Inequality Solver
How to solve inequalities
Enter the coefficients a, b, and c for ax + b < c (or >, ≤, ≥). The solver isolates x step by step and gives the answer in interval notation.
The sign-flip rule
When you multiply or divide both sides of an inequality by a negative number, the inequality sign flips: < becomes > and vice versa. For example, −2x < 6 becomes x > −3.
Worked example
Solve 2x + 3 < 11: subtract 3 → 2x < 8, divide by 2 → x < 4. Interval notation: (−∞, 4).
Frequently asked questions
When multiplying or dividing both sides by a negative number. Adding or subtracting never flips the sign.
A compact way to write solution sets: (−∞, 4) means all x less than 4. Square brackets include the endpoint: [2, ∞) means x ≥ 2.
Then it is either always true (e.g. 3 < 5) or never true (e.g. 7 < 2). The solver tells you which.
How isolating x works with inequalities
You isolate x with the same moves you use for equations: undo addition or subtraction first, then undo multiplication or division. The one extra job is watching the direction of the sign every time you multiply or divide, because a negative multiplier reverses the order of the two sides.
After solving, test one number inside your answer range and one outside it. The first should make the original inequality true, the second false.
Example: Solve -3x + 5 >= 14 for a budget limit
Step 1, Subtract 5 from both sides: -3x >= 9.
Step 2, Divide by -3 and flip the sign: Dividing by a negative reverses the symbol, so x <= -3.
Step 3, Write the interval: The solution is (-∞, -3], with a square bracket because -3 is included.
Step 4, Test the answer: At x = -4, -3(-4) + 5 = 17, which is at least 14, so it works. At x = 0 the value is 5, which fails, as expected.
Common mistakes
- Flipping at the wrong time: Adding or subtracting a negative number never flips the sign. Only multiplying or dividing by a negative does.
- Using the wrong bracket: A strict sign such as < takes a round bracket, while <= and >= include the endpoint and take a square bracket.
- Flipping only once in two steps: If you divide by a negative and later multiply by another negative, the sign flips twice and ends up back where it started.
When to use this calculator
Use it for limits and ranges rather than single values: a maximum spend, a minimum score, or the temperatures at which something stays safe. It is free to use, and the step list is handy when you need to show your working as well as the final range.
More questions
Mark the endpoint with a filled circle if it is included, or an open circle if it is not, then shade in the direction the inequality allows. For x <= -3, fill the point at -3 and shade everything to its left.
Yes. That is a compound inequality, meaning x sits between two values. Solve each side in turn, keeping x in the middle, and write the answer as the interval (1, 6).
An equation usually pins x to a single value, but an inequality only sets a boundary. Every number on the allowed side of that boundary is a valid solution, so the answer is a whole set of values.