Graphical Representation
A linear inequality is a mathematical statement that compares a linear expression to a value using one of the symbols <, >, ≤, or ≥. Unlike a linear equation, which has exactly one solution (or none), a linear inequality typically has infinitely many solutions — an entire range of numbers that satisfy the condition. To communicate this range clearly, mathematicians represent solution sets graphically on a number line, turning an abstract algebraic statement into a vivid visual picture. The number-line representation works by shading the portion of the line that corresponds to all values making the inequality true. A single point is marked to show where the boundary lies, and whether that boundary point is included or excluded is shown by the style of the dot: a closed (filled-in) circle means the endpoint IS part of the solution, while an open (hollow) circle means it is NOT. An arrow extending left or right — or both — indicates that the solution continues infinitely in that direction. Graphical representation is not merely a convenience; it builds genuine mathematical intuition. When you see the shaded ray on a number line, you immediately grasp that the solution is an infinite set, understand its direction, and can quickly read off whether specific test values are included. This skill connects directly to interval notation, set-builder notation, compound inequalities, and later to graphing solution regions for two-variable inequalities in the coordinate plane.
- Solve the inequality for the variable
- Locate the boundary point on the number line
- Choose open or closed circle
- Shade the correct ray
- Write the solution in interval notation
Solve the inequality for the variable
Isolate the variable exactly as you would for an equation, applying inverse operations to both sides. The critical rule: if you multiply or divide both sides by a negative number, you must reverse the direction of the inequality symbol. For example, −2x < 6 becomes x > −3 after dividing by −2 and flipping the sign.
Locate the boundary point on the number line
Draw a horizontal number line with enough scale to show the boundary and a comfortable range around it. Find the boundary value — the number the variable is being compared to — and mark it on the line. This is the critical threshold that separates solutions from non-solutions.
Choose open or closed circle
If the inequality is strict (< or >), draw an OPEN circle at the boundary point to show it is NOT included in the solution set. If the inequality is non-strict (≤ or ≥), draw a CLOSED (filled) circle to show the boundary point IS included. Mathematically, open circles correspond to strict inequalities because the set {x : x > 3} does not contain 3 itself.
Shade the correct ray
Shade the portion of the number line that represents all solutions. If x > a or x ≥ a, shade to the RIGHT with an arrow indicating the solution continues to +∞. If x < a or x ≤ a, shade to the LEFT with an arrow toward −∞. A useful check: pick a simple test value from the shaded region (not the boundary) and substitute it into the original inequality — it should make the inequality true.
Write the solution in interval notation
Translate the graph into interval notation to complete the representation. Use a parenthesis ( or ) for open endpoints and a bracket [ or ] for closed endpoints. Infinity is always written with a parenthesis because it is never actually reached. For example, x ≥ −3 becomes [−3, +∞), and x < 5 becomes (−∞, 5).
The Key Equation
- a — coefficient of x (dimensionless): The number multiplying the variable x. If a < 0, dividing by it reverses the inequality direction.
- b — constant term (dimensionless): The additive constant on the left side, moved to the right by subtraction.
- c — right-hand side value (dimensionless): The value the expression ax + b is being compared to; it defines the boundary after rearrangement.
- \lessgtr — inequality symbol (—): Represents any one of the four symbols: <, >, ≤, ≥. The direction may flip depending on operations performed.
Worked Example
Problem. Solve 3x − 4 ≤ 11 and represent the solution on a number line, then write the answer in interval notation.
Answer. All real numbers less than or equal to 5; interval notation: (−∞, 5]
Origins · 1 minRené Descartes and how this was discovered
Descartes developed the Cartesian coordinate system, published in his 1637 work La Géométrie. By assigning numerical coordinates to geometric points, he made it possible to represent algebraic relationships — including inequalities — as visual graphs. The number line itself is a one-dimensional instance of his coordinate idea, and shading solution sets on it descends directly from this union of algebra and geometry he pioneered. Descartes reportedly developed his habit of lying in bed until late morning while serving as a soldier — and legend has it that he conceived the idea of coordinates while watching a fly move across his bedroom ceiling.
Related Concepts
- Compound Inequalities — Compound inequalities combine two linear inequalities with 'and' or 'or', creating solution sets that appear on number lines as intersections or unions of intervals — a direct extension of single-inequality graphing.
- Interval Notation — Interval notation is the symbolic language used to express exactly what a number-line graph shows visually; the two representations are two sides of the same coin.
- Solving Linear Inequalities — Before graphing, you must solve the inequality algebraically. Understanding why the inequality symbol flips under division by a negative is essential background.
- Graphing Two-Variable Inequalities — The same open/closed boundary logic and shading technique used on number lines extends to the coordinate plane, where boundary lines replace boundary points.