Number Line Graph
Pre-AlgebraA 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.
Example
For $x > 3$: draw a number line, put an open circle at $3$ (not included), and draw an arrow pointing right (all numbers greater than $3$).
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.
Definition
A number line graph represents solution sets visually. Conventions: open circle at an excluded endpoint ($<$ or $>$), closed (filled) circle at an included endpoint ($\le$ or $\ge$). A shaded ray or segment shows the range of solutions. A single point is graphed as just one dot.
Example
$-1 \le x < 4$ is graphed with a closed circle at $-1$, an open circle at $4$, and a shaded segment between them. The compound "or" inequality $x < -2$ or $x \ge 1$ shows two separate rays.
Key Insight
Number line graphs let you visually identify intersections and unions of solution sets, making compound inequalities easier to understand at a glance.
Definition
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. A finite union of intervals (possibly non-connected) represents the general solution set of a polynomial inequality.
Example
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 $-2$, $1$, $3$. The number line graph makes the sign chart visual.
Key Insight
Sign analysis of a factored polynomial inequality generalizes the number line graph technique. The structure of the solution set reveals the topology of the complement of the zero set, a theme that recurs in algebraic topology.