Many real limits are inequalities: a budget you cannot go over, a goal you must reach, a weight an elevator can hold. To solve them, translate the words just as you do for equations, then solve the inequality. The last step is new: decide which answers make sense in the story.
| Words | Symbol | Example |
|---|---|---|
| at least, no less than, a minimum of | $\ge$ | at least $18$: $a \ge 18$ |
| at most, no more than, a maximum of | $\le$ | no more than $\$40$: $c \le 40$ |
| more than, exceeds | $>$ | more than $5$: $n > 5$ |
| less than, fewer than | $<$ | fewer than $30$: $s < 30$ |
Why it works
“At least $18$” means $18$ or anything more, so $18$ itself counts: the sign is $\ge$, not $>$. A quick test is to ask, “Is the boundary number itself allowed?” If yes, the sign includes “or equal to.” And when the unknown counts whole things, like tickets or people, only whole-number answers make sense, so you round in the direction the story allows.
Key idea
Name the unknown, write the inequality, solve it, then interpret: whole items round down under an “at most” limit and up for an “at least” goal. Check the answer in the story.
Worked examples
Worked example: A limit: round down
An elevator holds at most $1200$ kg. A worker weighs $80$ kg, and each box weighs $35$ kg. How many boxes can she take at once?
- Let $b$ be the number of boxes. The total weight is $80 + 35b$, and it must be at most $1200$: $80 + 35b \le 1200$.
- Subtract $80$: $35b \le 1120$. Divide by $35$: $b \le 32$.
- Check: $80 + 35(32) = 1200$, exactly the limit, which is allowed.
Answer: At most $32$ boxes.
Worked example: A goal: round up
A club needs at least $\$400$. It has $\$85$ and sells candles for $\$6$ each. What is the fewest candles it must sell?
- Let $c$ be the number of candles: $85 + 6c \ge 400$.
- Subtract $85$: $6c \ge 315$. Divide by $6$: $c \ge 52.5$.
- Half a candle cannot be sold, and $52$ would fall short ($85 + 312 = 397$). Round up.
Answer: $53$ candles.
Common mistake
Using $>$ for “at least,” and forgetting that answers must make sense. “At least $400$” includes $400$, so it is $\ge$. And $c \ge 52.5$ is not the final answer for candles: the fewest whole candles is $53$. Check by trying your answer and the one next to it in the story.