20 2b 20 26 | 6c 74 3b 49 26 67 74 3b |&gt; + &|lt;I&gt;|
|00001b40| 63 26 6c 74 3b 2f 49 26 | 67 74 3b 29 2e 0a 0a 3c |c&lt;/I&|gt;)...<|
|00001b50| 50 3e 0a 26 6c 74 3b 2f | 4c 49 26 67 74 3b 0a 26 |P>.&lt;/|LI&gt;.&|
|00001b60| 6c 74 3b 2f 4f 4c 26 67 | 74 3b 0a 0a 3c 50 3e 0a |lt;/OL&g|t;..<P>.|
|00001b70| 26 6c 74 3b 50 26 67 74 | 3b 0a 26 6c 74 3b 2f 4c |&lt;P&gt|;.&lt;/L|
|00001b80| 49 26 67 74 3b 0a 26 6c | 74 3b 4c 49 26 67 74 3b |I&gt;.&l|t;LI&gt;|
|00001b90| 28 31 2e 31 39 29 20 41 | 66 74 65 72 20 64 69 76 |(1.19) A|fter div|
|00001ba0| 69 64 69 6e 67 20 62 79 | 20 35 2c 20 77 65 20 73 |iding by| 5, we s|
|00001bb0| 65 65 20 74 68 61 74 20 | 26 6c 74 3b 21 26 6e 64 |ee that |&lt;!&nd|
|00001bc0| 61 73 68 3b 20 4d 41 54 | 48 0a 20 3c 21 2d 2d 20 |ash; MAT|H. <!-- |
|00001bd0| 4d 41 54 48 0a 20 24 61 | 20 3d 20 34 2f 35 2c 20 |MATH. $a| = 4/5, |
|00001be0| 62 20 3d 20 32 2f 35 2c | 20 63 20 3d 20 34 2f 35 |b = 2/5,| c = 4/5|
|00001bf0| 24 0a 20 2d 2d 3e 0a 3c | 49 3e 61 3c 2f 49 3e 20 |$. -->.<|I>a</I> |
|00001c00| 3d 20 34 2f 35 2c 20 3c | 49 3e 62 3c 2f 49 3e 20 |= 4/5, <|I>b</I> |
|00001c10| 3d 20 32 2f 35 2c 20 3c | 49 3e 63 3c 2f 49 3e 20 |= 2/5, <|I>c</I> |
|00001c20| 3d 20 34 2f 35 0a 20 26 | 6e 64 61 73 68 3b 26 67 |= 4/5. &|ndash;&g|
|00001c30| 74 3b 0a 26 6c 74 3b 49 | 26 67 74 3b 61 26 6c 74 |t;.&lt;I|&gt;a&lt|
|00001c40| 3b 2f 49 26 67 74 3b 20 | 3d 20 34 2f 35 2c 20 26 |;/I&gt; |= 4/5, &|
|00001c50| 6c 74 3b 49 26 67 74 3b | 62 26 6c 74 3b 2f 49 26 |lt;I&gt;|b&lt;/I&|
|00001c60| 67 74 3b 20 3d 20 32 2f | 35 2c 20 26 6c 74 3b 49 |gt; = 2/|5, &lt;I|
|00001c70| 26 67 74 3b 63 26 6c 74 | 3b 2f 49 26 67 74 3b 20 |&gt;c&lt|;/I&gt; |
|00001c80| 3d 20 34 2f 35 2e 20 20 | 54 68 75 73 20 74 68 65 |= 4/5.  |Thus the|
|00001c90| 20 73 6f 6c 75 74 69 6f | 6e 20 73 65 74 20 26 6c | solutio|n set &l|
|00001ca0| 74 3b 21 26 6e 64 61 73 | 68 3b 20 4d 41 54 48 0a |t;!&ndas|h; MATH.|
|00001cb0| 20 3c 21 2d 2d 20 4d 41 | 54 48 0a 20 24 5b 61 20 | <!-- MA|TH. $[a |
|00001cc0| 2d 20 63 2c 20 61 20 2d | 62 5d 20 5c 63 75 70 20 |- c, a -|b] \cup |
|00001cd0| 5b 61 20 2b 20 62 2c 20 | 61 20 2b 20 63 5d 24 0a |[a + b, |a + c]$.|
|00001ce0| 20 2d 2d 3e 0a 5b 3c 49 | 3e 61 3c 2f 49 3e 20 2d | -->.[<I|>a</I> -|
|00001cf0| 20 3c 49 3e 63 3c 2f 49 | 3e 2c 20 3c 49 3e 61 3c | <I>c</I|>, <I>a<|
|00001d00| 2f 49 3e 20 2d 20 3c 49 | 3e 62 3c 2f 49 3e 5d 26 |/I> - <I|>b</I>]&|
|00001d10| 23 38 37 34 36 3b 5b 3c | 49 3e 61 3c 2f 49 3e 20 |#8746;[<|I>a</I> |
|00001d20| 2b 20 3c 49 3e 62 3c 2f | 49 3e 2c 20 3c 49 3e 61 |+ <I>b</|I>, <I>a|
|00001d30| 3c 2f 49 3e 20 2b 20 3c | 49 3e 63 3c 2f 49 3e 5d |</I> + <|I>c</I>]|
