Making an inequality true

Question

$n > 10$ implies $n + 3 \leq \Box\times n$

Possible answers:

• $1$
• $2$
• $3$
• $4$

I answered "$2$" and got it wrong. Why? When $n=2$, $(11) + 3 \le 2(11)$.

$n > 1$ implies $n + 3 \leq \Box\times n$

Possible answers:

• $1$
• $2$
• $3$
• $4$

I answered "$4$" and got it right.

Why did I get the 2nd problem right, and the 1st wrong?

Answer 1

My first guess what that this was a question of the sort where you need to select every correct option, but if that were the case, then you would not have gotten the 2nd question correct. So I assume it is a mistake on the part of whoever is grading.

The statement $$n>10\implies n+3\leq \Box\times n$$ is true for all values of $\Box$ that are greater than or equal to $\frac{14}{11}$. Out of the provided options, this means that the correct ones are $\Box=2$, $\Box=3$, and $\Box=4$.

The statement $$n>1\implies n+3\leq \Box\times n$$ is true for all values of $\Box$ that are greater than or equal to $\frac{5}{2}$. Out of the provided options, this means that the correct ones are $\Box=3$ and $\Box=4$.