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Calculator Answers

Compound Inequality Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A compound inequality calculator is an algebraic tool that solves pairs of inequalities joined by the logical connectors 'and' or 'or'. The calculator solves each inequality independently, then determines the solution set by finding either the intersection (for 'and' statements) or the union (for 'or' statements) of the individual solution intervals. It outputs the result in inequality notation and interval notation. A number line is not drawn. Algebra students use this utility to verify homework and understand compound logical conditions.

Pick AND or OR mode. Enter the linear pieces and operators; the calculator returns each piece's solution and the combined interval (intersection for AND, union for OR).

Quick Answer

Solve compound inequalities joined by 'and' or 'or'. Enter the coefficients for each piece to get both solution intervals and the combined intersection or union, in interval notation.

Mode

Format: a (op₁) mx + b (op₂) c. Both operators should point in the same direction (e.g. both < or both ≤).

op₁

op₂

For a single linear inequality (no AND/OR), use the inequality calculator. This page is specifically for compound forms.

Compound inequality

Solution (intersection)

(-2, 2]

Split into two inequalities: 2x + 1 > -3 AND 2x + 1 <= 5. Intersect the two solution sets.

Piece 1(-2, ∞)
Piece 2(−∞, 2]
Intersection(-2, 2]

AND inequalities (a < expr < c) take the intersection of the two solutions. OR inequalities take the union. Empty intersection means no solution; a full union of (−∞, ∞) means all real numbers.

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Examples

−3 < 2x + 1 ≤ 5

x ∈ (−2, 2]

x < −1 OR x > 3

x ∈ (−∞, −1) ∪ (3, ∞)

0 ≤ x − 5 ≤ 10

x ∈ [5, 15]

2x + 1 < 3 OR x − 1 > 4

x ∈ (−∞, 1) ∪ (5, ∞)

How it works

The calculator solves each linear piece on its own using the same logic as the single inequality solver (it flips the operator when the coefficient is negative). Then it combines the two intervals.

AND · solution = piece₁ ∩ piece₂

OR · solution = piece₁ ∪ piece₂

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Inequality CalculatorSolve a one-variable linear inequality of the form ax + b ◊ cx + d for <, ≤, >, or ≥. Returns the solution interval, a number-line description, and the flip step when dividing by a negative.

Frequently asked questions

An inequality made of two pieces joined by AND or OR. AND means both pieces must hold (intersection); OR means at least one piece must hold (union). The classic AND form is a < x < b.

Split into two separate inequalities (one for each side), solve each, then take the intersection of the two solution sets. The calculator does both steps and shows each piece.

Solve each inequality on its own. The solution is the union of the two solution sets. Sometimes the union is the entire real line; sometimes it leaves a gap in the middle.

The intersection is empty, so there is no x satisfying both at once. The calculator reports 'no solution' in that case.

The standard AND form has both operators pointing the same way (a < x < c or c > x > a). The calculator follows this convention. To express opposite-direction conditions, use the OR mode instead.

Not directly. This calculator is for linear pieces. For absolute-value inequalities, split into two linear cases first, then enter those as AND or OR here.