Question 647533
You could actually physically see this, but the two smallest sides must add up to more than the third. This is called the triangle inequality. 

Notice that if x<12, x is the smallest side.

If {{{12<x<20}}}. 12 is the smallest side.

If x>20,  12 is the smallest side.

So really for x>12, 12 is the smallest side.

Case 1:  x<12

Then  x+12 > 20

x>8 and since we're only talking about integer lengths, {{{x>=9.}}}

Case 2:  {{{12<x<20}}}

Then 12 + x < 20

Same as case 1.
{{{x>=9}}}

Case 3: x>20

12+20 > x

x < 32 or {{{x<=31}}}

So our possible values range from 9 to 31.

31 is the largest, and it is the answer.

E.