Linear and quadratic inequality calculator guide
A linear inequality is isolated much like a linear equation, except multiplying or dividing by a negative coefficient reverses the inequality sign. The endpoint is included only for a non-strict operator such as less than or equal to.
A quadratic inequality is solved by finding the related quadratic zeros and determining the polynomial sign on the intervals they create. This exact sign analysis avoids arbitrary sampling limits and produces truly unbounded solution sets when appropriate.
How to solve an inequality in interval notation
- Choose the type: Select linear ax + b compared with c, or quadratic ax squared + bx + c compared with zero.
- Enter coefficients: Provide finite values and keep a nonzero in quadratic mode.
- Choose the operator: Select a strict or inclusive inequality sign.
- Solve: Select Solve inequality to receive exact boundary points and interval notation.
Formula and variables
Linear isolation creates one boundary. Real quadratic zeros partition the number line into regions where the polynomial sign is constant.
Linear boundary: x = (c - b)/a; quadratic boundaries: x = (-b +/- sqrt(b^2 - 4ac))/(2a)- a, b, c — Coefficients
- Finite real coefficients defining the linear or quadratic expression.
- <, <=, >, >= — Inequality operator
- Strict operators exclude zero boundaries; non-strict operators include them.
- D — Quadratic discriminant
- Determines whether the quadratic has two, one, or no real boundary points.
- x — Variable
- Real values tested by the inequality.
Solve x² - 4 > 0
Find the real x-values where the quadratic is positive.
- a
- 1
- b
- 0
- c
- -4
- Operator
- >
- Solve x² - 4 = 0 to get boundaries -2 and 2
- The upward-opening quadratic is positive outside its two zeros
- The strict inequality excludes both zeros
Result: (-∞, -2) ∪ (2, ∞).
Values farther than 2 units from zero make x squared greater than 4.
Understanding your results
Brackets and parentheses
A bracket includes a finite endpoint. A parenthesis excludes it, and infinity always uses a parenthesis.
Empty or all-real result
The empty-set symbol means no real value works. (-∞, ∞) means every real value works.
Assumptions
- The variable ranges over real numbers.
- Quadratic mode compares ax squared + bx + c with zero.
- Only the four order operators <, <=, >, and >= are supported.
- Coefficients are finite real numbers.
Limitations
- The calculator does not parse free-form expressions or compound AND/OR statements.
- It does not solve rational, absolute-value, exponential, or trigonometric inequalities.
- Near-zero floating-point discriminants are classified by their computed value.
- Symbolic exact radicals are displayed as decimal boundary values.
Common mistakes
- Forgetting to reverse the operator after dividing by a negative number.
- Including endpoints for a strict inequality.
- Excluding endpoints for a non-strict inequality.
- Testing only a finite window and mistaking it for the full real line.
- Using the quadratic formula without first comparing the polynomial with zero.
Practical use cases
Threshold constraints
Solve one-variable upper or lower bounds with a constant rate and offset.
Quadratic sign regions
Find where a parabola is above, below, or touching the x-axis.
Planning and decision guide
Check endpoints separately
First find exact equality boundaries, then decide whether each is included from the original operator. Use sign regions for the open intervals between and beyond the boundaries.
Frequently asked questions
When does the inequality sign reverse?
Reverse it whenever both sides are multiplied or divided by a negative number.
What is the difference between a bracket and parenthesis?
A bracket includes the finite endpoint; a parenthesis excludes it.
Why is infinity never included?
Infinity is not a real endpoint value, so interval notation always uses a parenthesis beside it.
Can a quadratic inequality have no boundary points?
Yes. A negative discriminant means the quadratic never reaches zero, so its sign is constant over all real x.
Does this calculator estimate solutions from graph samples?
No. It calculates linear boundaries or quadratic zeros and builds the exact sign intervals from them.
Sources and review
- Solve Linear Inequalities — OpenStax Intermediate Algebra 2e. Accessed 2026-08-29.
- Solve Quadratic Inequalities — OpenStax Intermediate Algebra 2e. Accessed 2026-08-29.
Reviewed 2026-08-29.