document.write( "Question 34660: If a,b, and c are consective positive integers, show that (6/5)<= (a/(b+c)) + (b/(a+c)) + (c/(a+b)) <= 2 \n" ); document.write( "
Algebra.Com's Answer #20995 by mukhopadhyay(490)\"\" \"About 
You can put this solution on YOUR website!
a,b, and c are consective positive integers
\n" ); document.write( "If a = x-1 (x-1 positive integer), b = x, and c = x+1;
\n" ); document.write( "(a/(b+c)) + (b/(a+c)) + (c/(a+b))
\n" ); document.write( "= (x-1)/(2x+1) + x/(2x) + (x+1)/(2x-1)
\n" ); document.write( "= (x-1)/(2x+1) + 1/2 + (x+1)/(2x-1)
\n" ); document.write( "= 1/2 + (x-1)/(2x+1) + (x+1)/(2x-1)
\n" ); document.write( "= 1/2 + [(x-1)(2x-1)+(x+1)(2x+1)]/[(2x-1)(2x+1)]
\n" ); document.write( "= 1/2 + [(2x^2-2x-x+1)+(2x^2+2x+x+1)]/[(2x-1)(2x+1)]
\n" ); document.write( "= 1/2 + [4x^2+2]/[4x^2-1]......... (Exp 0.5)
\n" ); document.write( "= 1/2 + [4x^2-1+3]/[4x^2-1]
\n" ); document.write( "= 1/2 + 1 + 3/[4x^2-1]
\n" ); document.write( "= 3/2 + 3/[4x^2-1].....(Exp 1)
\n" ); document.write( "......................
\n" ); document.write( "x-1 is a positive integer (>= 1)
\n" ); document.write( "=> x >= 2
\n" ); document.write( "=> x^2 >= 4
\n" ); document.write( "=> 4x^2 > 16
\n" ); document.write( "=> 4x^2 - 1 >= 15
\n" ); document.write( "=> 1/[4x^2 - 1] <= 1/15
\n" ); document.write( "=> (Exp 1) < 3/2 + 1/15 (which is <= 2)
\n" ); document.write( "The above proves the upper boundary of the expression in question.
\n" ); document.write( "..........................
\n" ); document.write( "Same way, 3/2 + 3/[4x^2-1] >= 3/2 (because 3/[4x^2-1] >=0)
\n" ); document.write( "3/2 >= 6/5
\n" ); document.write( "This proves the lower boundary of the expression. \n" ); document.write( "
\n" );