Compound Inequality Calculator
Solve two linear inequalities connected by AND or OR, then see the answer as an inequality, interval notation, and a number-line description.
What makes a compound inequality different
A compound inequality combines two inequalities into one statement. The connector matters. AND means a number must satisfy both inequalities at the same time, so you keep the overlap. OR means a number can satisfy either inequality, so you keep both regions. OpenStax explains this overlap-versus-union idea in its compound inequalities lesson, including how to solve, graph, and write answers in interval notation.
AND vs OR in plain English
Think of AND as a narrow gate. A value must pass both tests, so the answer is usually the middle overlap. Think of OR as two doors. A value can enter through either door, so the answer can be two separated intervals.
| Connector | Example | Interval notation |
|---|---|---|
| AND | x > 2 and x ≤ 8 | (2, 8] |
| OR | x < -3 or x ≥ 5 | (-∞, -3) ∪ [5, ∞) |
Worked example: an AND compound inequality
The final answer keeps only values that satisfy both statements. For example, 5.5 works, but 4 does not pass the first inequality and 7 does not pass the second.
Common mistakes with compound inequalities
The biggest mistake is treating AND and OR the same way. They are not the same. Another common mistake is writing two intervals for an AND problem even when the solution should be one overlapping interval. For OR problems, students often forget the union symbol ∪ when the answer has two separated pieces.
Common questions
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A compound inequality is a statement made from two inequalities connected by AND or OR. For example, x > 2 and x < 7 is a compound inequality because x must satisfy both conditions. Another example is x < -3 or x ≥ 5, where x can satisfy either condition. Compound inequalities are common when a solution is limited to a range or split into two separate ranges.
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AND means both inequalities must be true at the same time, so you look for the overlap of the two solution sets. OR means at least one inequality must be true, so you combine the allowed regions. In interval notation, AND often becomes one bounded interval, while OR can become two intervals joined with the union symbol.
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Solve each inequality first. For an AND inequality, shade only the part where the two graphs overlap. For an OR inequality, shade every part that belongs to either graph. Open circles represent strict inequalities, and closed circles represent inclusive inequalities.
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First solve each inequality separately. Then decide whether the connector asks for the overlap or the union. AND uses the overlap. OR uses the union of both solution sets. For example, x > 1 and x ≤ 4 becomes (1, 4], while x < 1 or x ≥ 4 becomes (-∞, 1) ∪ [4, ∞).
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Yes. An AND compound inequality can have no solution when the two conditions cannot be true at the same time. For example, x < 2 and x > 5 has no solution because no number is both less than 2 and greater than 5. The interval notation is ∅.
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Yes. This can happen especially with OR inequalities. For example, x < 5 or x ≥ 5 covers every real number because every number is either less than 5 or at least 5. The interval notation is (-∞, ∞).
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Absolute value measures distance from zero. A less-than absolute value inequality, such as |x| < 4, means x is within 4 units of zero, so it becomes -4 < x < 4. A greater-than absolute value inequality, such as |x| > 4, means x is outside that distance, so it becomes x < -4 or x > 4.
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Choose test values from the final interval and from outside the final interval. Substitute them into the original inequalities. Values inside the final solution should make the correct AND or OR statement true, while values outside should fail the condition.