# Math Help - Prove that if 2a+3b >= 12m+1, then a>=3m+1 or b>=2m+1

1. ## Prove that if 2a+3b >= 12m+1, then a>=3m+1 or b>=2m+1

Course: Foundations of Higher Math
Chapter: Proofs involving Real Numbers

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

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

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

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

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

2. ## Re: Prove that if 2a+3b >= 12m+1, then a>=3m+1 or b>=2m+1

$a<3m+1 \Rightarrow a\le 3m$ and $b<2m+1 \Rightarrow b\le 2m$. Hence $2a+3b \le 6m+6m = 12m < 12m+1$.