Is the perimeter of a certain triangle less than 24 inches?

\(0 < a \le b \le c\,\,\,\,\left[ {{\rm{inches}}} \right]\,\,\,\,\,\,\,\,\,\,\left( * \right)\)

\(a + b + c\,\,\mathop < \limits^? \,\,24\)

\(\left( 1 \right)\,\,\,a = 8{1 \over 4}\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,a + b + c\,\,\, \ge \,\,\,3 \cdot \left( {8{1 \over 4}} \right)\,\,\,\, > \,\,\,24\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle\)

\(\left( 2 \right)\,\,\,\left\{ \matrix{\\
a = b = x \hfill \cr \\
c = x + 1 \hfill \cr \\
\,\left( {\Delta \,\,{\rm{existence}}} \right)\,\,\,x + 1 < 2x\,\,\,\, \Leftrightarrow \,\,\,\,x > 1\,\,\,\left( {**} \right) \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\,\left( {a,b,c} \right) = \left( {2,2,3} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr \\
\,{\rm{Take}}\,\,\,\left( {a,b,c} \right) = \left( {8,8,9} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.\,\,\,\,\,\,\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.