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Inequality Solver

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About This Tool

🔢 Inequality Solver – Solve Any One-Variable Inequality Instantly

An inequality is a mathematical statement that compares two expressions using a relation other than equality — using symbols like <, ≤, >, or ≥. Unlike equations that yield discrete points, inequalities produce ranges of values called solution sets. This Inequality Solver handles five major types automatically, returning the solution in plain language, interval notation, and set-builder notation, together with a visual number-line graph.

🧮 Supported Inequality Types

TypeExampleSolution
Linear2x − 5 > 9x > 7 → (7, ∞)
Compound−3 ≤ 2x + 1 < 9−2 ≤ x < 4 → [−2, 4)
Quadratic / Polynomialx² − 5x + 6 ≤ 02 ≤ x ≤ 3 → [2, 3]
Rational(x+1)/(x−2) ≥ 0(−∞, −1] ∪ (2, ∞)
Absolute Value|2x − 1| < 5−2 < x < 3 → (−2, 3)

⚙️ How the Solver Works

The solver follows a sign-analysis (interval-testing) method, which is the standard algebraic approach taught in algebra and pre-calculus courses:

  1. Rearrange to standard form. The inequality is rewritten as f(x) op 0 by subtracting the right-hand side from the left. For example, 2x − 5 > 9 becomes 2x − 14 > 0.
  2. Find critical points. The solver locates all values where f(x) = 0 (roots) and where f(x) is undefined (denominator zeros for rational expressions). These critical points divide the real number line into intervals.
  3. Test each interval. A single test value from each interval is substituted into f(x). If the sign satisfies the original operator, the entire interval is part of the solution.
  4. Apply endpoint rules. Strict inequalities (<, >) exclude boundary points (open circles); non-strict inequalities (≤, ≥) include them (closed circles) — unless the point is a domain exclusion.
  5. Express the result. The solution is written as a plain-language inequality, standard interval notation, and formal set-builder notation. A number-line graph highlights the solution region.

📐 Interval Notation Explained

Interval notation is a compact way to describe solution sets. The key rules are:

  • Round brackets ( ) denote open endpoints — the value is not included (used for <, >, and infinity).
  • Square brackets [ ] denote closed endpoints — the value is included (used for ≤ and ≥).
  • Infinity symbols −∞ and ∞ always use round brackets because infinity is not a reachable value.
  • Union ∪ combines two or more separate solution intervals into one expression (common in rational and absolute-value inequalities).
  • Empty set ∅ means no real number satisfies the inequality.

⚠️ The Sign Flip Rule

One of the most common mistakes in inequality solving is forgetting to flip the inequality sign when multiplying or dividing by a negative number. For example:

−2x > 6
  ÷(−2) on both sides → flip sign
  x < −3

This solver avoids the pitfall entirely by using interval sign analysis instead of direct algebraic manipulation on the full expression. Critical points are found first, then each region is tested independently — no sign flip ambiguity arises.

📊 Special Cases

Rational inequalities require special care because the expression is undefined at denominator zeros. Even if substituting a denominator zero would "satisfy" the sign condition, that point must remain excluded and is shown with an open circle on the number line.

Absolute value inequalities are converted to equivalent compound or disjoint inequalities before solving. |expr| < k becomes −k < expr < k; |expr| > k becomes expr > k OR expr < −k.

📝 How to Enter Expressions

  • Use ^ for exponents: x^2, x^3
  • Use / for division: (x+1)/(x-2)
  • Use | for absolute value bars: |2x - 1|
  • Include the full inequality sign in the expression: 2x - 5 > 9
  • For compound inequalities write the full chain: -3 <= 2x + 1 < 9
  • Specify the variable you are solving for (default is x)

The Inequality Solver is useful for algebra students learning solution set analysis, for exam preparation, and for quickly verifying hand-solved inequalities. The step-by-step breakdown makes it easy to understand each stage of the solving process.

Frequently Asked Questions

Is the Inequality Solver free?

Yes, Inequality Solver is totally free :)

Can I use the Inequality Solver offline?

Yes, you can install the webapp as PWA.

Is it safe to use Inequality Solver?

Yes, any data related to Inequality Solver only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

What types of inequalities can this solver handle?

The Inequality Solver supports five types: linear inequalities (2x − 5 > 9), compound inequalities (−3 ≤ 2x + 1 < 9), quadratic and polynomial inequalities (x² − 5x + 6 ≤ 0), rational inequalities ((x + 1)/(x − 2) ≥ 0), and absolute-value inequalities (|2x − 1| < 5). Enter the full inequality in the expression field and the solver automatically detects the type.

How does the Inequality Solver work?

The solver rearranges the inequality into the form f(x) op 0, then finds all critical points — roots of the numerator and excluded points from any denominator. It sorts these critical points on the real line, picks a test value in each resulting interval, evaluates the sign of f at that point, and marks the interval as part of the solution when the sign satisfies the original operator. Endpoints are included or excluded based on the strict or non-strict inequality sign.

What does interval notation mean in the output?

Interval notation uses parentheses for open (excluded) endpoints and square brackets for closed (included) endpoints. For example, (7, ∞) means all real numbers strictly greater than 7, while [2, 3] means all real numbers between 2 and 3, including both endpoints. The union symbol ∪ combines separate solution intervals, and ∅ denotes no solution.

Why do rational inequalities show domain restrictions?

A rational expression is undefined wherever its denominator equals zero. Even if a critical point satisfies the sign condition, it cannot be included in the solution because the original expression is not defined there. The solver lists all such excluded values as domain restrictions (e.g., x ≠ 2) and uses open endpoints at those points in the interval notation.

How are absolute-value inequalities solved?

Absolute-value inequalities are split into equivalent cases. For |expr| < rhs, the solver converts to −rhs < expr < rhs (a compound inequality). For |expr| > rhs, it converts to expr > rhs OR expr < −rhs and takes the union of both solution sets. If rhs is negative, |expr| < negative has no solution, while |expr| > negative is always true.

How should I enter the inequality expression?

Type the full inequality as you would write it mathematically, for example: 2x - 5 > 9, x^2 - 4 <= 0, (x+1)/(x-2) >= 0, or |2x - 1| < 5. Use ^ for exponents, * for multiplication (or leave it implicit for simple cases), / for division, and | | for absolute value. Specify the variable you are solving for in the Variable field.