Linear Inequations in One Variable — Solution and the Number Line

A linear inequation in one variable has the form $ax+b<0$ (or with $>,\le,\ge$), where $a\ne0$ and $a,b$ are real. Its solution is always an interval — a ray of the real line — which we both state in interval form and draw on a number line.

Solving

Isolate $x$ using the three rules of Section 1. For example, $3x-5\le 7$ gives $3x\le 12$, so $x\le 4$. The solution set is $\{x\in\mathbb{R}: x\le 4\}=(-\infty,4]$.

Representing the solution on a number line

Mark the boundary value, then decide the dot and the direction:

For example, the solution $x\le 4$ is drawn as a closed (filled) dot at $4$ with the ray shaded to the left — through $3,2,1,0$ and continuing beyond — representing every real number $x\le 4$, i.e. the interval $(-\infty,4]$.

Domain matters: real vs. natural-number solutions …