Inequalities & Interval Grapher

Enter linear or quadratic inequalities to see solutions on a number line, or set up a system of linear inequalities to visualize the feasible region on a coordinate plane.

Presets

Inequality 1

Combine

Solution

(2, ∞)

Step-by-step

Reference Guide

Linear Inequalities

A linear inequality has the form ax + b compared to c using one of the four operators. Solving follows the same steps as solving an equation, with one key difference.

Standard form

ax + b < c

Operator flip rule

When you multiply or divide both sides by a negative number, the inequality direction reverses.

-2x > 6 \implies x < -3

Quadratic Inequalities

Solve by finding the roots and then using the sign of the leading coefficient to determine which regions satisfy the inequality.

Sign analysis

For a parabola opening upward (a > 0), the quadratic is negative between the roots and positive outside.

x^2 - 4 \le 0 \implies x \in [-2,\, 2]

Discriminant

\Delta = b^2 - 4ac

Compound Inequalities

Two inequalities can be combined with AND (intersection) or OR (union) to form a compound inequality.

AND (intersection)

Both conditions must be true at the same time.

x > -1 \text{ AND } x < 5 \implies x \in (-1,\, 5)

OR (union)

At least one condition must be true.

x < -3 \text{ OR } x > 3 \implies x \in (-\infty, -3) \cup (3, \infty)

Systems of Linear Inequalities

Each linear inequality in two variables defines a half-plane. The feasible region is the intersection of all half-planes.

Half-plane

The boundary line divides the plane into two halves. The inequality tells you which side to shade.

ax + by \le c

Feasible region

The overlapping area where all constraints are satisfied. Vertices occur where boundary lines intersect.