Free Inequality Calculator
Paste any inequality — linear, quadratic, absolute value, or compound — and see the full step-by-step solution with interval notation and sign-flip warnings.
💡 You can also paste an image with Ctrl+V or drag a file here.
Learn
A quadratic inequality — sign-test walkthrough
Below is one fully worked example plus a short primer so you can see exactly how our AI reasons through a problem.
Example Problem
DEMO- 1
Factor the quadratic
Two integers that multiply to 6 and add to −5: −2 and −3.
- 2
Find the critical points (roots)
Set each factor to zero: x = 2 and x = 3. These split the number line into three intervals.
- 3
Sign-test each interval
Pick a test point in each interval and evaluate (x − 2)(x − 3):
- 4
Pick the intervals matching the inequality sign
We need (x − 2)(x − 3) ≥ 0, so we want + or 0. That's x ≤ 2 or x ≥ 3. Include the endpoints because the inequality is non-strict.
- 5
Write in interval notation
Final Answer
How to solve an inequality — linear, quadratic and absolute-value
An inequality compares two expressions with <, >, ≤ or ≥ instead of =. Linear inequalities are solved almost like linear equations, with one critical difference: multiplying or dividing both sides by a negative number flips the inequality sign. Quadratic inequalities are solved by finding the roots, splitting the number line into intervals, and sign-testing each region — the parabola's shape tells you which intervals satisfy the inequality. Absolute-value inequalities split into two cases: |x| < k becomes −k < x < k, while |x| > k becomes x < −k or x > k. Compound inequalities (two conditions joined by AND or OR) are solved separately and intersected or unioned at the end. Interval notation — using (, ), [, ] and ∞ — is the cleanest way to write the solution set, and it's what SAT, GCSE Higher, and A-Level papers expect.
Inequalities to practise (SAT, GCSE, Class 9–12)
Tap any problem to solve it with full step-by-step working.
- Solve with AI →
1. 3x - 7 < 2x + 4
LinearClass 9Easy - Solve with AI →2.Sign flipClass 9 / GCSEEasy
- Solve with AI →
3. x^2 - 9 < 0
QuadraticClass 10 / GCSE HigherMedium - Solve with AI →4.Absolute valueSAT / A-LevelMedium
- Solve with AI →5.CompoundClass 10Medium
Go further with SolveGini
Frequently asked questions
When do I flip the inequality sign?+
Every time you multiply or divide both sides by a negative number. Adding, subtracting, or multiplying/dividing by a positive number does not flip the sign. Squaring or taking reciprocals needs extra care — SolveGini flags these moves explicitly.
How do I solve a quadratic inequality like x² − 5x + 6 ≥ 0?+
Factor the quadratic, find the roots (critical points), split the number line into intervals at those roots, and sign-test each interval. Pick the intervals where the expression matches the inequality. Include the endpoints only for ≥ or ≤, not for > or <.
How do absolute-value inequalities split into cases?+
|x| < k (with k > 0) means −k < x < k — a single interval between the two bounds. |x| > k (with k > 0) means x < −k OR x > k — two disjoint intervals. Always start by isolating the absolute value on one side before splitting.
What is interval notation and why use it?+
Interval notation writes a solution set compactly: (a, b) for a < x < b, [a, b] for a ≤ x ≤ b, (a, ∞) for x > a, and (−∞, a] ∪ [b, ∞) for split ranges. It's standard on SAT, A-Level and A-Level Further Maths papers, and less error-prone than shading a number line.
Is the inequality calculator free to use?+
SolveGini has a free plan. Guests get 1 solve per day; free accounts unlock 5 daily solves plus quizzes, flashcards and the study planner — all step-by-step working shown on the free plan.
Can I use this for SAT Heart of Algebra and GCSE Higher?+
Yes. SAT Heart of Algebra inequality questions, GCSE Higher inequality and quadratic-inequality problems, and Class 11 linear-inequality chapter (NCERT) are all covered with the phrasing each curriculum uses.
Keep mastering inequalities — sign up on the free plan
Create an account in 10 seconds for 5 daily solves across inequalities, algebra, quadratics and every other topic, plus quizzes and flashcards tuned to your board.
Next
Related topics to master next
Inequalities intersect with algebra, quadratics and graphing.
Algebra Solver
Algebra fluency is a prerequisite for every inequality technique.
Open solver →
Linear Equations Solver
Linear inequalities are solved almost identically — minus the sign-flip rule.
Open solver →
Quadratic Equation Solver
Quadratic inequalities rely on the same factoring and discriminant ideas.
Open solver →
System of Equations
Systems of inequalities define regions on the coordinate plane.
Open solver →