Graphing Inequalities on a Number Line Calculator

How to Graph Inequalities on a Number Line

A number line graph provides a visual representation of all real numbers that satisfy an algebraic inequality. Unlike single equations (which typically yield discrete individual points), inequalities possess infinite solution sets that form continuous rays or bounded segments along the real continuum.

Correctly plotting an inequality on a number line requires mastering two foundational visual conventions:

1. Open (Hollow) Circles: \(<\) and \(>\)

An open hollow circle means the boundary number itself is strictly excluded from the solution set. For example, \(x > 3\) starts immediately past 3, but does not include 3 itself.

2. Closed (Solid) Circles: \(\le\) and \(\ge\)

A solid filled circle means the boundary number is included in the solution set. For example, \(x \le 5\) includes 5 as a valid solution.

How to Use the Graphing Inequalities Calculator

Graph single and compound inequalities with automatic interval conversion in three steps:

1

Choose Inequality Mode

Select Single Linear (e.g. \(2x + 4 \le 12\)), Compound AND bounded segment (\(-3 < x \le 5\)), or Compound OR disjoint rays (\(x < -2 \text{ or } x \ge 4\)).

2

Enter Boundaries & Operators

Input your inequality expression or select boundary values and comparison operators (\(<, \le, >, \ge\)). Quick presets are available for immediate one-click testing.

3

View Vector SVG & Interval

The vector number line renders immediately with precise endpoint coordinates, shaded ray direction, and corresponding formal Interval and Set-Builder notations.

Problems This Number Line Grapher Solves

Eliminating Open vs Closed Circle Mistakes

Forgetting whether \(\le\) or \(<\) gets a hollow or solid dot is a persistent source of lost exam points. Our dynamic SVG graph visually enforces the exact standard rule.

Handling the Negative Sign Reversal Rule

Dividing or multiplying by a negative coefficient reverses the inequality direction (e.g. \(-2x \le 6 \implies x \ge -3\)). Our step-by-step solver explicitly highlights this sign-flip.

Unifying Compound AND vs OR Shading

Visually demonstrates the difference between the intersection of two bounds (bounded segment \([a, b]\)) versus the union of two opposing rays (\((-\infty, a] \cup [b, \infty)\)).

Translating to Calculus Interval Notation

Precalculus and calculus require reporting domains and ranges in interval notation. Seeing the number line alongside \((-\infty, 4]\) cements the connection between graphs and brackets.

Key Features & Capabilities

Dynamic Vector SVG

Crisp vector number line that dynamically rescales tick marks and arrowheads.

Compound Logic

Full support for three-part double inequalities and split "OR" union rays.

Triple Output

Outputs isolated inequality, interval notation, and formal set-builder notation.

Instant Reactive UI

Zero calculate button wait: number line redraws smoothly as you type.

Directional Shading & The Negative Division Rule

Variable on the Left Rule When the variable is isolated on the left side of the inequality (e.g. \(x < c\)), the inequality sign acts as an arrow pointing in the exact direction of shading. Less than (\(<, \le\)) points left toward \(-\infty\); greater than (\(>, \ge\)) points right toward \(+\infty\).

The Critical Negative Coefficient Rule When solving an inequality, multiplying or dividing both sides by a negative number reverses the inequality sign (e.g. \(-3x < 9 \implies x > -3\)). Forgetting to reverse the sign is the #1 most common error on algebra exams.

Compound Inequalities: AND vs. OR

Conjunctions (AND / Bounded Double Inequalities): Format \(-2 \le x < 5\). The solution is the interior overlapping line segment between the two boundary points.
Disjunctions (OR / Disjoint Unions): Format \(x < -3 \text{ or } x \ge 4\). The solution consists of two outward-pointing rays extending in opposite directions toward \(-\infty\) and \(+\infty\).

Frequently Asked Questions

When do you use an open circle versus a closed circle on a number line? ▼

An open (hollow) circle indicates that the boundary value is NOT part of the solution set; this is used for strict inequalities (< and >). A closed (solid) circle indicates that the boundary value IS included in the solution set; this is used for inclusive inequalities (≤ and ≥).

Which direction do you shade on the number line? ▼

When the variable is on the left side of the inequality (e.g. x < 5), the inequality sign points in the exact direction of shading: '<' points left (shade left toward negative infinity), while '>' points right (shade right toward positive infinity). If the variable is on the right side, flip the inequality first (e.g. 5 > x becomes x < 5).

How are compound inequalities graphed on a number line? ▼

Compound inequalities come in two types: Conjunctions ('AND', e.g. -2 ≤ x < 5) represent the overlap between two conditions, producing a bounded shaded segment between the two endpoints. Disjunctions ('OR', e.g. x < -1 or x ≥ 4) represent the union of two separate sets, producing two outward rays pointing in opposite directions.

Why does dividing by a negative number flip the inequality sign? ▼

Multiplying or dividing by a negative number reflects values across zero on the number line, reversing their relative order. For instance, while 2 < 5 is true, multiplying both sides by -1 produces -2 and -5; on the number line, -2 lies to the right of -5, requiring the inequality to reverse to -2 > -5.

How does this tool convert number line graphs to interval notation? ▼

Our calculator directly maps number line visual features to mathematical notation: an open circle or infinity maps to a round parenthesis '(', ')', a solid circle maps to a square bracket '[', ']', and multiple disconnected shaded rays are joined using the set union symbol '∪'.