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

  1. #1
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    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"
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  2. #2
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    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.
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