|00001d40| 0a 20 26 6e 64 61 73 68 | 3b 26 67 74 3b 0a 5b 26 |. &ndash|;&gt;.[&|
|00001d50| 6c 74 3b 49 26 67 74 3b | 61 26 6c 74 3b 2f 49 26 |lt;I&gt;|a&lt;/I&|
|00001d60| 67 74 3b 20 2d 20 26 6c | 74 3b 49 26 67 74 3b 63 |gt; - &l|t;I&gt;c|
|00001d70| 26 6c 74 3b 2f 49 26 67 | 74 3b 2c 20 26 6c 74 3b |&lt;/I&g|t;, &lt;|
|00001d80| 49 26 67 74 3b 61 26 6c | 74 3b 2f 49 26 67 74 3b |I&gt;a&l|t;/I&gt;|
|00001d90| 20 2d 20 26 6c 74 3b 49 | 26 67 74 3b 62 26 6c 74 | - &lt;I|&gt;b&lt|
|00001da0| 3b 2f 49 26 67 74 3b 5d | 26 61 6d 70 3b 23 38 37 |;/I&gt;]|&amp;#87|
|00001db0| 34 36 3b 5b 26 6c 74 3b | 49 26 67 74 3b 61 26 6c |46;[&lt;|I&gt;a&l|
|00001dc0| 74 3b 2f 49 26 67 74 3b | 20 2b 20 26 6c 74 3b 49 |t;/I&gt;| + &lt;I|
|00001dd0| 26 67 74 3b 62 26 6c 74 | 3b 2f 49 26 67 74 3b 2c |&gt;b&lt|;/I&gt;,|
|00001de0| 20 26 6c 74 3b 49 26 67 | 74 3b 61 26 6c 74 3b 2f | &lt;I&g|t;a&lt;/|
|00001df0| 49 26 67 74 3b 20 2b 20 | 26 6c 74 3b 49 26 67 74 |I&gt; + |&lt;I&gt|
|00001e00| 3b 63 26 6c 74 3b 2f 49 | 26 67 74 3b 5d 20 62 65 |;c&lt;/I|&gt;] be|
|00001e10| 63 6f 6d 65 73 20 26 6c | 74 3b 21 26 6e 64 61 73 |comes &l|t;!&ndas|
|00001e20| 68 3b 20 4d 41 54 48 0a | 20 3c 21 2d 2d 20 4d 41 |h; MATH.| <!-- MA|
|00001e30| 54 48 0a 20 24 5b 30 2c | 20 32 2f 35 5d 20 5c 63 |TH. $[0,| 2/5] \c|
|00001e40| 75 70 20 5b 36 2f 35 2c | 20 38 2f 35 5d 24 0a 20 |up [6/5,| 8/5]$. |
|00001e50| 2d 2d 3e 0a 5b 30 2c 20 | 32 2f 35 5d 26 23 38 37 |-->.[0, |2/5]&#87|
|00001e60| 34 36 3b 5b 36 2f 35 2c | 20 38 2f 35 5d 0a 20 26 |46;[6/5,| 8/5]. &|
|00001e70| 6e 64 61 73 68 3b 26 67 | 74 3b 0a 5b 30 2c 20 32 |ndash;&g|t;.[0, 2|
|00001e80| 2f 35 5d 26 61 6d 70 3b | 23 38 37 34 36 3b 5b 36 |/5]&amp;|#8746;[6|
|00001e90| 2f 35 2c 20 38 2f 35 5d | 2e 0a 0a 3c 50 3e 0a 26 |/5, 8/5]|...<P>.&|
|00001ea0| 6c 74 3b 50 26 67 74 3b | 0a 26 6c 74 3b 2f 4c 49 |lt;P&gt;|.&lt;/LI|
|00001eb0| 26 67 74 3b 0a 26 6c 74 | 3b 4c 49 26 67 74 3b 28 |&gt;.&lt|;LI&gt;(|
|00001ec0| 31 2e 31 37 29 20 49 66 | 20 26 6c 74 3b 21 26 6e |1.17) If| &lt;!&n|
|00001ed0| 64 61 73 68 3b 20 4d 41 | 54 48 0a 20 3c 21 2d 2d |dash; MA|TH. <!--|
|00001ee0| 20 4d 41 54 48 0a 20 24 | 63 20 5c 72 69 67 68 74 | MATH. $|c \right|
|00001ef0| 61 72 72 6f 77 20 2b 5c | 69 6e 66 74 79 24 0a 20 |arrow +\|infty$. |
|00001f00| 2d 2d 3e 0a 3c 49 3e 63 | 3c 2f 49 3e 26 23 38 35 |-->.<I>c|</I>&#85|
|00001f10| 39 34 3b 20 2b 20 26 23 | 38 37 33 34 3b 0a 20 26 |94; + &#|8734;. &|
|00001f20| 6e 64 61 73 68 3b 26 67 | 74 3b 0a 26 6c 74 3b 49 |ndash;&g|t;.&lt;I|
|00001f30| 26 67 74 3b 63 26 6c 74 | 3b 2f 49 26 67 74 3b 26 |&gt;c&lt|;/I&gt;&|
