Course: Foundations of Higher Math

Chapter: Proofs involving Real Numbers

Let a, b, and m be integers. Prove that if $\displaystyle 2a+3b\geq 12m+1$ , then $\displaystyle a\geq 3m+1$or$\displaystyle b\geq 2m+1$.

Contrapositive: If $\displaystyle a<3m+1$and$\displaystyle b<2m+1$, then $\displaystyle 2a+3b<12m+1$

Assume that $\displaystyle a<3m+1$ and $\displaystyle b<2m+1$. So, $\displaystyle 2a<6m+2$ and $\displaystyle 3b<6m+3$.

Then, $\displaystyle 2a+3b<6m+2+6m+3=12m+5$

$\displaystyle \Rightarrow 2a+3b<12m+5$ but being less than "12m+5" doesn't necessarily mean it's less than "12m+1"