Video: Identifying the Interval Represented on a Given Number Line

Which of the following expressions represents the set shown on the number line? [A] ℝ − (−4, 1] [B] ℝ − [−4, 1] [C] (−∞, −4] ∪ [1, ∞) [D] ℝ − (−4, 1) [E] ℝ − [−4, 1)

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Video Transcript

Which of the following expressions represents the set shown on the number line? Is it (a) the set of all real numbers minus the set of numbers greater than negative four and less than or equal to one? Is it (b) the set of all real numbers minus the set of numbers in the closed interval negative four to one? Is it (c) the union of the set of numbers greater than negative ∞ and less than or equal to negative four and the set greater than or equal to one and less than ∞? (d) is the set of all real members minus the set of numbers in the open interval negative four to one. And (e) is the set of all real numbers minus the set of numbers greater than or equal to negative four and less than one.

Let’s go right back to our number line. And we begin by recalling the meaning of these dots. An empty dot means greater than or less than whereas a solid dot means greater than or equal to or less than or equal to. And so, let’s imagine the set of numbers we’re interested in are the set of numbers for some unknown 𝑥. We can see that 𝑥 can take values less than or equal to negative four and greater than one. So how do we represent these using interval notation?

Well, we can see that our value for 𝑥 can take all real numbers except those in this underlined part. Those are the values including negative three, negative two, negative one, zero, and one. Well, we know that this letter ℝ represents the set of all real numbers. And we want to subtract some set from negative four to one. We don’t want to subtract negative four from all real numbers because we know that 𝑥 could be negative four. But we use the closed interval notation next to the one. And that’s to show that we don’t want to include the number one in our set.

And so, the expression that represents the set shown on the number line is the set of all real numbers minus the set containing numbers greater than negative four and less than or equal to one. And we can see that’s (a).

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