Summary

This chapter developed the algebra and geometry of linear inequalities, extending the equation-solving skills built earlier in the course.

• An inequality compares two expressions using $<, >, \le, \ge$; a numerical inequality involves only known numbers, while a literal inequality contains a variable and is called linear when that variable appears only to the first power. A solution is any value satisfying the inequality; the full collection is the solution set, typically an infinite interval rather than a single point.
• Solving a linear inequality in one variable algebraically uses the same "isolate the variable" method as an equation, with one crucial exception: adding/subtracting any quantity, or multiplying/dividing by a positive number, preserves the inequality's direction, but multiplying or dividing by a negative number reverses it — proved directly from the fact that a positive number times a positive number stays positive. Compound inequalities like $a < px+q < b$ are solved by operating on all three parts at once.
• The modulus function $|x|$ measures distance from $0$. Its inequalities reduce to two key equivalences: $|x| < a \iff -a < x < a$ (a single bounded interval) and $|x| > a \iff x < -a$ or $x > a$ (two disjoint unbounded rays) — both proved by splitting into the cases $x \ge 0$ and $x < 0$, and both extend directly to shifted expressions like $|x-c|$.
• Any one-variable solution set is pictured on the number line using an open circle for an excluded (strict) endpoint and a closed circle for an included (slack) endpoint, with an arrow for an unbounded direction and a shaded segment for a bounded interval. …