Graph the set \{x \mid -5 < x \le 1\}.
Answer (2)
Graph the set on a number line with an open circle at -5 and a closed circle at -2, shading the region between them.
Use a parenthesis for -5 since it's not included and a bracket for -2 since it is included.
Write the set in interval notation as ( − 5 , − 2 ] .
The final answer in interval notation is ( − 5 , − 2 ] .
Explanation
Understanding the Problem We are given the set { x ∣ − 5 < x ≤ − 2 } , which includes all numbers x that are greater than -5 but less than or equal to -2. Our task is to represent this set graphically on a number line and then express it using interval notation.
Graphing on the Number Line To graph the set on a number line, we first draw a number line and mark the points -5 and -2. Since -5"> x > − 5 , we use an open circle at -5 to indicate that -5 is not included in the set. Since x ≤ − 2 , we use a closed circle (or a filled-in circle) at -2 to indicate that -2 is included in the set. Then, we shade the region between -5 and -2 to represent all the numbers between these two points.
Expressing in Interval Notation To write the set in interval notation, we use parentheses and brackets to indicate whether the endpoints are included or not. Since -5 is not included, we use a parenthesis. Since -2 is included, we use a bracket. Therefore, the interval notation for the set is ( − 5 , − 2 ] .
Final Answer The set { x ∣ − 5 < x ≤ − 2 } graphed on a number line includes all numbers greater than -5 (but not including -5) up to and including -2. In interval notation, this is represented as ( − 5 , − 2 ] .
Examples
Understanding sets and intervals is crucial in many real-world applications. For example, if you are measuring the temperature range for a specific chemical reaction to occur, you might find that the reaction only happens within a temperature range of -5 degrees Celsius (exclusive) to -2 degrees Celsius (inclusive). This would be represented as the interval (-5, -2]. Similarly, in finance, if you are analyzing stock prices, you might define a set of prices within a certain range to identify buying or selling opportunities.
To graph the set {x \mid -5 < x \leq 1}, draw a number line with an open circle at − 5 and a closed circle at 1 , then shade the region between them. The corresponding interval notation for this set is ( − 5 , 1 ] . This notation signifies that all values greater than − 5 and up to 1 are included in the set.
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