Algebra / Linear Inequalities
Linear Inequalities
Linear inequalities use symbols such as $<$, $>$, $\le$, and $\ge$ to describe ranges of values. Supplemental practice should emphasize integer boundary questions, compound inequalities, strict versus inclusive endpoints, context constraints, and parameter values that determine a solution set.
Least you need to know
- Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.
- Strict inequalities do not include the endpoint, while $\le$ and $\ge$ do.
- Integer-solution questions require translating the final interval into actual integer values.
Key notation
- $x<5$ — All values of $x$ less than 5
- $-2<x\le 7$ — All values greater than -2 and less than or equal to 7
Worked example
- Consider $-3x+5<20$.
- Subtract 5 from both sides: $-3x<15$.
- Divide by $-3$ and reverse the inequality: $x>-5$.
- If the question asks for the least integer solution, the answer is $-4$.
Common mistakes
- Reverse the inequality symbol only when multiplying or dividing by a negative number.
- For integer counts, list or carefully count the integers in the interval.
- A boundary point may fail a strict inequality even if it lies on the boundary line.
How to recognize it
- The problem asks for a least or greatest integer value.
- The problem gives a compound inequality or a real-world maximum/minimum constraint.
- The problem asks which ordered pair satisfies a two-variable inequality.
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