If each book is 2 cm thick, what is the total thickness of $n$ books? How must that compare to 90 cm? Do the same when each book is 3 cm thick.

Once you have two inequalities involving $n$ (one from 2 cm books, one from 3 cm books), solve each and then combine them into a single statement about $n$.

In the Desmos calculator, type 90/3 and evaluate it. Interpret this quotient as the smallest number of books needed when each book is as thick as allowed (3 cm).

Next type 90/2 and evaluate it. Interpret this quotient as the largest number of books that can still fit when each book is as thin as allowed (2 cm). Use these two values as the endpoints of a closed interval for $n$, and choose the answer option whose inequality matches that interval.

The shelf is 90 cm long, and books are placed side by side until they fill the shelf.

If there are $n$ books, each book has thickness between 2 cm and 3 cm. That means:

• The total thickness of all $n$ books must be exactly 90 cm.
• Each individual book is at least 2 cm thick and at most 3 cm thick.

To get as many books as possible, each book should be as thin as possible (2 cm), because thinner books take less space.

If all $n$ books are 2 cm thick, their total thickness is $2n$ cm. They must fit on the 90 cm shelf, so

Solving this inequality:

So the number of books cannot be more than 45.

To get as few books as possible, each book should be as thick as possible (3 cm), because thicker books take more space.

If all $n$ books are 3 cm thick, their total thickness is $3n$ cm. Since they must cover 90 cm of shelf, there must be at least enough total thickness to reach 90 cm, so

Solving this inequality:

So the number of books must be at least 30.

From the thin-book case we found $n \le 45$, and from the thick-book case we found $n \ge 30$.

Putting these together, the possible values of $n$ must satisfy

This matches answer choice A.