document.write( "Question 1171809: If x+y<16 but x-y>6, where x and y are integers, what is the largest integral value of xy? \n" ); document.write( "
Algebra.Com's Answer #796727 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The product of two POSITIVE numbers is greatest when their difference in\r\n" );
document.write( "absolute value is as small as possible.  Since x-y > 6, the smallest their\r\n" );
document.write( "difference can be is 7, and since they can both be negative integers, their sum\r\n" );
document.write( "will always be less than 16, so their POSITIVE product gets larger and larger\r\n" );
document.write( "without bound.\r\n" );
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document.write( "For instance if -10000 and y=-10007, x+y=-20007 < 16 and x-y=7>6,\r\n" );
document.write( "then xy=100070000.  And we can get xy larger and larger.\r\n" );
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document.write( "There is no solution because there can be no largest value of xy.\r\n" );
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document.write( "Perhaps you meant that x and y are both POSITIVE integers, and cannot be\r\n" );
document.write( "negative?  If so, then there is a solution.\r\n" );
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document.write( "Since to get the largest product, their difference must be as small as\r\n" );
document.write( "possible, and the smallest their difference can be is 7, then x = y+7,\r\n" );
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document.write( "x+y < 16\r\n" );
document.write( "Substituting y+7 for x\r\n" );
document.write( "y+7+y < 16\r\n" );
document.write( " 2y+7 < 16\r\n" );
document.write( "   2y < 9\r\n" );
document.write( "    y < 4.5 and since y is an integer, the largest y can be is 4, and\r\n" );
document.write( "the largest x can be is 4+7 or 11.  So the largest possible product xy\r\n" );
document.write( "is (4)(11) = 44.  That is the solution if x and y are POSITIVE integers.\r\n" );
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document.write( "Edwin
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