A closed interval is a set of real numbers that includes both the lower and upper bounds. It is denoted by square brackets and represents a range of values that are greater than or equal to the lower bound and less than or equal to the upper bound.
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Closed intervals are important in the context of solving rational inequalities, as the solutions often involve finding the values of the variable that satisfy the inequality.
The endpoints of a closed interval are considered part of the interval, meaning they are included in the set of solutions.
Closed intervals can be used to represent the domain or range of a rational function, which is crucial for understanding the behavior and properties of the function.
Graphically, a closed interval is represented by a line segment that includes the endpoints.
Closed intervals are often used in conjunction with other interval types, such as open intervals or half-open intervals, to describe the complete set of solutions to a rational inequality.
Review Questions
Explain how closed intervals are used to represent the solutions to a rational inequality.
When solving a rational inequality, the solutions often involve finding the values of the variable that satisfy the inequality. These solutions can be represented using closed intervals, which include the lower and upper bounds. The closed interval notation, such as [\$a\$, \$b\$], indicates that the solutions include all the values between \$a\$ and \$b\$, including \$a\$ and \$b\$ themselves. This is important because the endpoints of the interval are considered part of the solution set, unlike in the case of open intervals.
Describe how closed intervals are used to represent the domain and range of a rational function.
The domain and range of a rational function can be represented using closed intervals. The domain of a rational function is the set of all real numbers for which the denominator is not equal to zero. This can be expressed as a closed interval that includes all the values of the variable for which the function is defined. Similarly, the range of a rational function, which is the set of all possible output values, can also be represented using a closed interval. Understanding the closed interval representation of the domain and range is crucial for analyzing the behavior and properties of a rational function.
Evaluate how closed intervals can be used in conjunction with other interval types to describe the complete set of solutions to a rational inequality.
When solving rational inequalities, the complete set of solutions may involve a combination of closed intervals and other interval types, such as open intervals or half-open intervals. For example, the solution to a rational inequality may be expressed as the union of multiple closed intervals, or as the intersection of a closed interval and an open interval. This flexibility in representing the solutions allows for a more comprehensive and accurate description of the values that satisfy the inequality. By understanding how closed intervals can be used in conjunction with other interval types, you can better interpret and communicate the complete set of solutions to a rational inequality.