Interval Notation

Definition

Interval notation is a mathematical notation used to represent a range of numbers between two endpoints. This system allows for the concise expression of intervals, where the type of brackets indicates whether the endpoints are included or excluded, making it essential for understanding solutions to linear inequalities.

5 Must Know Facts For Your Next Test

1. Interval notation uses brackets [ ] to indicate inclusive endpoints and parentheses ( ) to indicate exclusive endpoints.
2. When solving linear inequalities, solutions can often be expressed using interval notation to show all possible values that satisfy the inequality.
3. The notation can represent infinite intervals, such as (-∞, a) or (b, ∞), indicating all numbers less than a or greater than b, respectively.
4. It is crucial to use proper notation when combining multiple intervals; for instance, the union of intervals can be expressed as [a, b) ∪ (c, d].
5. When graphing inequalities on a number line, interval notation provides a clear way to visualize the solution set by marking included or excluded values.

Review Questions

• How do you convert a linear inequality into interval notation?
  • To convert a linear inequality into interval notation, first solve the inequality for the variable to determine the range of values that satisfy it. Identify whether the endpoints are included or excluded based on the inequality symbols: use brackets for ≤ or ≥ and parentheses for < or >. Finally, write the interval using the determined endpoints and correct symbols.
• In what scenarios would you use open versus closed intervals in interval notation?
  • You would use closed intervals when you want to include the endpoints in your solution set, typically when solving inequalities that use ≤ or ≥. Conversely, open intervals are appropriate when you exclude the endpoints from the solution set, which occurs with strict inequalities using < or >. Choosing between these intervals accurately reflects the nature of the inequality you are working with.
• Evaluate how interval notation enhances understanding when working with compound inequalities.
  • Interval notation enhances understanding of compound inequalities by providing a clear and concise way to express multiple ranges of solutions simultaneously. For example, if you have a compound inequality like x < 3 and x ≥ 1, using interval notation allows you to combine these into a single expression: [1, 3). This visual representation simplifies identifying valid solutions and allows for easier manipulation and comparison with other intervals.

Related terms

Inequality: A mathematical statement that shows the relationship between two expressions that are not equal, often represented with symbols such as <, >, ≤, or ≥.

Closed Interval: An interval that includes its endpoints, denoted by square brackets [a, b], indicating that both a and b are part of the interval.

Open Interval: An interval that does not include its endpoints, denoted by parentheses (a, b), meaning that neither a nor b is part of the interval.