|00001f40| 61 6d 70 3b 23 38 35 39 | 34 3b 20 2b 20 26 61 6d |amp;#859|4; + &am|
|00001f50| 70 3b 23 38 37 33 34 3b | 2c 20 74 68 65 6e 20 74 |p;#8734;|, then t|
|00001f60| 68 65 20 73 6f 6c 75 74 | 69 6f 6e 20 73 65 74 20 |he solut|ion set |
|00001f70| 62 65 63 6f 6d 65 73 20 | 26 6c 74 3b 21 26 6e 64 |becomes |&lt;!&nd|
|00001f80| 61 73 68 3b 20 4d 41 54 | 48 0a 20 3c 21 2d 2d 20 |ash; MAT|H. <!-- |
|00001f90| 4d 41 54 48 0a 20 24 28 | 2d 5c 69 6e 66 74 79 2c |MATH. $(|-\infty,|
|00001fa0| 20 61 20 2d 62 5d 20 5c | 63 75 70 20 5b 61 20 2b | a -b] \|cup [a +|
|00001fb0| 20 62 2c 20 2b 5c 69 6e | 66 74 79 29 24 0a 20 2d | b, +\in|fty)$. -|
|00001fc0| 2d 3e 0a 28 2d 20 26 23 | 38 37 33 34 3b 2c 20 3c |->.(- &#|8734;, <|
|00001fd0| 49 3e 61 3c 2f 49 3e 20 | 2d 20 3c 49 3e 62 3c 2f |I>a</I> |- <I>b</|
|00001fe0| 49 3e 5d 26 23 38 37 34 | 36 3b 5b 3c 49 3e 61 3c |I>]&#874|6;[<I>a<|
|00001ff0| 2f 49 3e 20 2b 20 3c 49 | 3e 62 3c 2f 49 3e 2c 20 |/I> + <I|>b</I>, |
|00002000| 2b 20 26 23 38 37 33 34 | 3b 29 0a 20 26 6e 64 61 |+ &#8734|;). &nda|
|00002010| 73 68 3b 26 67 74 3b 0a | 28 2d 20 26 61 6d 70 3b |sh;&gt;.|(- &amp;|
|00002020| 23 38 37 33 34 3b 2c 20 | 26 6c 74 3b 49 26 67 74 |#8734;, |&lt;I&gt|
|00002030| 3b 61 26 6c 74 3b 2f 49 | 26 67 74 3b 20 2d 20 26 |;a&lt;/I|&gt; - &|
|00002040| 6c 74 3b 49 26 67 74 3b | 62 26 6c 74 3b 2f 49 26 |lt;I&gt;|b&lt;/I&|
|00002050| 67 74 3b 5d 26 61 6d 70 | 3b 23 38 37 34 36 3b 5b |gt;]&amp|;#8746;[|
|00002060| 26 6c 74 3b 49 26 67 74 | 3b 61 26 6c 74 3b 2f 49 |&lt;I&gt|;a&lt;/I|
|00002070| 26 67 74 3b 20 2b 20 26 | 6c 74 3b 49 26 67 74 3b |&gt; + &|lt;I&gt;|
|00002080| 62 26 6c 74 3b 2f 49 26 | 67 74 3b 2c 20 2b 20 26 |b&lt;/I&|gt;, + &|
|00002090| 61 6d 70 3b 23 38 37 33 | 34 3b 29 2e 20 20 49 6e |amp;#873|4;).  In|
|000020a0| 65 71 75 61 6c 69 74 79 | 20 26 6c 74 3b 21 26 6e |equality| &lt;!&n|
|000020b0| 64 61 73 68 3b 20 4d 41 | 54 48 0a 20 3c 21 2d 2d |dash; MA|TH. <!--|
|000020c0| 20 4d 41 54 48 0a 20 24 | 7c 31 20 2d 20 33 78 7c | MATH. $||1 - 3x||
|000020d0| 20 3e 20 32 24 0a 20 2d | 2d 3e 0a 7c 20 31 20 2d | > 2$. -|->.| 1 -|
|000020e0| 20 33 3c 49 3e 78 3c 2f | 49 3e 7c 20 26 67 74 3b | 3<I>x</|I>| &gt;|
|000020f0| 20 32 0a 20 26 6e 64 61 | 73 68 3b 26 67 74 3b 0a | 2. &nda|sh;&gt;.|
|00002100| 7c 20 31 20 2d 20 33 26 | 6c 74 3b 49 26 67 74 3b || 1 - 3&|lt;I&gt;|
|00002110| 78 26 6c 74 3b 2f 49 26 | 67 74 3b 7c 20 26 61 6d |x&lt;/I&|gt;| &am|
|00002120| 70 3b 67 74 3b 20 32 20 | 63 61 6e 20 62 65 20 72 |p;gt; 2 |can be r|
|00002130| 65 77 72 69 74 74 65 6e | 20 61 73 20 26 6c 74 3b |ewritten| as &lt;|
|00002140| 21 26 6e 64 61 73 68 3b | 20 4d 41 54 48 0a 20 3c |!&ndash;| MATH. <|
