Algebra / Linear Inequalities
Compound inequalities and integer counts
Compound inequalities require the same operation on each part. Endpoint type matters when counting integers.
Least you need to know
- Strict endpoints are not included.
- Inclusive endpoints are included.
- When dividing a three-part inequality by a negative, reverse both signs.
Key notation
- < — Strict endpoint, not included
- \le — Inclusive endpoint, included
Worked example
For $0<2x\le 6$, divide all parts by 2 to get $0<x\le 3$. The integer solutions are 1, 2, and 3.
Common mistakes
- Do not count a strict endpoint.
- Rewrite intervals in increasing order before counting.
How to recognize it
- The problem asks how many integers satisfy an inequality.
- The inequality has two comparison signs.
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