Question 256379
<pre><font size = 4 color = "indigo"><b>
In any triangle, the sum of any two sides must be greater
than the third side.

That is, these three inequalities must hold:

{{{system(2&1/2+5>x, 2&1/2+x>5, 5+x>=2&1/2)}}}

The third inequality is bound to hold, since regardless
of what positive integer x is the left side is larger
than 5 so obviously it's larger than {{{2&1/2}}}.

So we only need to concentrate on the first two inequalities.

The first inequality abounts to {{{7.5>x}}} and the second
inequality, when solved for x amounts to {{{x>2.5}}}.

Thus x is an integer such that {{{2.5<x<7.5}}} and the only
integers between 2.5 and 7.5 are 3,4,5,6,and 7.

So those are the only integers x can take on.  The answer is 5.

Edwin</pre>