|00002150| 21 2d 2d 20 4d 41 54 48 | 0a 20 24 32 2f 33 20 3c |!-- MATH|. $2/3 <|
|00002160| 20 7c 78 20 2d 20 31 2f | 33 7c 24 0a 20 2d 2d 3e | |x - 1/|3|$. -->|
|00002170| 0a 32 2f 33 20 26 6c 74 | 3b 20 7c 20 3c 49 3e 78 |.2/3 &lt|; | <I>x|
|00002180| 3c 2f 49 3e 20 2d 20 31 | 2f 33 7c 0a 20 26 6e 64 |</I> - 1|/3|. &nd|
|00002190| 61 73 68 3b 26 67 74 3b | 0a 32 2f 33 20 26 61 6d |ash;&gt;|.2/3 &am|
|000021a0| 70 3b 6c 74 3b 20 7c 20 | 26 6c 74 3b 49 26 67 74 |p;lt; | |&lt;I&gt|
|000021b0| 3b 78 26 6c 74 3b 2f 49 | 26 67 74 3b 20 2d 20 31 |;x&lt;/I|&gt; - 1|
|000021c0| 2f 33 7c 2c 20 73 6f 20 | 26 6c 74 3b 21 26 6e 64 |/3|, so |&lt;!&nd|
|000021d0| 61 73 68 3b 20 4d 41 54 | 48 0a 20 3c 21 2d 2d 20 |ash; MAT|H. <!-- |
|000021e0| 4d 41 54 48 0a 20 24 61 | 20 3d 20 31 2f 33 2c 20 |MATH. $a| = 1/3, |
|000021f0| 62 20 3d 20 32 2f 33 24 | 0a 20 2d 2d 3e 0a 3c 49 |b = 2/3$|. -->.<I|
|00002200| 3e 61 3c 2f 49 3e 20 3d | 20 31 2f 33 2c 20 3c 49 |>a</I> =| 1/3, <I|
|00002210| 3e 62 3c 2f 49 3e 20 3d | 20 32 2f 33 0a 20 26 6e |>b</I> =| 2/3. &n|
|00002220| 64 61 73 68 3b 26 67 74 | 3b 0a 26 6c 74 3b 49 26 |dash;&gt|;.&lt;I&|
|00002230| 67 74 3b 61 26 6c 74 3b | 2f 49 26 67 74 3b 20 3d |gt;a&lt;|/I&gt; =|
|00002240| 20 31 2f 33 2c 20 26 6c | 74 3b 49 26 67 74 3b 62 | 1/3, &l|t;I&gt;b|
|00002250| 26 6c 74 3b 2f 49 26 67 | 74 3b 20 3d 20 32 2f 33 |&lt;/I&g|t; = 2/3|
|00002260| 2e 20 20 4e 6f 74 69 6e | 67 20 74 68 65 20 73 74 |.  Notin|g the st|
|00002270| 72 69 63 74 20 69 6e 65 | 71 75 61 6c 69 74 69 65 |rict ine|qualitie|
|00002280| 73 20 69 6e 20 74 68 69 | 73 20 70 72 6f 62 6c 65 |s in thi|s proble|
|00002290| 6d 2c 20 74 68 65 20 73 | 6f 6c 75 74 69 6f 6e 20 |m, the s|olution |
|000022a0| 73 65 74 20 62 65 63 6f | 6d 65 73 20 26 6c 74 3b |set beco|mes &lt;|
|000022b0| 21 26 6e 64 61 73 68 3b | 20 4d 41 54 48 0a 20 3c |!&ndash;| MATH. <|
|000022c0| 21 2d 2d 20 4d 41 54 48 | 0a 20 24 28 2d 5c 69 6e |!-- MATH|. $(-\in|
|000022d0| 66 74 79 2c 20 2d 31 2f | 33 29 20 5c 63 75 70 28 |fty, -1/|3) \cup(|
|000022e0| 31 2c 20 2b 5c 69 6e 66 | 74 79 29 24 0a 20 2d 2d |1, +\inf|ty)$. --|
|000022f0| 3e 0a 28 2d 20 26 23 38 | 37 33 34 3b 2c 20 2d 31 |>.(- &#8|734;, -1|
|00002300| 2f 33 29 26 23 38 37 34 | 36 3b 28 31 2c 20 2b 20 |/3)&#874|6;(1, + |
|00002310| 26 23 38 37 33 34 3b 29 | 0a 20 26 6e 64 61 73 68 |&#8734;)|. &ndash|
|00002320| 3b 26 67 74 3b 0a 28 2d | 20 26 61 6d 70 3b 23 38 |;&gt;.(-| &amp;#8|
|00002330| 37 33 34 3b 2c 20 2d 31 | 2f 33 29 26 61 6d 70 3b |734;, -1|/3)&amp;|
|00002340| 23 38 37 34 36 3b 28 31 | 2c 20 2b 20 26 61 6d 70 |#8746;(1|, + &amp|
