Linear inequality word problems

These are linear word problems whose answer is a range rather than a single value. The modelling is identical; the extra step is turning a phrase into the right inequality sign.

The distinction that matters is whether the boundary value itself counts. At least and no more than include it, giving $\geq$ and $\leq$. More than and fewer than exclude it, giving $>$ and $<$.

So "a budget of at most $\$200$" becomes $\text{cost} \leq 200$, while "more than 10 attendees" becomes $n > 10$.

Solve exactly as you would an equation, with one exception: multiplying or dividing by a negative number flips the sign.

Finally, check the answer against the situation. Counts of people or items must be whole numbers, so a solution of $n > 10.4$ means $11$ or more — and the answer choices will include $10.4$ for anyone who stops at the algebra.

Tickets cost $\$15$ each and the budget is at most $\$200$. How many can be bought?
At most includes the boundary Tickets are whole, so at most $13$.

A van holds more than $8$ boxes but no more than $15$. Write this as an inequality.
More than excludes $8$; no more than includes $15$