Mathematics ยท Algebra
Compound Inequalities and Interval Notation
Reference entry · last updated September 20, 2026
A compound inequality joins two inequalities with the word and or the word or. The solution set is the intersection of the two individual sets for and, and the union for or. Interval notation writes the result without repeating the variable.
1. First principles
An and statement requires both conditions to hold. It is an intersection of two sets, written \( A \cap B \). An or statement requires at least one condition to hold. It is a union of sets, written \( A \cup B \) [1, 2]. On the number line, an intersection is the shared overlap and a union is everything covered by either part.
2. Interval notation
| Inequality | Interval | Bound type |
|---|---|---|
| \( a < x < b \) | \( (a, b) \) | Both endpoints open |
| \( a \le x \le b \) | \( [a, b] \) | Both endpoints closed |
| \( a \le x < b \) | \( [a, b) \) | Mixed |
| \( x \le b \) | \( (-\infty, b] \) | Unbounded below |
| \( x > a \) | \( (a, \infty) \) | Unbounded above |
Infinity is never a number, so \(-\infty\) and \(\infty\) always take parentheses, never brackets.
3. Conjunctions (and)
Solve each inequality, then keep only the values satisfying both. The compact form \( a < x < b \) is a shorthand for \( x > a \) and \( x < b \), and it can be solved in one pass by applying the same operation to all three parts.
Example. Solve \( -1 < 3x + 2 \le 8 \). Subtract \(2\) from all parts, then divide by \(3\):
4. Disjunctions (or)
Solve each inequality separately and take the union. The solution set is usually two separate rays on the number line.
Example. Solve \( 2x < -6 \) or \( x - 1 > 4 \). The first gives \( x < -3 \), the second gives \( x > 5 \). The solution is \( (-\infty, -3) \cup (5, \infty) \).
5. Worked examples
- \( 5 \le 2x + 1 < 11 \) gives \( 2 \le x < 5 \), the interval \( [2, 5) \).
- \( x + 4 < 3 \) or \( 2x \ge 10 \) gives \( x < -1 \) or \( x \ge 5 \), the union \( (-\infty, -1) \cup [5, \infty) \).
- \( x - 3 < 2 \) and \( x - 3 > -2 \) gives \( 1 < x < 5 \), the interval \( (1, 5) \).
6. Edge cases and the sign flip
- Multiplying or dividing an inequality by a negative number reverses its direction. This applies to every part of a three-part compound inequality.
- An empty intersection, such as \( x < 1 \) and \( x > 4 \), has no solution; write \( \varnothing \).
- A union that covers the whole line, such as \( x < 3 \) or \( x > 0 \), has solution \( (-\infty, \infty) \).
- A closed endpoint belongs to the set and uses a bracket; an open endpoint does not and uses a parenthesis.
See also
- Absolute Value Equations and Inequalities
- Domain and Range of Real Functions
- Forms of Linear Equations
References
- ^OpenStax, College Algebra 2e, Rice University (Linear inequalities and absolute value inequalities).
- ^James Stewart, Lothar Redlin, and Saleem Watson, Precalculus: Mathematics for Calculus, 7th ed., Cengage Learning, 2015.