|00002350| 3b 23 38 37 33 34 3b 29 | 2e 0a 0a 3c 50 3e 0a 26 |;#8734;)|...<P>.&|
|00002360| 6c 74 3b 50 26 67 74 3b | 0a 26 6c 74 3b 2f 4c 49 |lt;P&gt;|.&lt;/LI|
|00002370| 26 67 74 3b 0a 26 6c 74 | 3b 4c 49 26 67 74 3b 28 |&gt;.&lt|;LI&gt;(|
|00002380| 31 2e 31 35 29 20 20 49 | 66 20 26 6c 74 3b 49 26 |1.15)  I|f &lt;I&|
|00002390| 67 74 3b 62 26 6c 74 3b | 2f 49 26 67 74 3b 20 3d |gt;b&lt;|/I&gt; =|
|000023a0| 20 30 2c 20 74 68 65 6e | 20 74 68 65 20 73 6f 6c | 0, then| the sol|
|000023b0| 75 74 69 6f 6e 20 73 65 | 74 20 26 6c 74 3b 21 26 |ution se|t &lt;!&|
|000023c0| 6e 64 61 73 68 3b 20 4d | 41 54 48 0a 20 3c 21 2d |ndash; M|ATH. <!-|
|000023d0| 2d 20 4d 41 54 48 0a 20 | 24 5b 61 20 2d 20 63 2c |- MATH. |$[a - c,|
|000023e0| 20 61 20 2d 20 62 5d 20 | 5c 63 75 70 20 5b 61 20 | a - b] |\cup [a |
|000023f0| 2b 20 62 2c 20 61 20 2b | 20 63 5d 24 0a 20 2d 2d |+ b, a +| c]$. --|
|00002400| 3e 0a 5b 3c 49 3e 61 3c | 2f 49 3e 20 2d 20 3c 49 |>.[<I>a<|/I> - <I|
|00002410| 3e 63 3c 2f 49 3e 2c 20 | 3c 49 3e 61 3c 2f 49 3e |>c</I>, |<I>a</I>|
|00002420| 20 2d 20 3c 49 3e 62 3c | 2f 49 3e 5d 26 23 38 37 | - <I>b<|/I>]&#87|
|00002430| 34 36 3b 5b 3c 49 3e 61 | 3c 2f 49 3e 20 2b 20 3c |46;[<I>a|</I> + <|
|00002440| 49 3e 62 3c 2f 49 3e 2c | 20 3c 49 3e 61 3c 2f 49 |I>b</I>,| <I>a</I|
|00002450| 3e 20 2b 20 3c 49 3e 63 | 3c 2f 49 3e 5d 0a 20 26 |> + <I>c|</I>]. &|
|00002460| 6e 64 61 73 68 3b 26 67 | 74 3b 0a 5b 26 6c 74 3b |ndash;&g|t;.[&lt;|
|00002470| 49 26 67 74 3b 61 26 6c | 74 3b 2f 49 26 67 74 3b |I&gt;a&l|t;/I&gt;|
|00002480| 20 2d 20 26 6c 74 3b 49 | 26 67 74 3b 63 26 6c 74 | - &lt;I|&gt;c&lt|
|00002490| 3b 2f 49 26 67 74 3b 2c | 20 26 6c 74 3b 49 26 67 |;/I&gt;,| &lt;I&g|
|000024a0| 74 3b 61 26 6c 74 3b 2f | 49 26 67 74 3b 20 2d 20 |t;a&lt;/|I&gt; - |
|000024b0| 26 6c 74 3b 49 26 67 74 | 3b 62 26 6c 74 3b 2f 49 |&lt;I&gt|;b&lt;/I|
|000024c0| 26 67 74 3b 5d 26 61 6d | 70 3b 23 38 37 34 36 3b |&gt;]&am|p;#8746;|
|000024d0| 5b 26 6c 74 3b 49 26 67 | 74 3b 61 26 6c 74 3b 2f |[&lt;I&g|t;a&lt;/|
|000024e0| 49 26 67 74 3b 20 2b 20 | 26 6c 74 3b 49 26 67 74 |I&gt; + |&lt;I&gt|
|000024f0| 3b 62 26 6c 74 3b 2f 49 | 26 67 74 3b 2c 20 26 6c |;b&lt;/I|&gt;, &l|
|00002500| 74 3b 49 26 67 74 3b 61 | 26 6c 74 3b 2f 49 26 67 |t;I&gt;a|&lt;/I&g|
|00002510| 74 3b 20 2b 20 26 6c 74 | 3b 49 26 67 74 3b 63 26 |t; + &lt|;I&gt;c&|
|00002520| 6c 74 3b 2f 49 26 67 74 | 3b 5d 20 62 65 63 6f 6d |lt;/I&gt|;] becom|
|00002530| 65 73 20 26 6c 74 3b 21 | 26 6e 64 61 73 68 3b 20 |es &lt;!|&ndash; |
|00002540| 4d 41 54 48 0a 20 3c 21 | 2d 2d 20 4d 41 54 48 0a |MATH. <!|-- MATH.|
|00002550| 20 24 5b 61 20 2d 20 63 | 2c 20 61 20 5d 20 5c 63 | $[a - c|, a ] \c|
