AI Inequality Solver
An inequality compares two quantities instead of setting them equal, so its answer is a range of values rather than one number. This AI inequality solver rearranges the inequality step by step and tells you precisely when the sign must be reversed.
How inequalities differ from equations
Solving 3x − 5 < 10 uses the same moves as solving an equation: add, subtract, multiply or divide both sides. There is one extra rule that causes most lost marks. When you multiply or divide both sides by a negative number, the inequality sign flips. The solver highlights that step whenever it happens.
Why it flips: 2 < 5 is true, but multiplying both by −1 gives −2 and −5, and −2 is greater than −5. The order of the numbers reverses, so the sign must too.
Ways to write the solution
| Inequality | Number line | Interval notation |
|---|---|---|
| x < 5 | Open circle at 5, arrow left | (−∞, 5) |
| x ≥ −2 | Filled circle at −2, arrow right | [−2, ∞) |
| −2 < x ≤ 4 | Open at −2, filled at 4, line between | (−2, 4] |
Strict signs (< and >) use open circles and round brackets because the end value is not included. Signs with a bar (≤ and ≥) use filled circles and square brackets.
Compound and quadratic inequalities
A compound inequality such as −3 < 2x + 1 ≤ 9 is solved by applying each step to all three parts at once. Quadratic inequalities need a different approach: solve the related equation first, then test the regions between the roots. See our full guide on how to solve inequalities for those cases, or use the linear equation solver when the sign is an equals sign.
Worked examples
Example 1: Solve 3x − 5 < 10
- Add 5 to both sides: 3x < 15.
- Divide by 3, a positive number, so the sign stays: x < 5.
Final answer: x < 5
Example 2: Solve −2x + 4 > 10
- Subtract 4 from both sides: −2x > 6.
- Divide by −2. The number is negative, so flip the sign: x < −3.
Final answer: x < −3
Example 3: Solve −3 < 2x + 1 ≤ 9
- Subtract 1 from all three parts: −4 < 2x ≤ 8.
- Divide all parts by 2: −2 < x ≤ 4.
- In interval notation this is (−2, 4].
Final answer: −2 < x ≤ 4
Example 4: Solve x ÷ (−3) ≥ 2
- Multiply both sides by −3, which is negative, so flip the sign.
- x ≤ −6.
Final answer: x ≤ −6
Common mistakes to avoid
- Forgetting to flip the sign after dividing or multiplying by a negative number.
- Flipping the sign when subtracting a negative-looking term; only multiplication and division matter.
- Using a filled circle for a strict inequality, or an open circle for ≤ and ≥.
- Turning the answer into a single value; an inequality has infinitely many solutions.
- Not testing a value from your answer back in the original inequality.
Frequently asked questions
Why does the inequality sign flip?
Multiplying by a negative reverses the order of numbers on the number line, so “less than” becomes “greater than”. Adding or subtracting never changes the direction.
What is the difference between < and ≤?
The sign < excludes the boundary value; ≤ includes it. On a graph, that is an open versus a filled circle.
Can the solver do inequalities with x on both sides?
Yes. It moves all x-terms to one side first, then isolates x, flipping the sign if the coefficient ends up negative.
How do I check an inequality answer?
Pick any number that satisfies your answer, put it in the original inequality, and confirm it is true. Then try a number outside the range and confirm it is false.
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AI-generated solutions are for learning purposes. Always verify important answers.