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Solve AND/OR compound inequalities with number line visualization and interval notation
A compound inequality combines two inequalities using AND or OR. Understanding the difference between these types is crucial for solving them correctly.
AND inequalities require BOTH conditions to be true simultaneously. Written as a < x < b, they represent values between two numbers. The solution is the intersection (overlap) of both conditions.
OR inequalities require AT LEAST ONE condition to be true. Written as x < a OR x > b, they represent values outside a range. The solution is the union (combination) of both conditions.
AND means both conditions must be true (intersection), while OR means at least one condition must be true (union). AND creates a range between two values, OR creates two separate ranges.
AND inequalities are written in the form a < x < b. This is shorthand for "a < x AND x < b". The variable is in the middle with boundary values on each side.
Yes! If the lower bound is greater than or equal to the upper bound (like 7 < x < 3), there are no values that satisfy both conditions, so there's no solution.
The ∪ symbol means "union" and is used in OR inequalities to combine two separate intervals. For example, (-∞, 2) ∪ (5, ∞) represents x < 2 OR x > 5.
For AND, shade the region between the two boundary points. For OR, shade the regions on either side of the boundary points. Use open circles for < or >, closed circles for ≤ or ≥.
Yes! If the OR conditions overlap or cover the entire number line (like x < 5 OR x > 2), the solution is all real numbers.
AND uses a single interval like (a, b), while OR uses two intervals connected with ∪, like (-∞, a) ∪ (b, ∞).
Use ≤ or ≥ (brackets in interval notation) when the endpoint is included in the solution. Use < or > (parentheses) when the endpoint is not included.