What this calculator does
Find every value of x that makes your inequality true, with each step shown. An inequality says one side is bigger (or smaller) than the other. So its answer is usually a whole stretch of the number line, not a single number. This solver handles linear ones, where x appears only to the first power, such as 3x − 7 ≤ 2x + 1. It also handles quadratic ones, with an x² term, such as x² − 5x + 6 > 0. You get the answer as an inequality and in interval notation, a short way to write ranges of numbers.
It works exactly, so fractions stay fractions. Boundary points that aren't neat numbers appear with square roots, such as (5 − √13)/2, with a decimal beside them. The working shows every move. That includes the moments when multiplying or dividing by a negative number flips the sign. For quadratics, it also shows the test point used in each piece of the number line.
Where it's used
Why this math matters outside the classroom:
- Budget limits. A shopper with $50 buying a $14 item plus $4 snacks solves 14 + 4x ≤ 50 to find they can buy at most 9 snacks.
- Phone plan choice. Comparing a $20 plan plus $0.10 a minute with a flat $45 plan, 20 + 0.1x < 45 shows the first is cheaper below 250 minutes.
- Projectile heights. A physics student solves −5t² + 20t > 15 to find a ball stays above 15 m between 1 and 3 seconds after the throw.
- Profitable output. A workshop with profit −2q² + 40q − 150 solves profit > 0 to find the range of weekly output that makes money.
- Proofing dough. A baker who needs dough at least 25 °C solves 18 + 1.4m ≥ 25 to learn it must proof for at least 5 minutes in the warmer.
Grade level and standards: Grades 9–10; Common Core State Standards for Mathematics HSA.REI.B.3. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Type the inequality using x as the variable, ^ for powers and one of <, >, <= or >= (≤ and ≥ also work).
- Brackets and products are fine. The calculator multiplies out (x − 1)(x + 4) < 0 and 2(x + 3) > 4(x − 1) for you.
- Read the solution in interval notation. Below it, you'll see the same answer written as an inequality, and the boundary points.
- Open the working to see how the inequality was rearranged and tested.
What the inputs mean
- Inequality in x
- Use only one inequality sign at a time. A double inequality such as 1 < 2x + 3 < 9 is really two inequalities joined by 'and'. Solve 1 < 2x + 3 and 2x + 3 < 9 separately, then keep the values that work for both. Once everything is moved to one side, the highest power of x can be at most 2.
Formula and method
Linear: bx + c > 0 → x > −c ÷ b when b > 0, x < −c ÷ b when b < 0 Quadratic: ax² + bx + c with a > 0 is positive outside its roots and negative between them roots x = (−b ± √(b² − 4ac)) ÷ 2a ( ) excludes an endpoint, [ ] includes it, ∪ joins two pieces, ∅ means no solution
In these lines, a, b and c are the plain numbers in front of x², in front of x, and on their own. The ± means do it once with plus and once with minus. Adding or subtracting the same amount on both sides never changes an inequality. Neither does multiplying or dividing by a positive number. But multiplying or dividing by a negative number reverses the order, so 3 > 1 becomes −3 < −1. That's why the sign must flip. To solve a linear inequality, you get x on its own with these moves.
For a quadratic, move everything to one side and make the number in front of x² positive. If you multiply by −1 to do that, flip the sign. The graph of the left side is then a U-shaped curve opening upward, called a parabola. It can only change sign where it crosses zero, at the roots from the quadratic formula. So it's negative between the two roots and positive outside them. The discriminant, b² − 4ac, tells you how many roots there are. If it's zero, the curve just touches zero at one point. If it's negative, the curve never reaches zero and is positive everywhere.
The calculator checks the sign in each piece with a test point. That's the same idea as reading the graph. Then it keeps the pieces that match the inequality. Strict signs (< and >) leave the boundary points out, and ≤ and ≥ include them.
Worked example
A ball thrown upward has height −5x² + 20x meters after x seconds. During which times is it higher than 15 meters? Solve −5x² + 20x > 15.
- Move everything to the left: −5x² + 20x − 15 > 0.
- Multiply by −1 to make the x² term positive, which flips the sign: 5x² − 20x + 15 < 0.
- Discriminant: D = (−20)² − 4 × 5 × 15 = 400 − 300 = 100, so there are two roots, x = 1 and x = 3.
- Test a point in each piece: at x = 0 the left side is 15 (positive), at x = 2 it is −5 (negative), and at x = 4 it is 15 (positive).
- Keep the piece where it is negative: 1 < x < 3.
Result: The ball is above 15 meters for 1 < x < 3, written (1, 3) in interval notation. That's from 1 second to 3 seconds after the throw, not counting those exact moments.
Understanding the result
Interval notation lists the left end, then the right end. A round bracket means the endpoint is left out, and a square bracket means it's included. ∞ (infinity) always gets a round bracket, because it isn't a number you can reach. (−∞, 2) ∪ (3, ∞) means every number below 2, together with every number above 3.
Two special answers are worth knowing. (−∞, ∞) means every real number works, and ∅ (the empty set) means none does. A single point such as {2} shows up when a perfect square has to be at most zero.
Assumptions and limitations
- x is a real number, and the numbers in your inequality are exact decimals or fractions you typed.
- Once every term is moved to one side, the highest power of x is 1 or 2 (degree 1 or 2).
- It doesn't handle inequalities with x³ or higher powers, x in a denominator (rational inequalities), or absolute values.
- Double inequalities must be split into two single ones.
- Numbers involving π or e aren't accepted, because the solver works with exact fractions.
Frequently asked questions
Why does the inequality sign flip when I divide by a negative number?
Multiplying by a negative number mirrors the number line, so the order of any two numbers reverses: 2 < 5, but −2 > −5. Dividing is the same as multiplying by the flipped fraction (the reciprocal), so it reverses the order too. Adding, subtracting, or multiplying and dividing by a positive number keeps the sign.
How do I write the answer in interval notation?
Write the smallest and largest values of each piece in brackets. Use round brackets if the endpoint is left out, and square ones if it's included. x ≥ 4 is [4, ∞), and 2 < x ≤ 7 is (2, 7]. x < −1 or x > 1 is (−∞, −1) ∪ (1, ∞).
How do I solve a quadratic inequality by hand?
Get zero on one side, find the roots of the quadratic, and mark them on a number line. They split it into at most three pieces. Pick any number inside each piece, put it in for x, and note whether the result is positive or negative. Keep the pieces whose sign matches the inequality.
What does it mean when the answer is all real numbers or no solution?
It means the x terms either cancel out completely or leave an expression that never changes sign. For example, x² + 1 is always at least 1. So x² + 1 > 0 is true for every x, and x² + 1 < 0 is true for none.
How this calculator is checked
Method: Order properties of the real numbers (adding, and multiplying by positive or negative numbers), the quadratic formula and the sign of a quadratic between and outside its roots; exact rational arithmetic and simplified radicals. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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