Compound Inequality Calculator — Solve AND & OR Inequalities Instantly

Solve compound inequalities with step-by-step breakdowns and interval notation. Free online compound inequality calculator supporting AND (conjunction) and OR (disjunction) types with instant results.

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Compound Inequality Calculator

Solve AND (conjunction) or OR (disjunction) compound inequalities with full step-by-step solutions and interval notation.

Lower Bound [sign] ax + b [sign] Upper Bound
Enter values and click Calculate Inequality to see the solution.

Compound Inequality Formulas Explained

A compound inequality combines two or more simple inequalities using the logical operators AND (conjunction) or OR (disjunction). The solution set depends on how the inequalities are joined.

AND (Conjunction): lower ≤ ax + b < upper → both conditions must be true simultaneously
OR (Disjunction): a₁x + b₁ < c₁ OR a₂x + b₂ > c₂ → at least one condition must be true

Key Variable Definitions

  • a, a₁, a₂ — Coefficients of the variable x in each inequality part
  • b, b₁, b₂ — Constant terms added to the variable term
  • lower, upper, c₁, c₂ — Bound values that constrain the expression
  • ≤, <, ≥, > — Inequality signs defining the relationship

For AND inequalities, the solution is the intersection of both solution sets. For OR inequalities, the solution is the union of both solution sets.

How to Solve Compound Inequalities

Solving compound inequalities requires careful attention to inequality signs and the direction of operations. Follow these steps:

  1. Identify the compound type — Determine if it's an AND (conjunction) or OR (disjunction) inequality.
  2. Split into separate inequalities — For AND, separate into two inequalities. For OR, they're already separate.
  3. Solve each inequality for x — Isolate x in each part using algebraic operations.
  4. Flip signs when needed — When multiplying or dividing by a negative number, reverse the inequality sign.
  5. Combine solutions — For AND, find the intersection (overlap). For OR, take the union of both sets.
  6. Write in interval notation — Express the final solution using parentheses (exclusive) and brackets [inclusive].

Compound Inequality Calculator Examples

Example 1: AND Compound Inequality

Solve: -3 < 2x + 1 < 7

Split: -3 < 2x + 1 AND 2x + 1 < 7
Solve left: -3 < 2x + 1 → -4 < 2x → x > -2
Solve right: 2x + 1 < 7 → 2x < 6 → x < 3
Intersection: -2 < x < 3
Interval: (-2, 3)

Example 2: AND with Negative Coefficient

Solve: -4 ≤ -2x + 6 < 2

Left: -4 ≤ -2x + 6 → -10 ≤ -2x → x ≤ 5 (flip: divide by -2)
Right: -2x + 6 < 2 → -2x < -4 → x > 2 (flip)
Intersection: 2 < x ≤ 5
Interval: (2, 5]

Example 3: OR Compound Inequality

Solve: x - 2 < -5 OR 3x + 1 > 10

Inequality 1: x - 2 < -5 → x < -3
Inequality 2: 3x + 1 > 10 → 3x > 9 → x > 3
Union: x < -3 OR x > 3
Interval: (-∞, -3) ∪ (3, ∞)

Real-World Compound Inequality Applications

  • Temperature Ranges: Expressing acceptable operating temperatures for machinery or storage conditions.
  • Grade Boundaries: Defining letter grade cutoffs where scores fall within specific compound ranges.
  • Dosage Calculations: Determining safe medication dosage windows between minimum effective and maximum safe levels.
  • Manufacturing Tolerances: Specifying acceptable dimension ranges where parts must fall within upper and lower limits.
  • Budget Constraints: Modeling spending limits that must stay above a minimum need but below a maximum cap.
  • pH Levels: Defining acceptable acidity ranges for swimming pools, aquariums, or chemical processes.
  • Speed Limits: Representing legal speed ranges on highways with both minimum and maximum limits.

