Is 10m > 5n - k?

Here is the My approach
Here we are not given much info in the question stem apart from 10m>5m-k to be proven
Statement 1 => no clue of k => not sufficient
statement 2 => it can be seen from this that k is negative but we have no clue of m and n => not sufficient
combining them => to be proven => 10m>5m-k as n=2m => 10m>10m-k or the question is asking us is k>0 or not which as per statement 2 is false.
hence C is sufficient

Why does statement 2 imply that k can be 0? I understand that it can imply that k is negative, but not understanding why k can be 0.

Thanks!

If k =0, then |k| = |0| = 0 and -k = -0 = 0, so |k| = -k = 0.

\(10m\,\,\mathop > \limits^? \,\,5n - k\)


\(\left( 1 \right)\,\,\,n = 2m\,\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\,\left( {n,m,k} \right) = \left( {0,0,0} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\, \hfill \cr \\
\,{\rm{Take}}\,\,\,\left( {n,m,k} \right) = \left( {2,1,1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\, \hfill \cr} \right.\,\)


\(\left( 2 \right)\,\,\,\,\left| k \right| = - k\,\,\,\, \Leftrightarrow \,\,\,\,k \le 0\)

\(\left\{ \matrix{\\
\,{\rm{Take}}\,\,\,\left( {n,m,k} \right) = \left( {0,0,0} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\, \hfill \cr \\
\,{\rm{Take}}\,\,\,\left( {n,m,k} \right) = \left( {0,1, - 1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\, \hfill \cr} \right.\)


\(\left( {1 + 2} \right)\,\,\,\left\{ \matrix{\\
10m\,\,\mathop > \limits^? \,\,5\left( {2m} \right) - k\,\,\,\,\, \Leftrightarrow \,\,\,\,k\,\,\mathop > \limits^? \,\,0 \hfill \cr \\
k \le 0 \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{SUFF}}.\)


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

Regards,
Fabio.

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