Inequalities on a Number Line

A number line graph makes an inequality visual. Instead of describing a solution set with symbols — x > 3, -2 ≤ x < 5 — the graph shows the same information as a shaded region on a horizontal axis. Every point in the shaded region is a value that satisfies the inequality. Every unshaded point does not.

The graph has two parts: the endpoint marker and the shading. The endpoint marker is a circle placed at the boundary value. An open circle means the boundary is not in the solution. A closed circle means it is. The shading extends from the circle in the direction of all values that satisfy the inequality — left for less than, right for greater than, or between two circles for a bounded interval.

Reading a number line graph is the reverse of drawing one. Given a graph, identify the circle type to determine whether the symbol is strict or inclusive, then identify the shading direction to determine whether the symbol is greater than or less than.

A concept becomes durable only when you can move from the rule back into a fresh problem. The calculator is useful here because it lets you test the exact pattern from this article, compare your work with the step list, and verify the final graph or notation.

Start with the number line inequality calculator when you want to check circle placement and shading fast. If you want a focused visual primer before that, review how to graph inequalities on a number line, and if the symbol itself is the source of confusion, revisit inequality symbols.

For split-region practice, compare the same input with the compound inequality calculator. That makes it easier to see when a graph should become one bounded interval and when it should become two separate rays.

Reading a number line graph means converting the visual information back into an inequality or interval notation. Start with the circle type: an open circle means the symbol is strict (> or <), and a closed circle means the symbol is inclusive (≥ or ≤). Then look at the shading direction: shading to the right means greater than, shading to the left means less than.

For a bounded interval, read both circles and the region between them. A closed circle at -2 and an open circle at 5 with shading between them reads as -2 ≤ x < 5 or [-2, 5). For a union graph with two separate rays, read each ray independently and join them with OR: a ray going left from an open circle at -1 and a ray going right from a closed circle at 3 reads as x < -1 or x ≥ 3, or (-∞, -1) ∪ [3, +∞).

When a number line graph shows two separate rays with a gap between them, the gap represents every value that satisfies neither part of the OR inequality. For x < -1 or x ≥ 3, the gap is the region from -1 to 3. A value like x = 0 is not less than -1 and not greater than or equal to 3, so it is correctly excluded from the shaded region.

Students sometimes shade the gap instead of the outer regions, which produces the graph for a compound AND inequality rather than an OR inequality. The test is simple: pick a value in the gap and check whether it satisfies either part of the original inequality. If it does not, the gap should be unshaded.

Every number line graph has an exact interval notation equivalent, and every interval notation expression has an exact number line graph. They are two representations of the same solution set, not two different answers.

This means you can always check your interval notation by sketching the corresponding number line graph, or check your graph by writing the interval notation and verifying the bracket types match the circle types. If the graph shows a closed circle at -2 but the interval notation writes (-2, 5), the representations disagree and one of them contains an error.

Concept pages are useful only if they transfer back into actual problem solving. After reading this guide, the best next step is to practice both directions: drawing a graph from an inequality and reading an inequality from a graph. The two skills reinforce each other.

The calculator pages linked here are meant to shorten that feedback loop. You can test a new inequality, inspect the number line output, and compare the graph with the interval notation to confirm that your visual model is consistent.

After you sketch a graph, rewrite it in interval notation and check whether the bracket types match the circle types. The visual graph and the notation should describe the same set without contradiction.