People Also Ask

A compound inequality consists of two or more simple inequalities joined by AND (conjunction) or OR (disjunction). AND inequalities require both conditions true simultaneously (intersection of solutions). OR inequalities require at least one condition true (union of solutions). For example, -3 < x < 5 is an AND compound inequality meaning x is between -3 and 5.
Split the AND compound inequality into two separate inequalities. Solve each one independently for x, remembering to flip inequality signs when multiplying or dividing by a negative number. The final solution is the intersection (overlap) of both individual solution sets, representing values that satisfy both conditions simultaneously.
Solve each inequality in the OR compound statement independently for x. The final solution is the union of both individual solution sets, meaning any x that satisfies either inequality is part of the solution. OR solutions often extend in opposite directions on the number line and use the union symbol ∪ in interval notation.
Flip the inequality sign whenever you multiply or divide both sides by a negative number. For example, solving -2x > 6 requires dividing by -2, flipping > to <, yielding x < -3. This rule applies to both AND and OR compound inequalities and is critical for obtaining correct solutions.
Interval notation compactly represents solution sets. Use parentheses () for exclusive bounds (< or >) and brackets [] for inclusive bounds (≤ or ≥). For AND solutions like -2 < x ≤ 5, write (-2, 5]. For OR solutions like x < -3 OR x > 4, write (-∞, -3) ∪ (4, ∞). The ∪ symbol denotes the union of disjoint intervals.

Frequently Asked Questions

AND inequalities require both conditions to be true at the same time, yielding the intersection of solution sets (a single connected interval or no solution). OR inequalities require at least one condition to be true, yielding the union of solution sets (often two separate intervals). The AND solution is typically a bounded range, while OR solutions often extend to infinity.
Yes. An AND compound inequality has no solution when the two individual solution sets do not overlap. For example, x < 2 AND x > 5 has no solution because no number can be simultaneously less than 2 and greater than 5. The result is the empty set, denoted as ∅ or written as "no solution."
Yes. The calculator correctly handles negative coefficients by automatically flipping inequality signs when dividing by negative values. This ensures accurate solutions for all compound inequalities regardless of the sign of the coefficient of x.
When a = 0, the expression becomes a constant (just b). The calculator evaluates whether the constant satisfies the inequality bounds. For AND inequalities, if the constant falls within the bounds, the solution is all real numbers; otherwise, there is no solution. For OR inequalities, the solution depends on whether each constant satisfies its respective inequality.
Interval notation uses parentheses () for strict inequalities (< or >) and brackets [] for inclusive inequalities (≤ or ≥). For unbounded intervals, use -∞ or ∞ with parentheses. The calculator automatically generates the correct interval notation based on the inequality signs in your compound inequality.
Absolute value inequalities can be rewritten as compound inequalities. For example, |x - 3| < 5 is equivalent to -5 < x - 3 < 5, an AND compound inequality. Similarly, |x + 1| > 4 becomes x + 1 < -4 OR x + 1 > 4. You can use this calculator after converting the absolute value form.

Compound Inequality Glossary

Compound Inequality

Two or more simple inequalities joined by AND or OR, expressing a combined condition on the variable.

Conjunction (AND)

A compound inequality where both conditions must be true simultaneously. The solution is the intersection of individual solutions.

Disjunction (OR)

A compound inequality where at least one condition must be true. The solution is the union of individual solutions.

Intersection

The set of all values that satisfy both inequalities in an AND compound statement. Represented by the overlap on a number line.

Union

The set of all values that satisfy either inequality in an OR compound statement. Denoted by ∪ in interval notation.

Interval Notation

A compact representation of solution sets using parentheses for exclusive bounds and brackets for inclusive bounds.

Inequality Sign Flip

The rule requiring reversal of inequality direction when multiplying or dividing both sides by a negative number.

Empty Set

Denoted ∅, the result when an AND compound inequality has no overlapping solution (e.g., x < 2 AND x > 5).

Editorial Review & Methodology

This compound inequality calculator was built and reviewed by the NumbrWiz Editorial Team. The methods for solving compound inequalities are foundational concepts in algebra, verified against standard mathematics curricula including Common Core algebra standards and college-level precalculus textbooks.

  • Formula verification: Cross-checked against multiple authoritative algebra and precalculus sources.
  • Edge case testing: Tested with zero coefficients, negative coefficients, identical bounds, no-solution scenarios, and all-real-number solutions.
  • UX review: Designed for intuitive input with clear error messaging, mode toggling, and comprehensive step-by-step breakdown.

Transparency note: All calculations run client-side in your browser. No data is ever collected, stored, or transmitted. Results are for educational purposes; verify critical calculations independently.

Page last reviewed: May 2026 · NumbrWiz Editorial Team