Write the inequality $3
Answer (2)
The inequality 3 < x ≤ 7 can be represented in interval notation as ( 3 , 7 ] . On a number line, this is shown with an open circle at 3 and a closed circle at 7, connected by a line segment. This indicates all numbers greater than 3 and up to and including 7.
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Convert the inequality 3 < x ≤ 7 to interval notation.
Use parenthesis '(' for values not included in the interval and bracket ']' for values included.
The interval notation is ( 3 , 7 ] .
On a number line, represent this interval with an open circle at 3 and a closed circle at 7, connected by a line segment. The final answer is ( 3 , 7 ] .
Explanation
Understanding the Problem We are given the inequality 3 < x ≤ 7 and asked to express it in interval notation and sketch it on a number line.
Expressing in Interval Notation Interval notation is a way to represent a set of numbers using brackets and parentheses. Parentheses indicate that the endpoint is not included in the interval, while brackets indicate that the endpoint is included. Since x is strictly greater than 3, we use a parenthesis at 3. Since x is less than or equal to 7, we use a bracket at 7. Therefore, the interval notation for 3 < x ≤ 7 is ( 3 , 7 ] .
Sketching on a Number Line To sketch the interval on a number line, we draw a number line and mark the points 3 and 7. At 3, we draw an open circle to indicate that 3 is not included in the interval. At 7, we draw a closed circle to indicate that 7 is included in the interval. Then, we draw a line segment connecting the open circle at 3 and the closed circle at 7 to represent all the numbers between 3 and 7, including 7 but not including 3.
Final Answer The interval notation for the inequality 3 < x ≤ 7 is ( 3 , 7 ] . On a number line, this is represented by an open circle at 3, a closed circle at 7, and a line segment connecting them.
Examples
Interval notation and number line representation are used in various fields, such as statistics, calculus, and computer science, to define ranges of values. For example, in statistics, a confidence interval might be expressed as ( a , b ) , representing a range of values within which a population parameter is expected to lie. In calculus, intervals are used to define the domain of a function or the limits of integration. In computer science, intervals can represent ranges of valid inputs for a program or the bounds of a data structure. Understanding how to represent inequalities using interval notation and number lines is a fundamental skill for these applications.