|00002560| 75 70 20 5b 61 2c 20 61 | 20 2b 20 63 5d 24 0a 20 |up [a, a| + c]$. |
|00002570| 2d 2d 3e 0a 5b 3c 49 3e | 61 3c 2f 49 3e 20 2d 20 |-->.[<I>|a</I> - |
|00002580| 3c 49 3e 63 3c 2f 49 3e | 2c 20 3c 49 3e 61 3c 2f |<I>c</I>|, <I>a</|
|00002590| 49 3e 5d 26 23 38 37 34 | 36 3b 5b 3c 49 3e 61 3c |I>]&#874|6;[<I>a<|
|000025a0| 2f 49 3e 2c 20 3c 49 3e | 61 3c 2f 49 3e 20 2b 20 |/I>, <I>|a</I> + |
|000025b0| 3c 49 3e 63 3c 2f 49 3e | 5d 0a 20 26 6e 64 61 73 |<I>c</I>|]. &ndas|
|000025c0| 68 3b 26 67 74 3b 0a 5b | 26 6c 74 3b 49 26 67 74 |h;&gt;.[|&lt;I&gt|
|000025d0| 3b 61 26 6c 74 3b 2f 49 | 26 67 74 3b 20 2d 20 26 |;a&lt;/I|&gt; - &|
|000025e0| 6c 74 3b 49 26 67 74 3b | 63 26 6c 74 3b 2f 49 26 |lt;I&gt;|c&lt;/I&|
|000025f0| 67 74 3b 2c 20 26 6c 74 | 3b 49 26 67 74 3b 61 26 |gt;, &lt|;I&gt;a&|
|00002600| 6c 74 3b 2f 49 26 67 74 | 3b 5d 26 61 6d 70 3b 23 |lt;/I&gt|;]&amp;#|
|00002610| 38 37 34 36 3b 5b 26 6c | 74 3b 49 26 67 74 3b 61 |8746;[&l|t;I&gt;a|
|00002620| 26 6c 74 3b 2f 49 26 67 | 74 3b 2c 20 26 6c 74 3b |&lt;/I&g|t;, &lt;|
|00002630| 49 26 67 74 3b 61 26 6c | 74 3b 2f 49 26 67 74 3b |I&gt;a&l|t;/I&gt;|
|00002640| 20 2b 20 26 6c 74 3b 49 | 26 67 74 3b 63 26 6c 74 | + &lt;I|&gt;c&lt|
|00002650| 3b 2f 49 26 67 74 3b 5d | 20 6f 72 20 73 69 6d 70 |;/I&gt;]| or simp|
|00002660| 6c 79 20 26 6c 74 3b 21 | 26 6e 64 61 73 68 3b 20 |ly &lt;!|&ndash; |
|00002670| 4d 41 54 48 0a 20 3c 21 | 2d 2d 20 4d 41 54 48 0a |MATH. <!|-- MATH.|
|00002680| 20 24 5b 61 20 2d 20 63 | 2c 20 20 61 20 2b 20 63 | $[a - c|,  a + c|
|00002690| 5d 24 0a 20 2d 2d 3e 0a | 5b 3c 49 3e 61 3c 2f 49 |]$. -->.|[<I>a</I|
|000026a0| 3e 20 2d 20 3c 49 3e 63 | 3c 2f 49 3e 2c 20 3c 49 |> - <I>c|</I>, <I|
|000026b0| 3e 61 3c 2f 49 3e 20 2b | 20 3c 49 3e 63 3c 2f 49 |>a</I> +| <I>c</I|
|000026c0| 3e 5d 0a 20 26 6e 64 61 | 73 68 3b 26 67 74 3b 0a |>]. &nda|sh;&gt;.|
|000026d0| 5b 26 6c 74 3b 49 26 67 | 74 3b 61 26 6c 74 3b 2f |[&lt;I&g|t;a&lt;/|
|000026e0| 49 26 67 74 3b 20 2d 20 | 26 6c 74 3b 49 26 67 74 |I&gt; - |&lt;I&gt|
|000026f0| 3b 63 26 6c 74 3b 2f 49 | 26 67 74 3b 2c 20 26 6c |;c&lt;/I|&gt;, &l|
|00002700| 74 3b 49 26 67 74 3b 61 | 26 6c 74 3b 2f 49 26 67 |t;I&gt;a|&lt;/I&g|
|00002710| 74 3b 20 2b 20 26 6c 74 | 3b 49 26 67 74 3b 63 26 |t; + &lt|;I&gt;c&|
|00002720| 6c 74 3b 2f 49 26 67 74 | 3b 5d 2e 20 20 52 65 77 |lt;/I&gt|;].  Rew|
|00002730| 72 69 74 65 20 20 26 6c | 74 3b 21 26 6e 64 61 73 |rite  &l|t;!&ndas|
|00002740| 68 3b 20 4d 41 54 48 0a | 20 3c 21 2d 2d 20 4d 41 |h; MATH.| <!-- MA|
|00002750| 54 48 0a 20 24 7c 33 20 | 2d 20 32 78 7c 20 3c 20 |TH. $|3 |- 2x| < |
