document.write( "Question 1083811: Find out the maximum and Minimum value of (a) |z-2|+|z-3|
\n" ); document.write( "(b) |2z-1|+|3z-2| \n" ); document.write( "
Algebra.Com's Answer #697883 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
(a) |z-2|+|z-3|
\n" ); document.write( "
\r\n" );
document.write( "We consider 4 potential cases\r\n" );
document.write( "\r\n" );
document.write( " Case 1:  both expressions within absolute value bars \r\n" );
document.write( "are non-negative:\r\n" );
document.write( "\r\n" );
document.write( "  \r\n" );
document.write( "\r\n" );
document.write( "That's the same as \"z%3E=3\"\r\n" );
document.write( "\r\n" );
document.write( "So |z-2|+|z-3| becomes  \r\n" );
document.write( "\r\n" );
document.write( "(z-2)+(z-3) = z-2+z-3 = 2z-5\r\n" );
document.write( "\r\n" );
document.write( "We build up 2z-5 from \"z%3E=3\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%3E=3\"\r\n" );
document.write( "\"2%2Az%3E=2%2A3\"\r\n" );
document.write( "\"2z%3E=6\"\r\n" );
document.write( "\"2z-5%3E=6-5\"\r\n" );
document.write( "\"2z-5%3E=1\"\r\n" );
document.write( "\r\n" );
document.write( "So the minimum value is 1\r\n" );
document.write( "and there is no maximum value.\r\n" );
document.write( "\r\n" );
document.write( "Case 2:  the first expressions within absolute value bars \r\n" );
document.write( "is non-negative and the second expression within absolute\r\n" );
document.write( "value bars is non-positive\r\n" );
document.write( "\r\n" );
document.write( "\"matrix%282%2C3%2C%0D%0Az-2%3E=0%2C+and%2C+z-3%3C=0%2C%0D%0Az%3E=2%2C+++and%2C++++z%3C=3%29\"\r\n" );
document.write( "\r\n" );
document.write( "That's the same as \"2%3C=z%3C=3\".\r\n" );
document.write( "\r\n" );
document.write( " So  |z-2|+|z-3| becomes \r\n" );
document.write( "\r\n" );
document.write( "(z-2)-(z-3) = z-2-z+3 = 1\r\n" );
document.write( "\r\n" );
document.write( "So the minimum (and maximum) value for case 2 is 1.\r\n" );
document.write( "\r\n" );
document.write( "Case 3:  the first expressions within absolute value bars \r\n" );
document.write( "is non-positive and the second expression within absolute\r\n" );
document.write( "value bars is non-negative:\r\n" );
document.write( "\r\n" );
document.write( "\"matrix%282%2C3%2C%0D%0Az-2%3C=0%2C+and%2C+z-3%3E=0%2C%0D%0Az%3C=2%2C+++and%2C++++z%3E=3%29\"\r\n" );
document.write( "\r\n" );
document.write( "That is impossible, so we rule out case 3\r\n" );
document.write( "\r\n" );
document.write( "Case 4:  both expressions within absolute value bars \r\n" );
document.write( "are non-positive:\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "That's the same as \"z%3C=2\"\r\n" );
document.write( "\r\n" );
document.write( "So  |z-2|+|z-3| becomes  \r\n" );
document.write( "\r\n" );
document.write( " -(z-2)-(z-3) = -z+2-z+3 = -2z+5\r\n" );
document.write( "\r\n" );
document.write( "We build up -2z+5 from \"z%3E=3\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%3C=2\"\r\n" );
document.write( "\"-2z%3C=-2%2A2\"  <--inequality reverses\r\n" );
document.write( "\"-2z%3C=-4\"\r\n" );
document.write( "\"-2z%2B5%3C=-4%2B5\"\r\n" );
document.write( "\"-2z%2B5%3E=1\"\r\n" );
document.write( "\r\n" );
document.write( "So for case 4, the minimum value is 1 and there is no maximum value.\r\n" );
document.write( "\r\n" );
document.write( "Therefore for all the cases the minimum value is 1\r\n" );
document.write( "and there is no maximum value. \r\n" );
document.write( "\r\n" );
document.write( "--------------------------------------\r\n" );
document.write( "

\n" ); document.write( "(b) |2z-1|+|3z-2|
\n" ); document.write( "
