Let $\alpha$ and $\beta$ be positive integers such that:

Find the smallest possible value of $\beta$.

Step 1: Take the reciprocals of the inequality:

Step 2: Write these fractions as mixed numbers:

Step 3: Subtract 2 from all parts of the inequality:

Step 4: Let $k=\beta -2\alpha$. Since $\beta$ and $\alpha$ are integers, $k$ must also be an integer:

Step 5: We test small positive integers for $k$ to find one that allows an integer value for $\alpha$:

• If $k=1$: $3.2<\alpha <3.5⟹$ no integer solution.

• If $k=2$: $6.4<\alpha <7⟹$ no integer solution.

• If $k=3$: $9.6<\alpha <10.5⟹\alpha =10$.

Step 6: For $\alpha =10$ and $k=3$, we solve for $\beta$:

Thus, the smallest possible value of $\beta$ is $23$.