Interval Notation for Inequalities

Key Concepts

• Interval Notation: A way to represent intervals using parentheses and brackets.
• Parentheses: Indicates that the endpoint is not included.
• Brackets: Indicates that the endpoint is included.
• Infinity (∞): Always associated with parentheses.
• Open Circle: Endpoint not included.
• Closed Circle: Endpoint included.

Examples

1. x > 4

• Number Line: Open circle at 4, shade to the right.
• Interval Notation: (4, ∞)

2. x ≥ 2

• Number Line: Closed circle at 2, shade to the right.
• Interval Notation: [2, ∞)](streamdown:incomplete-link)

3. x < 3

• Number Line: Open circle at 3, shade to the left.
• Interval Notation: (-∞, 3)

4. x ≤ -1

• Number Line: Closed circle at -1, shade to the left.
• Interval Notation: (-∞, -1]

Compound Inequalities

5. 2 < x ≤ 6

• Number Line: Open circle at 2, closed circle at 6, shade between.
• Interval Notation: (2, 6]

6. -3 ≤ x < 4

• Number Line: Closed circle at -3, open circle at 4, shade between.
• Interval Notation: [-3, 4)](streamdown:incomplete-link)

7. x < -2 or x ≥ 5

• Number Line:
  • Open circle at -2, shade to the left.
  • Closed circle at 5, shade to the right.
• Interval Notation: (-∞, -2) ∪ [5, ∞)](streamdown:incomplete-link)

8. x ≤ 1 or x > 2

• Number Line:
  • Closed circle at 1, shade to the left.
  • Open circle at 2, shade to the right.
• Interval Notation: (-∞, 1] ∪ (2, ∞)

Conclusion

• Always graph the solution on a number line first.
• Translate the number line into interval notation using the appropriate symbols for open and closed circles and intervals including or excluding endpoints.