Inequality Step Solver Calculator

Solve an inequality

Example data table

Input inequality	Type	Solution set (interval notation)
2x + 3 ≤ 7	Linear	(-∞, 2]
x^2 - 5x + 6 > 0	Quadratic	(-∞, 2) ∪ (3, ∞)
\|2x - 3\| < 5	Absolute value	(-1, 4)
1 < 2x + 3 ≤ 9	Compound	(-1, 3]

Values are shown for demonstration and quick testing.

Formula used

Standard form: Move all terms to one side so you solve f(x) < 0, f(x) ≤ 0, f(x) > 0, or f(x) ≥ 0.
Linear: For ax + b compared to 0, isolate x. If you divide by a negative number, the inequality sign flips.
Quadratic: For ax² + bx + c, compute D = b² − 4ac, find real roots, then use a sign chart based on whether the parabola opens up or down.
Absolute value: For |u| < c, use −c < u < c. For |u| > c, use u < −c or u > c.

How to use this calculator

Type your inequality using x and operators like <= or >=.
Keep expressions polynomial-style: x^2, x, constants, plus and minus.
For absolute value, use |2x-3| or abs(2x-3).
For compound inequalities, write like 1 < 2x+3 <= 9.
Press Solve inequality to view steps and the final interval.
Use Download CSV or Download PDF to save your results.

Notes and limits

This solver focuses on one variable x, up to quadratic expressions.
Use and / or to combine two inequalities (basic set combination).
Parentheses and non-polynomial functions are intentionally restricted for clarity.

Article

Why step solving matters in inequality work

Inequalities need more care than equations because each operation affects the truth set. Adding or subtracting the same value preserves equivalence. Multiplying or dividing by a negative reverses direction, so the solver records the sign flip and the final interval notation used in most marking schemes.

Linear inequalities and sign changes

For ax + b < 0, isolating x creates one boundary point. If a is positive, solutions lie left for < and right for >. If a is negative, the division step flips the sign. That single rule explains many mistakes, and the exported steps form a clear audit trail.

Quadratic inequalities with discriminant data

Quadratics are handled by writing f(x) = ax² + bx + c and comparing it to zero. The discriminant D = b² − 4ac indicates the number of real roots. If D < 0, the parabola never crosses the axis, so the solution is either all reals or none, based on a and the inequality symbol.

Intervals from root ordering and sign charts

When D ≥ 0, the roots are computed and ordered x₁ ≤ x₂. For a > 0, f(x) is negative between roots and positive outside; for a < 0 the pattern reverses. Endpoint inclusion follows < versus ≤. Results are returned as one interval or a union, matching standard sign-chart conclusions.

Absolute value inequalities as bounded regions

Absolute value inputs such as |2x − 3| ≤ 5 become −5 ≤ 2x − 3 ≤ 5, then two linked linear constraints. For |u| > c, the region splits into u < −c or u > c, producing a union. This method is widely used in entrance testing because it is easy to verify.

Graphing and exports for faster validation

The Plotly number-line view marks boundary points and shades solution spans on a consistent scale. For unbounded intervals, the graph expands around the nearest finite bound to show direction. CSV stores the input, summary, interval, and steps as rows, while PDF provides a printable record for revision packs and tutoring notes. The visualization is intended for quick checking, not symbolic proof, but it helps confirm whether endpoints are open or closed and whether a union has two separate rays. Using the same input, students can compare manual work to the automated steps and learn consistent notation very quickly.

FAQs

1) What types of inequalities can this solver handle?

It solves single‑variable linear and quadratic inequalities, basic absolute value inequalities with linear interiors, and simple compound forms like 1 < expr <= 9. It also supports combining two inequalities using “and” or “or”.

2) Why does the inequality sign flip sometimes?

The sign flips only when you multiply or divide both sides by a negative number. That operation reverses the order on the number line. The step list calls this out explicitly to prevent common calculation errors.

3) How are quadratic solutions determined?

The solver computes the discriminant D = b² − 4ac, finds real roots when D ≥ 0, then applies a sign chart based on whether the parabola opens up or down. The final answer is returned in interval notation.

4) What does the Plotly graph represent?

It is a number‑line style visualization of the solution set. Boundary points are marked as open or closed based on the inequality symbol, and valid regions are shaded. It’s intended for quick checking and clearer interpretation.

5) Can I solve expressions with parentheses or functions?

This calculator is intentionally limited to polynomial‑style inputs (x^2, x, constants, plus and minus) and linear absolute value interiors. For complex functions, simplify first or use a dedicated CAS tool.

6) What is included in CSV and PDF exports?

CSV stores the input, summary, interval answer, and each step as rows for analysis or record‑keeping. PDF provides a printable copy of the same information, useful for study notes and classroom handouts.