Algebra / Linear Inequalities
Linear Inequality Constraints from Contexts
Inequality questions use symbols to represent limits, minimums, maximums, and ranges. Contexts often require translating phrases such as at least, at most, fewer than, or no more than.
Least you need to know
- At least means $\ge$ and at most means $\le$.
- Fewer than means $<$ and more than means $>$.
- When solving an inequality, dividing by a negative reverses the inequality symbol.
Key notation
- $12+5x\le47$ — A fixed amount plus a variable amount is at most 47
Worked example
- If a plan costs $12 plus $5 per item and the total can be at most $47, write $12+5x\le47$.
- Then $5x\le35$, so $x\le7$.
Common mistakes
- Match the inequality symbol to the wording before solving.
- For integer contexts, round in the direction allowed by the inequality.
How to recognize it
- The problem uses phrases such as at least, at most, minimum, maximum, or no more than.
- The answer is a range or an integer boundary.
- The problem asks which inequality represents a situation.
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