# Inequality Calculator - Solve Linear, Quadratic and Absolute Value Inequalities

> Free online inequality solver: linear, quadratic and absolute-value inequalities with the answer in set and interval notation, a number line and the reasoning.

- URL: https://qilingtools.com/en/math/inequality
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Type the coefficients, pick > ≥ < or ≤, and read off the solution set. You get set notation and interval notation side by side, a number line showing the solution, and an explanation of how the discriminant, the roots and the direction of the parabola decide the answer.

## How to use

1. Pick the type: linear, quadratic or absolute value
2. Enter the coefficients and choose the inequality sign
3. Read the solution in set and interval notation, with the number line below
4. Click “Plot the matching function” to check the answer against the graph

## Features

- Linear: the sign flip when a is negative is handled for you
- Quadratic: discriminant, both roots and the vertex, with the “outside or inside the roots” rule explained
- Absolute value: automatically split into −c < ax+b < c or two separate pieces
- Answers in set notation (x < 1 or x > 3) and interval notation ((−∞, 1) ∪ (3, +∞))
- Number line picture with filled dots for ≤ and ≥, hollow dots for < and >
- Always-true and no-solution cases are stated plainly instead of being fudged

## FAQ

### How do I know whether to take the outside or the inside of the roots?

First make the x² coefficient positive (multiply by −1 and flip the sign if needed). The parabola then opens upwards: “> 0” takes the region outside the two roots, “< 0” takes the region between them. The tool spells this step out.

### What if the discriminant is negative?

The parabola never crosses the x-axis and stays above it, so ax²+bx+c > 0 holds for every real number while ax²+bx+c < 0 has no solution at all.

### How do I remove the absolute value bars?

|u| < c is the same as −c < u < c, and |u| > c is the same as u < −c or u > c. When c is negative the first has no solution and the second is always true; the tool handles each case.

### What is the difference between square and round brackets?

A square bracket means the endpoint is included (it came from ≤ or ≥); a round bracket means it is excluded (from < or >). Infinity always gets a round bracket.

### Can I verify the answer?

Yes — click the plot link to draw y = f(x). The stretch of the curve above the x-axis is exactly where f(x) > 0, which you can compare with the interval you were given.

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