Education
Compound Inequality Calculator
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).
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.
Solution (intersection)
(-2, 2]
Split into two inequalities: 2x + 1 > -3 AND 2x + 1 <= 5. Intersect the two solution sets.
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.
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
Related calculators
- Inequality calculator for a single one-variable linear inequality.
- Proportion calculator for ratio relationships that sometimes lead to inequalities.
- System of equations calculator for the equality cousin of compound conditions.
- All education calculators.
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.
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