Question 1196425
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Suppose you are setting up a full-mesh network for x users; 
and n, the number of two-way connections required to link all users pairwise, 
must be no greater than 132. For what range of x values can you set up your network?
x(x-1)/2=n
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<pre>
They want you solve this inequality

    {{{(x*(x-1))/2}}} <= 132

in positive integer numbers.


So, you multiply both sides by 2

    x*(x-1) <= 2*132 = 264.


At this point, you can estimate x by noticing that  {{{sqrt(264)}}} = 16.24...  (rounded.


Hence, integer x  must be closest to it lesser integer {{{highlight(highlight(16))}}}.


        <U>CHECK</U>.  16*(16-1) = 16*15 = 240,  while 17*16 = 272, confirming this answer.



Alternatively, you can solve quadratic inequality

    x^2 - x - 264 <= 0,


and for x you will get


    {{{x[1,2]}}} = {{{(1 +- sqrt(1^2 + 4*1*264))/2}}} = {{{(1 +- sqrt(1057))/2}}} = {{{(1 +- 32.51)/2}}}.


Thus  -15.755 <= x <= 16.755,  and since we want x be positive integer number, it gives the same answer x <= 16.
</pre>

Solved in two ways for your better understanding, &nbsp;giving the &nbsp;&nbsp;<U>ANSWER</U> &nbsp;&nbsp;n <= 16.