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Inequality Calculator
Type any inequality in x and get the exact solution in interval and set-builder notation, a number line graph and every step: critical points, a sign chart and which endpoints are included.
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Examples:
Type <, >, <= or >= (or ≤ ≥). Powers with ^, absolute value with |…| or abs(…), roots with sqrt(…). Chains like −3 < 2x + 1 ≤ 7 work too.
Interval notation
Set-builder notation
Inequality form
Number line
Step-by-step solution
Sign chart
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How to use the Inequality Calculator
- Type your inequality in x, for example x^2 - 5x + 6 < 0, |2x - 3| <= 5 or (x + 1)/(x - 2) >= 0. Or tap an example.
- Use the symbol buttons to insert ≤, ≥, absolute value bars, powers and square roots.
- Press Solve (or Enter). The answer appears in interval notation, set-builder notation and inequality form.
- Check the number line: shaded parts are solutions, filled dots are included endpoints and open dots are excluded.
- Read the steps and sign chart to see how the critical points and test values lead to the answer, then copy the notation you need.
The critical point method
Almost every inequality can be solved with the same four moves, and they are exactly what the step-by-step panel shows:
- Get zero on one side. Rewrite the inequality as g(x) < 0, g(x) ≤ 0, g(x) > 0 or g(x) ≥ 0. For rational inequalities, combine everything into a single fraction. Do not multiply both sides by an expression containing x, because you do not know whether it is positive or negative.
- Find the critical points: the x values where g(x) = 0 and the x values where g(x) is undefined (a denominator is zero).
- Test each interval. The critical points cut the number line into intervals where g keeps one sign. One test value per interval tells you the sign of the whole interval.
- Decide the endpoints. Zeros are included for ≤ and ≥ and excluded for < and >. Points where g is undefined are always excluded.
Quick reference
| Inequality | Rule | Example | Solution |
|---|---|---|---|
| ax + b > c | Isolate x; flip the sign if you divide by a negative a | −2x + 1 > 7 | x < −3, (−∞, −3) |
| (x − r₁)(x − r₂) < 0, r₁ < r₂ | Upward parabola is negative between its roots | x² − 5x + 6 < 0 | (2, 3) |
| (x − r₁)(x − r₂) > 0 | Positive outside the roots | x² − 4 > 0 | (−∞, −2) ∪ (2, ∞) |
| (x − r)² ≤ 0 | A square is never negative; only equality works | (x − 1)² ≤ 0 | {1} |
| |A| < c (c > 0) | −c < A < c | |2x − 3| ≤ 5 | [−1, 4] |
| |A| > c (c ≥ 0) | A < −c or A > c | |x| > 2 | (−∞, −2) ∪ (2, ∞) |
| P(x)/Q(x) ≥ 0 | Critical points from P = 0 and Q = 0; Q = 0 always excluded | (x + 1)/(x − 2) ≥ 0 | (−∞, −1] ∪ (2, ∞) |
| a < f(x) < b | Solve both parts and intersect | −3 < 2x + 1 ≤ 7 | (−2, 3] |
Common mistakes to avoid
- Cross-multiplying a rational inequality. Turning (x + 1)/(x − 2) ≥ 0 into x + 1 ≥ 0 silently assumes x − 2 > 0 and loses half of the answer.
- Taking a square root of both sides. x² > 9 is not just x > 3; the negative side x < −3 also works. Factor or use |x| > 3 instead.
- Forgetting double roots. At a repeated root such as (x − 1)² the sign does not change, so neighbouring intervals have the same sign. The sign chart makes this visible.
- Mixing up brackets. Infinity always gets a round bracket, because ∞ is not a number that can be included.
Frequently asked questions
How do you solve a quadratic inequality?
Move everything to one side so it reads ax² + bx + c < 0 (or >, ≤, ≥). Find the roots of ax² + bx + c = 0, which split the number line into intervals. Test one value in each interval: the intervals where the sign matches the inequality are the solution. For example x² − 5x + 6 < 0 factors as (x − 2)(x − 3) < 0, which is negative only between the roots, so the answer is (2, 3).
When do you flip the inequality sign?
Flip it whenever you multiply or divide both sides by a negative number. For example −2x > 6 becomes x < −3. Adding or subtracting any number, or multiplying by a positive number, never changes the direction.
How do you solve absolute value inequalities?
For c ≥ 0, |A| < c means −c < A < c (a single interval), while |A| > c means A < −c or A > c (two separate pieces). With ≤ and ≥ the endpoints are included. If c is negative, |A| < c has no solution and |A| > c is true for every x.
Why is a point excluded in a rational inequality even with ≥?
A rational expression is undefined where its denominator equals zero, so that x value can never be part of the solution. In (x + 1)/(x − 2) ≥ 0 the zero of the numerator, x = −1, is included, but x = 2 is excluded, giving (−∞, −1] ∪ (2, ∞).
What is the difference between interval and set-builder notation?
Interval notation lists the pieces of the solution with brackets: [ ] for included endpoints and ( ) for excluded ones, joined by ∪. Set-builder notation describes the rule instead, for example {x | x ≤ −1 or x > 2}. Both describe the same set.
Can it solve inequalities with sin, cos, ln or square roots?
Yes. Expressions with roots, logarithms, exponentials and trig functions are solved numerically by locating the zeros and break points and testing each interval. Trig inequalities repeat forever, so their solution is shown for −4π ≤ x ≤ 4π, and non-exact endpoints are marked with ≈.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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