|00002760| 35 24 0a 20 2d 2d 3e 0a | 7c 20 33 20 2d 20 32 3c |5$. -->.|| 3 - 2<|
|00002770| 49 3e 78 3c 2f 49 3e 7c | 20 26 6c 74 3b 20 35 0a |I>x</I>|| &lt; 5.|
|00002780| 20 26 6e 64 61 73 68 3b | 26 67 74 3b 0a 7c 20 33 | &ndash;|&gt;.| 3|
|00002790| 20 2d 20 32 26 6c 74 3b | 49 26 67 74 3b 78 26 6c | - 2&lt;|I&gt;x&l|
|000027a0| 74 3b 2f 49 26 67 74 3b | 7c 20 26 61 6d 70 3b 6c |t;/I&gt;|| &amp;l|
|000027b0| 74 3b 20 35 20 61 73 20 | 26 6c 74 3b 21 26 6e 64 |t; 5 as |&lt;!&nd|
|000027c0| 61 73 68 3b 20 4d 41 54 | 48 0a 20 3c 21 2d 2d 20 |ash; MAT|H. <!-- |
|000027d0| 4d 41 54 48 0a 20 24 7c | 78 20 2d 20 33 2f 32 7c |MATH. $||x - 3/2||
|000027e0| 20 3c 20 35 2f 32 24 0a | 20 2d 2d 3e 0a 7c 20 3c | < 5/2$.| -->.| <|
|000027f0| 49 3e 78 3c 2f 49 3e 20 | 2d 20 33 2f 32 7c 20 26 |I>x</I> |- 3/2| &|
|00002800| 6c 74 3b 20 35 2f 32 0a | 20 26 6e 64 61 73 68 3b |lt; 5/2.| &ndash;|
|00002810| 26 67 74 3b 0a 7c 20 26 | 6c 74 3b 49 26 67 74 3b |&gt;.| &|lt;I&gt;|
|00002820| 78 26 6c 74 3b 2f 49 26 | 67 74 3b 20 2d 20 33 2f |x&lt;/I&|gt; - 3/|
|00002830| 32 7c 20 26 61 6d 70 3b | 6c 74 3b 20 35 2f 32 20 |2| &amp;|lt; 5/2 |
|00002840| 74 6f 20 69 64 65 6e 74 | 69 66 79 20 26 6c 74 3b |to ident|ify &lt;|
|00002850| 21 26 6e 64 61 73 68 3b | 20 4d 41 54 48 0a 20 3c |!&ndash;| MATH. <|
|00002860| 21 2d 2d 20 4d 41 54 48 | 0a 20 24 61 20 3d 20 33 |!-- MATH|. $a = 3|
|00002870| 2f 32 2c 20 63 20 3d 20 | 35 2f 32 24 0a 20 2d 2d |/2, c = |5/2$. --|
|00002880| 3e 0a 3c 49 3e 61 3c 2f | 49 3e 20 3d 20 33 2f 32 |>.<I>a</|I> = 3/2|
|00002890| 2c 20 3c 49 3e 63 3c 2f | 49 3e 20 3d 20 35 2f 32 |, <I>c</|I> = 5/2|
|000028a0| 0a 20 26 6e 64 61 73 68 | 3b 26 67 74 3b 0a 26 6c |. &ndash|;&gt;.&l|
|000028b0| 74 3b 49 26 67 74 3b 61 | 26 6c 74 3b 2f 49 26 67 |t;I&gt;a|&lt;/I&g|
|000028c0| 74 3b 20 3d 20 33 2f 32 | 2c 20 26 6c 74 3b 49 26 |t; = 3/2|, &lt;I&|
|000028d0| 67 74 3b 63 26 6c 74 3b | 2f 49 26 67 74 3b 20 3d |gt;c&lt;|/I&gt; =|
|000028e0| 20 35 2f 32 2e 20 20 20 | 54 68 65 20 73 6f 6c 75 | 5/2.   |The solu|
|000028f0| 74 69 6f 6e 20 73 65 74 | 20 69 73 20 74 68 65 6e |tion set| is then|
|00002900| 20 28 2d 20 31 2c 20 34 | 29 2e 20 20 20 49 66 20 | (- 1, 4|).   If |
|00002910| 77 65 20 68 61 76 65 20 | 73 74 72 69 63 74 20 69 |we have |strict i|
|00002920| 6e 65 71 75 61 6c 69 74 | 79 20 61 74 20 74 68 65 |nequalit|y at the|
|00002930| 20 6c 6f 77 65 72 20 65 | 6e 64 20 28 26 6c 74 3b | lower e|nd (&lt;|
|00002940| 21 26 6e 64 61 73 68 3b | 20 4d 41 54 48 0a 20 3c |!&ndash;| MATH. <|
|00002950| 21 2d 2d 20 4d 41 54 48 | 0a 20 24 30 20 3c 20 7c |!-- MATH|. $0 < ||
|00002960| 33 20 2d 20 32 78 7c 20 | 3c 20 35 24 0a 20 2d 2d |3 - 2x| |< 5$. --|
