Number Line Graph
A number line graph is a visual representation of the solution set of an equation or inequality, plotted as points or rays on a number line.
Definition
A number line graph shows the answer to an inequality as a picture: you draw a line, mark important numbers, use dots for specific values, and shade or draw arrows for ranges. The conventions are an open circle at an excluded endpoint ($<$ or $>$), a closed (filled) circle at an included endpoint ($\le$ or $\ge$), and a shaded ray or segment to show the range of solutions, letting you visually identify intersections and unions when graphing compound inequalities. More formally, a number line graph is a topological embedding of the solution set (a subset of $\mathbb{R}$) into the real line, with the inherited subspace topology; points, closed intervals, open intervals, and rays are the four types of connected subsets, and a finite union of intervals (possibly non-connected) represents the general solution set of a polynomial inequality.
Example
For $x > 3$: draw an open circle at $3$ (not included) with an arrow pointing right. $-1 \le x < 4$ is graphed with a closed circle at $-1$, an open circle at $4$, and a shaded segment between them, while the compound "or" inequality $x < -2$ or $x \ge 1$ shows two separate rays. The solution set of $(x-1)(x-3)(x+2) > 0$ consists of two open intervals and one ray, determined by sign analysis at the roots $-2$, $1$, $3$.
Key Insight
A picture of the solution makes it easy to see "how many" answers there are and roughly where they are on the number line. Sign analysis of a factored polynomial inequality generalizes the number line graph technique, and the structure of the solution set reveals the topology of the complement of the zero set, a theme that recurs in algebraic topology.