With two properly chosen weights and a balance scale, it is possible to determine the weight of an unknown object known to weigh an integer number of pounds from 1 to $n$. Find the largest possible value of $n$.

Remark. The balance scale tells whether the weights placed on each side are equal and, if not, which side is heavier. It may be used an unlimited number of times.
Let $a$ and $b$ be the known weights. The balance scale allows one to compare the unknown weight with four known weights: $a, b, a+b$, and $a-b$ (the last of these is gotten by balancing $x+b$ on one side with $a$ on the other). After the comparisons are done, there are at most 4 values that $x$ can be known to equal and at most 5 gaps ( 3 between and 2 on the ends) for $x$ to lie in. Thus at most 9 values of $x$ can be distinguished.

To show $n=9$ is achievable, take the weights to be $a=6 \mathrm{lb}$ and $b=2 \mathrm{lb}$. Then the nine possibilities can be distinguished by

$$
1a+b
$$