Interval Notation

Definition

Interval notation is a way to represent a set of real numbers that satisfy a specific condition or range. It is commonly used to describe and visualize the solutions to linear inequalities and their applications.

5 Must Know Facts For Your Next Test

1. Interval notation uses symbols such as $[a, b]$, $(a, b)$, $[a, b)$, and $(a, b]$ to represent sets of real numbers.
2. The square brackets $[a, b]$ indicate that the interval includes the endpoints $a$ and $b$, while the parentheses $(a, b)$ indicate that the interval excludes the endpoints.
3. Interval notation is particularly useful in solving and representing the solutions to linear inequalities, as it provides a concise and visual way to display the range of acceptable values.
4. In the context of applications with linear inequalities, interval notation can be used to describe the feasible region or the set of all possible solutions that satisfy the given constraints.
5. Interval notation can also be used to represent the domain and range of functions, as well as to describe the intervals where a function is increasing, decreasing, or constant.

Review Questions

• Explain how interval notation is used to represent the solutions to a linear inequality.
  • Interval notation is used to represent the solutions to a linear inequality by describing the range of values that satisfy the inequality. For example, the solution to the inequality $x \geq 3$ can be represented using the interval notation $[3, \infty)$, which indicates that the set of all real numbers greater than or equal to 3 is the solution. Similarly, the solution to the inequality $-2 < x \leq 5$ can be represented using the interval notation $(-2, 5]$, which includes all real numbers greater than -2 and less than or equal to 5.
• Describe how interval notation can be used to represent the feasible region in the context of applications with linear inequalities.
  • In the context of applications with linear inequalities, interval notation can be used to represent the feasible region, which is the set of all possible solutions that satisfy the given constraints. The feasible region is often represented using a combination of intervals, where each interval corresponds to the solution set of a particular linear inequality. For example, if a problem involves two linear inequalities, such as $x + y \leq 10$ and $x - y \geq 0$, the feasible region can be described using the interval notation $\{(x, y) | x + y \leq 10, x - y \geq 0\}$. This interval notation provides a concise and visual way to describe the set of all points that satisfy both inequalities.
• Analyze how interval notation can be used to describe the domain and range of a function, as well as the intervals where the function is increasing, decreasing, or constant.
  • Interval notation can be used to describe the domain and range of a function, as well as the intervals where the function is increasing, decreasing, or constant. The domain of a function represents the set of input values for which the function is defined, and this can be expressed using interval notation. For example, the domain of the function $f(x) = \sqrt{x}$ can be represented as $[0, \infty)$, indicating that the function is defined for all non-negative real numbers. Similarly, the range of a function represents the set of output values, and this can also be described using interval notation. Furthermore, interval notation can be used to describe the intervals where a function is increasing, decreasing, or constant, which is crucial for understanding the behavior of the function. For instance, the function $f(x) = x^2$ is increasing on the interval $(-\infty, 0)$ and $[0, \infty)$, and constant on the interval $\{0\}$.

Related terms

Real Numbers: The set of all numbers that can be represented on the number line, including positive and negative integers, fractions, and irrational numbers.

Linear Inequality: An inequality that can be expressed in the form $ax + b \geq c$ or $ax + b > c$, where $a$, $b$, and $c$ are real numbers.

Bounded Interval: An interval with a finite lower and upper bound, represented using square brackets or parentheses.