Compound Inequality Solver

Compound Inequalities: Logic in Math

In Algebra, solving for x isn't always about finding a single number. Often, it's about defining a range of possibilities. Compound inequalities combine two simple inequalities with the words AND or OR.

The Difference Between AND & OR

AND (Intersection) ∩

For an "AND" statement to be true, both conditions must be met at the same time. The solution is the overlap (intersection) of the two shaded regions.
Example: x > 2 AND x < 5
Solution: The numbers strictly between 2 and 5 (e.g., 3, 4, 2.1).
Interval Notation: (2, 5)
Empty Set: If there is no overlap (e.g., x < 2 AND x > 5), the answer is "No Solution" (∅).

OR (Union) U

For an "OR" statement to be true, at least one condition must be met. The solution is the combination (union) of both regions.
Example: x < 2 OR x > 5
Solution: Any number smaller than 2, PLUS any number bigger than 5.
Interval Notation: (-∞, 2) U (5, ∞)
All Real Numbers: If the two graphs cover the entire number line (e.g., x > 2 OR x < 5), the answer is "All Real Numbers" (-∞, ∞).

How to Write Interval Notation

• Parentheses ( ): Use for strict inequalities (<, >) or infinity. Means "up to but not including".
• Brackets [ ]: Use for inclusive inequalities (≤, ≥). Means "including".
• Union (U): The symbol used to join two separate intervals in an OR problem.