Graph the set \{x \mid x \leq 5\} on the number line. Then, write the set using interval notation.
Answer (2)
Graph the set on the number line, including a closed circle at x = 5 .
Recognize that the interval includes all numbers less than or equal to 5.
Express the set in interval notation using negative infinity as the lower bound and 5 as the upper bound, inclusive.
The set in interval notation is ( − ∞ , 5 ] .
Explanation
Understanding the Problem We are asked to graph the set of all real numbers x such that x is less than or equal to 5 on the number line and then to express this set using interval notation.
Graphing the Set To graph the set on the number line, we shade all values less than or equal to 5. This includes a closed circle at x = 5 to indicate that 5 is included in the set.
Writing in Interval Notation To write the set in interval notation, we note that the set includes all numbers less than or equal to 5. The interval extends from negative infinity to 5, inclusive. This is represented as ( − ∞ , 5 ] .
Examples
Understanding sets and intervals is crucial in many areas of mathematics and real-life applications. For example, when analyzing the domain of a function, you might find that the function is only defined for x ≤ 5 , representing a set similar to the one in this problem. In economics, you might describe the range of possible prices for a product as an interval. In computer science, you might define the valid range of input values for a program using interval notation. These concepts provide a concise and precise way to describe collections of numbers.
To graph the set {x \mid x \leq 5} on a number line, place a closed circle at 5 and shade to the left. The set in interval notation is written as ( − ∞ , 5 ] . This signifies all real numbers less than or equal to 5.
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