Softonic

Inequality Calculator

Step 1 / 3 Solution

Solve ax + b ⧋ c for x.

Next

Inequalities feel like equations until you divide by a negative and have to flip the symbol. This calculator solves linear (ax + b < c), quadratic (ax² + bx + c ≤ 0) and absolute value |ax + b| > c inequalities step by step, returns the solution in interval notation, set notation and plotted form, and reminds you of every sign-flip along the way.

How to solve an inequality

  1. 1

    Enter the inequality

    Type as you would write it: `2x + 3 > 7`, `x^2 - 4 <= 0`, `|3x + 1| < 5`.

  2. 2

    The tool classifies the form

    Linear, quadratic or absolute-value: different strategies apply.

  3. 3

    Step-by-step solution

    Every algebraic step is shown, with sign-flips highlighted.

  4. 4

    Read the result

    Interval notation, set-builder notation, and a number-line plot.

Solving each inequality type

Linear inequalities

ax + b < c (or ≤, >, ≥)

  1. Subtract b from both sides.
  2. Divide by a.
  3. If a is negative, flip the inequality.

Example: -2x + 5 > 9 → -2x > 4 → x < -2 (division by -2 flips > to <).

Quadratic inequalities

ax² + bx + c ≤ 0

  1. Find the roots with the quadratic formula.
  2. Classify by a’s sign and inequality direction:
Case ≤ 0 solution ≥ 0 solution
a > 0 between roots outside roots
a < 0 outside roots between roots
  1. Strict (<, >) excludes roots; non-strict (≤, ≥) includes them.

Absolute value inequalities

  • |ax + b| < c (with c > 0) → -c < ax + b < c. Solve as a double inequality.
  • |ax + b| > c → ax + b > c OR ax + b < -c. Solve each branch separately and union the results.
  • If c is negative, the “< c” form has no solution and the “> c” form is all real numbers.

Interval notation conventions

  • (a, b), open interval, endpoints excluded.
  • [a, b], closed interval, endpoints included.
  • (a, b], half-open.
  • (-∞, 3), unbounded on the left.
  • (-∞, 3) ∪ (5, ∞), union of two disjoint intervals.

Common traps

  • Forgetting to flip when multiplying or dividing by a negative.
  • Using a strict symbol (<) but including the endpoint in the answer.
  • Solving |x - 3| > 5 and combining the two cases with AND instead of OR.

Frequently Asked Questions

Whenever you multiply or divide both sides by a negative number. Also when you take reciprocals of both sides (if both sides have the same sign). Never when adding or subtracting.

A quadratic with a > 0 (parabola opening up) is negative only between its roots, so ax² + bx + c < 0 gives one interval. But > 0 gives the two outer intervals, written as a union.

Yes, enter them separated by commas. The tool intersects (AND) the solution sets by default; use the “union” toggle if you want OR semantics.

Yes. The step-by-step view explains every algebraic operation and justifies each sign-flip, in a form suitable for homework submission or exam practice.

Related Tools

Social Security Calculator

Estimate a US Social Security retirement benefit from your PIA and claiming age. Compare monthly and annual amounts at 62, FRA 67 and 70.

BMI Calculator

Calculate body mass index from height and weight. Shows WHO category, healthy-weight range and limitations of BMI.

Age Calculator

Calculate exact age in years, months and days from a birth date, plus total days, hours and the next birthday countdown.

Dog Age Calculator

Convert your dog's age to human years using modern veterinary research. Breed-size aware: small, medium and large dogs age differently.

Fahrenheit to Celsius Converter

Convert Fahrenheit temperatures to Celsius with the exact formula, Kelvin output and common reference points for weather, cooking and health.

Time Zone Converter

Convert a time between any two time zones, with daylight-saving shifts applied automatically.

Tool available in other languages