document.write( "Question 824981: Prove that:
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Algebra.Com's Answer #497041 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I prefer working with ≦ than with ≧, so I will prove the equivalent\r\n" );
document.write( "inequality:\r\n" );
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document.write( "                   ||x|-|y|| ≦ |x-y|\r\n" );
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document.write( "Any number either equals to its absolute value or to the negative of its\r\n" );
document.write( "absolute value, so  -|z| ≦ z ≦ |z| is true for any number z.\r\n" );
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document.write( "     Let z = x-y       also       let z = y\r\n" );
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document.write( "-|x-y| ≦ x-y ≦ |x-y|   and    -|y| ≦ y ≦ |y|\r\n" );
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document.write( "Add those two inequalities:\r\n" );
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document.write( "          -|x-y|-|y| ≦ x-y+y ≦ |x-y|+|y|\r\n" );
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document.write( "           -|x-y|-|y| ≦ x ≦ |x-y|+|y|\r\n" );
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document.write( "         -(|x-y|+|y|) ≦ x ≦ |x-y|+|y| \r\n" );
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document.write( "                      |x| ≦ |x-y|+|y| \r\n" );
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document.write( "That is true because -a≦b≦a is equivalent to |b|≦a             \r\n" );
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document.write( "Subtract |y| from both sides:\r\n" );
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document.write( "                 |x|-|y| ≦ |x-y| \r\n" );
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document.write( "--------------------------\r\n" );
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document.write( "Similarly we can interchange x and y in the above, and get\r\n" );
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document.write( "                |y|-|x| ≦ |y-x|\r\n" );
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document.write( "The right sides of both those inequalities are equal since\r\n" );
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document.write( "|y-x|=|-(x-y)|=|x-y|\r\n" );
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document.write( "      So we have |x|-|y| ≦ |x-y|\r\n" );
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document.write( "and also we have |y|-|x| ≦ |x-y|.\r\n" );
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document.write( "||x|-|y|| is either equal to |x|-|y| or to |y|-|x|,\r\n" );
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document.write( "and in either case it is ≦ |x-y|.\r\n" );
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document.write( "Therefore we have proved that\r\n" );
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document.write( "            ||x|-|y|| ≦ |x-y|\r\n" );
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document.write( "Edwin
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