Algebra / Linear Inequalities
Inequality Traps and Integer Boundaries
Inequality questions are error-prone because strict versus inclusive endpoints, negative division, compound inequalities, and integer boundaries all affect the final answer.
Least you need to know
- Dividing or multiplying by a negative reverses the inequality sign.
- Strict inequalities do not include the endpoint.
- $\le$ and $\ge$ include the endpoint.
- For integer questions, translate the interval into actual integer values before answering.
Key notation
- $-2<x\le5$ — Values greater than -2 and less than or equal to 5
- $-3x+7<22$ — An inequality that requires reversing the symbol when dividing by -3
Worked example
- Solve $-4x+3\le19$.
- Subtract 3: $-4x\le16$.
- Divide by $-4$ and reverse the inequality: $x\ge -4$.
Common mistakes
- A boundary may satisfy a non-strict inequality but fail a strict inequality.
- The least integer greater than $-3$ is $-2$, not $-3$.
- Context constraints may require a whole number even if the algebraic solution is not an integer.
How to recognize it
- The problem asks for least or greatest integer values.
- The inequality has a negative coefficient on the variable.
- The problem uses phrases such as fewer than, at least, at most, or no more than.
Következő ajánlott lecke
Folytasd ezt a témát ezzel: Inequality sign reversal.
Inequality sign reversalKapcsolódó leckék
Folytasd közeli leckékkel ugyanabban a témában.