Inequality Calculator: Solve the Inequality
Type an inequality and this calculator will solve the inequality step by step, write the answer in inequality and interval notation, and graph it on a number line. It handles linear, two-step, double and absolute value inequalities.
Result
Enter values and press Calculate to see the result.
Formula
Multiplying or dividing both sides by a negative number reverses the inequality sign.
An absolute value inequality becomes a double inequality (less than) or two separate ones joined by “or” (greater than).
How to use this calculator
- Type the inequality, for example 2x + 3 > 7, 5 - 3x <= 11 or |2x - 1| >= 5. Type <= for ≤ and >= for ≥.
- Press Calculate. You get the answer in inequality notation and interval notation, and a number line graph.
- On the graph, a filled dot means the end point is included (≤ or ≥) and an open dot means it isn’t (< or >).
Worked examples
Easy: a two-step inequality, 2x + 3 > 7
Subtract 3 from both sides: 2x > 4.
Divide both sides by 2 (a positive number, so the sign stays the same): x > 2.
Check: x = 3 gives 9 > 7 ✓, and x = 1 gives 5 > 7 ✗, as expected. In interval notation the answer is (2, ∞).
Medium: dividing by a negative, 5 − 3x ≤ 11
Subtract 5 from both sides: −3x ≤ 6.
Divide by −3. Dividing by a negative number reverses the sign: x ≥ −2.
Check: x = 0 gives 5 ≤ 11 ✓, and x = −3 gives 14 ≤ 11 ✗. The answer is [−2, ∞), with a filled dot at −2.
Hard: an absolute value inequality, |2x − 1| ≥ 5
“Greater than” splits into two inequalities joined by “or”: 2x − 1 ≤ −5 or 2x − 1 ≥ 5.
Solve each: 2x ≤ −4 gives x ≤ −2, and 2x ≥ 6 gives x ≥ 3.
Check: x = 0 gives |−1| = 1, not ≥ 5 ✗; x = 4 gives |7| = 7 ≥ 5 ✓; x = −3 gives |−7| = 7 ≥ 5 ✓. The answer is (−∞, −2] ∪ [3, ∞).
Common mistakes
- Forgetting to reverse the sign when multiplying or dividing by a negative number: −3x ≤ 6 gives x ≥ −2, not x ≤ −2.
- Mixing up open and closed dots: < and > use an open dot; ≤ and ≥ use a filled dot.
- Solving only one case of an absolute value inequality. |2x − 1| ≥ 5 has two parts: x ≤ −2 or x ≥ 3.
- Using “and” instead of “or”: |u| < c is one interval between −c and c; |u| > c is two pieces joined by “or”.
Frequently asked questions
When do I flip the inequality sign?
Only when you multiply or divide both sides by a negative number. Adding or subtracting, or multiplying by a positive number, never changes the sign.
How do I solve absolute value inequalities?
Get the absolute value on its own. If it is less than c, write −c < inside < c and solve. If it is greater than c, solve inside < −c or inside > c. If c is negative, “less than” has no solution and “greater than” is true for every number.
What is interval notation?
A short way to write the answer: round brackets mean the end point is not included and square brackets mean it is. x ≥ −2 is [−2, ∞), and ∞ always gets a round bracket.
What do open and closed circles mean on the graph?
A closed (filled) circle means the number is part of the answer (≤ or ≥). An open circle means it isn’t (< or >). The shaded part of the line shows every other number in the answer.
Can it solve two-step and double inequalities?
Yes. Two-step inequalities like 2x + 3 > 7 and double inequalities like −1 < 2x + 1 ≤ 9 are solved step by step, doing the same thing to every part.
Does the inequality calculator graph the answer?
Yes — every answer is drawn on a number line, and given in both inequality and interval notation.