|00002970| 3e 0a 30 20 26 6c 74 3b | 20 7c 20 33 20 2d 20 32 |>.0 &lt;| | 3 - 2|
|00002980| 3c 49 3e 78 3c 2f 49 3e | 7c 20 26 6c 74 3b 20 35 |<I>x</I>|| &lt; 5|
|00002990| 0a 20 26 6e 64 61 73 68 | 3b 26 67 74 3b 0a 30 20 |. &ndash|;&gt;.0 |
|000029a0| 26 61 6d 70 3b 6c 74 3b | 20 7c 20 33 20 2d 20 32 |&amp;lt;| | 3 - 2|
|000029b0| 26 6c 74 3b 49 26 67 74 | 3b 78 26 6c 74 3b 2f 49 |&lt;I&gt|;x&lt;/I|
|000029c0| 26 67 74 3b 7c 20 26 61 | 6d 70 3b 6c 74 3b 20 35 |&gt;| &a|mp;lt; 5|
|000029d0| 29 2c 20 74 68 65 6e 20 | 74 68 65 20 70 6f 69 6e |), then |the poin|
|000029e0| 74 20 26 6c 74 3b 49 26 | 67 74 3b 61 26 6c 74 3b |t &lt;I&|gt;a&lt;|
|000029f0| 2f 49 26 67 74 3b 20 3d | 20 33 2f 32 20 69 73 20 |/I&gt; =| 3/2 is |
|00002a00| 60 60 6d 69 73 73 69 6e | 67 27 27 20 66 72 6f 6d |``missin|g'' from|
|00002a10| 20 74 68 65 20 73 6f 6c | 75 74 69 6f 6e 20 73 65 | the sol|ution se|
|00002a20| 74 3a 20 26 6c 74 3b 21 | 26 6e 64 61 73 68 3b 20 |t: &lt;!|&ndash; |
|00002a30| 4d 41 54 48 0a 20 3c 21 | 2d 2d 20 4d 41 54 48 0a |MATH. <!|-- MATH.|
|00002a40| 20 24 28 2d 31 2c 20 33 | 2f 32 29 20 5c 63 75 70 | $(-1, 3|/2) \cup|
|00002a50| 20 28 33 2f 32 2c 20 34 | 29 24 0a 20 2d 2d 3e 0a | (3/2, 4|)$. -->.|
|00002a60| 28 2d 20 31 2c 20 33 2f | 32 29 26 23 38 37 34 36 |(- 1, 3/|2)&#8746|
|00002a70| 3b 28 33 2f 32 2c 20 34 | 29 0a 20 26 6e 64 61 73 |;(3/2, 4|). &ndas|
|00002a80| 68 3b 26 67 74 3b 0a 28 | 2d 20 31 2c 20 33 2f 32 |h;&gt;.(|- 1, 3/2|
|00002a90| 29 26 61 6d 70 3b 23 38 | 37 34 36 3b 28 33 2f 32 |)&amp;#8|746;(3/2|
|00002aa0| 2c 20 34 29 2e 20 20 53 | 75 63 68 20 60 60 70 75 |, 4).  S|uch ``pu|
|00002ab0| 6e 63 74 75 72 65 64 27 | 27 20 69 6e 74 65 72 76 |nctured'|' interv|
|00002ac0| 61 6c 73 20 70 6c 61 79 | 20 61 20 6b 65 79 20 72 |als play| a key r|
|00002ad0| 6f 6c 65 20 69 6e 20 74 | 68 65 20 73 74 75 64 79 |ole in t|he study|
|00002ae0| 20 6f 66 20 6c 69 6d 69 | 74 73 2e 0a 0a 3c 50 3e | of limi|ts...<P>|
|00002af0| 0a 26 6c 74 3b 50 26 67 | 74 3b 0a 26 6c 74 3b 2f |.&lt;P&g|t;.&lt;/|
|00002b00| 4c 49 26 67 74 3b 0a 26 | 6c 74 3b 2f 4f 4c 26 67 |LI&gt;.&|lt;/OL&g|
|00002b10| 74 3b 0a 26 6c 74 3b 48 | 52 26 67 74 3b 0a 0a 3c |t;.&lt;H|R&gt;..<|
|00002b20| 50 3e 0a 26 6c 74 3b 2f | 42 4f 44 59 26 67 74 3b |P>.&lt;/|BODY&gt;|
|00002b30| 0a 26 6c 74 3b 2f 48 54 | 4d 4c 26 67 74 3b 0a 0a |.&lt;/HT|ML&gt;..|
|00002b40| 3c 48 52 3e 0a 0a 3c 2f | 42 4f 44 59 3e 0a 3c 2f |<HR>..</|BODY>.</|
|00002b50| 48 54 4d 4c 3e 0a       |                         |HTML>.  |        |
+--------+-------------------------+-------------------------+--------+--------+