 \r\n" );
document.write( "Case 1:  both expressions within absolute value bars \r\n" );
document.write( "are non-negative:\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "That's the same as \"z%3E=2%2F3\"\r\n" );
document.write( "\r\n" );
document.write( "So |2z-1|+|3z-2| becomes  \r\n" );
document.write( "\r\n" );
document.write( "(2z-1)+(3z-2) = 2z-1+3z-2 = 5z-3\r\n" );
document.write( "\r\n" );
document.write( "We build up 5z-3 from \"z%3E=2%2F3\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%3E=2%2F3\"\r\n" );
document.write( "\"5%2Az%3E=5%2Aexpr%282%2F3%29\"\r\n" );
document.write( "\"5z%3E=10%2F3\"\r\n" );
document.write( "\"5z-3%3E=10%2F3-3\"\r\n" );
document.write( "\"5z-3%3E=10%2F3-3%2F3\"\r\n" );
document.write( "\"5z-3%3E=7%2F3\"\r\n" );
document.write( "\r\n" );
document.write( "That has a minimum value of 7/3\r\n" );
document.write( "and no maximum value.\r\n" );
document.write( " \r\n" );
document.write( "Case 2:  the first expressions within absolute value bars \r\n" );
document.write( "is non-negative and the second expression within absolute\r\n" );
document.write( "value bars is non-positive\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "z>=1/2,  and     x<=2/3)}}}\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "That's the same as \"1%2F2%3C=z%3C=2%2F3\".\r\n" );
document.write( "\r\n" );
document.write( "So  |2z-1|+|3z-2| becomes \r\n" );
document.write( "    (2z-1)-(3z-2) = 2z-1-3z+2 = -z+1 \r\n" );
document.write( " \r\n" );
document.write( "We build up -z+1 from \"1%2F2%3C=z%3C=2%2F3\".\r\n" );
document.write( "\r\n" );
document.write( "\"1%2F2%3C=z%3C=2%2F3\" \r\n" );
document.write( "\"-1%2F2%3E=-z%3E=-2%2F3\"   <---the inequality symbol reverses \r\n" );
document.write( "\"-1%2F2%2B1%3E=-z%2B1%3E=-2%2F3%2B1\"\r\n" );
document.write( "\"-1%2F2%2B2%2F2%3E=-z%2B2%3E=-2%2F3%2B3%2F3\"\r\n" );
document.write( "\"1%2F2%3E=-z%2B2%3E=1%2F3\"\r\n" );
document.write( "So the minimum value for case 2 is 1/3\r\n" );
document.write( "And the maximum value for case 2 is 1/2\r\n" );
document.write( "\r\n" );
document.write( "Case 3:  the first expressions within absolute value bars \r\n" );
document.write( "is non-positive and the second expression within absolute\r\n" );
document.write( "value bars is non-negative:\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "z<=1/2,  and,    z>=2/3}}}\r\n" );
document.write( " \r\n" );
document.write( "That is impossible, so we rule out case 3.\r\n" );
document.write( "\r\n" );
document.write( "Case 4:  both expressions within absolute value bars \r\n" );
document.write( "are non-positive:\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "That's the same as \"z%3C=1%2F2\"\r\n" );
document.write( "\r\n" );
document.write( "So |2z-1|+|3z-2| becomes  \r\n" );
document.write( "\r\n" );
document.write( "-(2z-1)-(3z-2) = -2z+1-3z+2 = -5z+3\r\n" );
document.write( "\r\n" );
document.write( "We build up -5z+3 from \"z%3C=1%2F2\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%3C=1%2F2\"\r\n" );
document.write( "\"-5%2Az%3E=-5%2Aexpr%281%2F2%29\"   <--inequality symbol reverses\r\n" );
document.write( "\"-5z%3E=-5%2F2\"\r\n" );
document.write( "\"-5z%2B3%3E=-5%2F2%2B3\"\r\n" );
document.write( "\"-5z%2B3%3E=-5%2F2%2B6%2F2\"\r\n" );
document.write( "\"-5z%2B3%3E=1%2F2\"\r\n" );
document.write( "\r\n" );
document.write( "That has a minimum value of 1/2\r\n" );
document.write( "and no maximum value.\r\n" );
document.write( "\r\n" );
document.write( "The minimum value from all the cases is 1/3 and\r\n" );
document.write( "there is no maximum value.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );