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.

下一节推荐课程

通过 Linear Inequality Constraints from Contexts 继续学习此主题。

Linear Inequality Constraints from Contexts