Algebra - Linear Inequalities

A linear inequality is a mathematical statement where two algebraic expressions are compared using inequality symbols such as $<, >, \leq,$ or $\geq$. Unlike equations, these often result in a range of values rather than a single point.

The Replacement Set (or Universal Set) is the set of values from which the variable $x$ can be chosen (e.g., Natural numbers $\mathbb{N}$, Whole numbers $\mathbb{W}$, Integers $\mathbb{Z}$, or Real numbers $\mathbb{R}$). The Solution Set is the subset of the replacement set that satisfies the inequality.

Rules of Transformation: Adding or subtracting the same number on both sides does not change the inequality sign. Multiplying or dividing by a positive number also keeps the sign the same. Crucially, multiplying or dividing by a negative number reverses the direction of the inequality (e.g., if $-x < 5$, then $x > -5$).

Interval Notation: For real numbers, solutions are expressed using brackets. Square brackets $[a, b]$ represent closed intervals where endpoints are included ($a \leq x \leq b$), while parentheses $(a, b)$ represent open intervals where endpoints are excluded ($a < x < b$). A mix like $[a, b)$ indicates $a \leq x < b$.

Number Line Representation: For strict inequalities ($<$ or $>$), use a hollow circle $\circ$ at the endpoint to show it is excluded. For non-strict inequalities ($\leq$ or $\geq$), use a solid darkened circle $\bullet$ to show it is included. For $x \in \mathbb{R}$, shade the entire line between points; for $x \in \mathbb{Z}$ or $\mathbb{N}$, only mark the specific discrete dots.

Compound Inequalities: When solving double inequalities of the form $a < f(x) < b$, solve them as two separate parts ($a < f(x)$ and $f(x) < b$) and find the intersection of the two solution sets. Visually, the solution is the overlapping region of both inequalities on the number line.