What an Inequality Calculator Solves
An inequality compares two expressions with <, >, ≤ or ≥. Where an equation like 2x + 1 = 7 has one answer, 2x + 1 < 7 is true for every x below 3.
This calculator covers linear, compound, absolute value, quadratic, cubic and rational inequalities, plus systems of two inequalities in x and y.
How to Use the Inequality Calculator
- Type an inequality such as x^2 - 5x + 6 > 0 or |2x - 3| < 5.
- Join conditions with and or or, or write a chain like -2 < 3x + 1 ≤ 7.
- Read the interval, check the number line, and type a value in the “Is x a solution?” box to test it.
For two variables, switch to Graph in x and y, put one inequality on each line, and the overlap is shaded with its exact corner points.
The One Rule That Trips Everyone: Flip the Sign
−4x + 8 > 20
- Subtract 8 from both sides: −4x > 12.
- Divide by −4 and flip the sign: x < −3.
Answer: x < −3, or (−∞, −3)
Why? 2 < 5, but multiplying by −1 gives −2 and −5, and now −2 is larger. Negatives reverse order.
Interval Notation and the Number Line
| Inequality | Interval | Number line |
|---|---|---|
| x > 4 | (4, ∞) | open circle, shade right |
| x ≥ −3 | [−3, ∞) | closed circle, shade right |
| −2 < x ≤ 4 | (−2, 4] | open at −2, closed at 4 |
| x < 1 or x ≥ 6 | (−∞, 1) ∪ [6, ∞) | two shaded pieces |
Round brackets leave the end out, square brackets keep it, and infinity always takes a round one.
Quadratic and Rational Inequalities: Use a Sign Chart
x² − 5x + 6 > 0
- Factor: (x − 2)(x − 3) > 0, so the cut points are 2 and 3.
- Test each interval: x = 0 gives 6, x = 2.5 gives −0.25, x = 4 gives 2.
- Keep the positive intervals.
Answer: (−∞, 2) ∪ (3, ∞)
(x − 1) / (x + 2) ≥ 0
- The top is zero at 1 and the bottom at −2, so those are the cut points.
- x = 1 is included because the inequality allows equality. x = −2 is never included, because it divides by zero.
- Test each interval: both outer ones are positive.
Answer: (−∞, −2) ∪ [1, ∞)
Across the 361 monic quadratics x² + bx + c with b and c from −9 to 9, only 281 have two real roots, 7 are perfect squares and 73 have none. So x² + 1 > 0 holds for every real number.
An opinion worth stating: multiplying both sides by the denominator is the most overrated shortcut in algebra. It fails whenever the denominator's sign is unknown, as in (x − 1)/(x + 2). The sign chart is safer.
Absolute Value and Compound Inequalities
| Form | Becomes | Example |
|---|---|---|
| |A| < k | −k < A < k | |2x − 3| < 5 → (−1, 4) |
| |A| > k | A < −k or A > k | |x − 1| ≥ 2 → (−∞, −1] ∪ [3, ∞) |
| |A| < negative | no solution | |x| < −2 → ∅ |
“And” keeps what two conditions share, which is an intersection. “Or” keeps everything either one allows, which is a union.
Original Data: Quadratic Inequalities Often Have No Boundary
We checked all 6,498 quadratics ax² + bx + c with integer coefficients from −9 to 9 and a ≠ 0. About 37.8% have a negative discriminant, so the parabola never crosses the x-axis. For those, the inequality ax² + bx + c > 0 is true for every real number or for none, depending only on the sign of a. Only about 62% of cases have real boundary points where a sign chart is needed. Check the discriminant first and you can answer more than a third of quadratic inequalities in one line.
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: shading the number line before solving. It is a way to present an answer, not a way to find one.
Underrated: the test-point method. Pick a number from each interval, plug it in and read off true or false. It works for quadratics, rational and higher-degree inequalities alike.
Plain wrong: multiplying both sides by an expression with x in it. If you do not know whether it is positive or negative, you do not know whether to flip the sign. Bring everything to one side and use a sign chart.
Systems of Inequalities in Two Variables
Graph each boundary line, dashed for < and >, solid for ≤ and ≥, shade the side that passes a test point, and keep the overlap. A system such as x ≥ 0, y ≥ 0, x + y ≤ 6 gives a triangle with corners (0, 0), (6, 0) and (0, 6). Add an objective like max 3x + 2y to find the best corner, the idea behind linear programming.
Why Students Get Stuck on Inequalities, and What a Good Calculator Should Do
Searching for help with inequalities usually starts late at night with one problem that will not work out. Multiplying by a negative flips the sign, and interval notation adds another layer. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to solve, graph and write the answer correctly, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Forgetting to flip the sign when dividing by a negativeSolve 3x - 5 < 7↗
- Mixing up open and closed endpointsSolve -2x + 4 >= 10↗
- Writing the interval notation backwardsSolve x^2 - 4 < 0↗
- Not testing a value to confirm the region
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants the flip explained when it happens
- Wants the number line and interval notation
- Wants a test value as a check
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Solve 3x - 5 < 7↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.