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Inequality Solver with Steps

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Single-variable inequalities on the real number line ℝ

Math Display :
x^2 - 5x + 6 < 0

Quick examples

Solving Inequalities Step by Step

An inequality states that two expressions are not equal, expressing an ordering relationship. By applying balanced operations — and flipping the sign when multiplying or dividing by a negative — we isolate the variable and express the solution as an interval.

Linear

ax+b<c  ⟹  x<c−baax + b < c \implies x < \frac{c - b}{a}

First-degree inequalities, solved by isolating x and flipping the sign when dividing by a negative.

Quadratic

ax2+bx+c≤0ax^2 + bx + c \leq 0

Solved using the discriminant Δ and sign analysis of the parabola.

Interval Notation

x∈(a,b)∪[c,+∞)x \in (a, b) \cup [c, +\infty)

Solutions expressed as intervals, with strict/non-strict brackets depending on the operator.

Sign Analysis for Quadratic Inequalities

For a quadratic f(x) = ax² + bx + c, the sign of f(x) depends on the discriminant Δ = b² − 4ac and the sign of a. When a > 0, the parabola opens upward — f(x) is negative between the roots and positive outside.

Δ > 0 (Two Roots)

Two roots x₁ < x₂. For a > 0: f < 0 between roots, f > 0 outside.

x₁ < x < x₂ or x < x₁, x > x₂

Δ = 0 (One Double Root)

Parabola touches x-axis at one point. For a > 0: f ≥ 0 everywhere.

x₀ = -b / 2a

The Golden Rule: Flip the Sign

The most important rule in inequality solving: whenever you multiply or divide both sides by a negative number, the inequality sign must flip. This is the key difference from solving equations.

−2x>4  ⇒  x<−2(sign flips when dividing by -2)-2x > 4 \;\Rightarrow\; x < -2 \quad \text{(sign flips when dividing by -2)}

⚠️ Always flip the inequality sign when multiplying or dividing both sides by a negative number.

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Inequality Examples & Solutions Library

Explore step-by-step solved inequalities across all categories with key solving strategies.

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Frequently Asked Questions

How do you solve a linear inequality step by step?

Isolate the variable by applying the same operations to both sides (add, subtract, multiply, divide). The critical rule: if you multiply or divide by a negative number, flip the inequality sign. Express the result as an interval, e.g. x ∈ (-∞, 3).

How do you solve a quadratic inequality?

1. Move everything to one side: ax² + bx + c op 0. 2. Find the roots using the discriminant Δ = b² − 4ac. 3. Perform sign analysis: if a > 0, the parabola opens upward — f(x) < 0 between roots and f(x) > 0 outside. Express the solution as a union of intervals.

What is the difference between < and ≤ in inequalities?

The strict operator < (less than) excludes the boundary value — represented by an open parenthesis ( in interval notation. The non-strict operator ≤ (less than or equal to) includes the boundary — represented by a square bracket [. For example, x < 3 gives (-∞, 3), while x ≤ 3 gives (-∞, 3].

Is this inequality solver free to use?

Yes, completely free. It runs entirely in your browser, requires no account or signup, and your inputs are never sent to any server — your data stays private on your device.