Graph the set \{x \mid x<-5\} on the number line. Then, write the set using interval notation.
Answer (2)
Graph the inequality x < − 5 on a number line using an open circle at -5 and shading to the left.
Represent the set in interval notation, indicating all numbers less than -5.
The interval notation for the set is ( − ∞ , − 5 ) .
The final answer is: ( − ∞ , − 5 )
Explanation
Understanding the Problem We are asked to graph the set of all numbers x such that x < − 5 on a number line and then express this set using interval notation.
Graphing on the Number Line First, let's visualize this set on a number line. We draw a number line and mark the point -5 on it. Since the inequality is x < − 5 , we use an open circle at -5 to indicate that -5 is not included in the set. Then, we shade the region to the left of -5 to represent all numbers less than -5.
Expressing in Interval Notation Now, let's express the set using interval notation. The set includes all numbers less than -5, so the interval starts at negative infinity and goes up to -5, not including -5. Therefore, the interval notation is ( − ∞ , − 5 ) .
Examples
Understanding sets and inequalities is crucial in many real-world scenarios. For example, if a company states that it only hires employees who score in the top 10% on a certain test, this can be represented as a set of scores greater than a certain value. Similarly, if a store offers a discount for customers over 65 years old, this can be represented as a set of ages greater than or equal to 65. Interval notation and number lines help visualize and understand these conditions clearly.
To graph the set {x \mid x < -5}, draw a number line, place an open circle on -5, and shade to the left. The interval notation for this set is ( − ∞ , − 5 ) . This indicates all numbers less than -5, not including -5 itself.
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