Video: Identifying the Inequality That Describes a Given Interval

Given that π‘₯ ∈ [βˆ’35, ∞), which of the following inequalities holds? [A] π‘₯ < βˆ’35 [B] π‘₯ ≀ βˆ’35 [C] π‘₯ β‰₯ βˆ’35 [D] π‘₯ > βˆ’35

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Video Transcript

Given that π‘₯ is an element of negative 35, ∞, which of the following inequalities holds? Option A) π‘₯ is less than negative 35. Option B) π‘₯ is less than or equal to negative 35. Option C) π‘₯ is greater than or equal to negative 35. Option D) π‘₯ is greater than negative 35.

Let’s start this question by having a look at the notation that’s used. The symbol here that looks like an E indicates that π‘₯ is an element or a member of the set from negative 35 to ∞. We can see that there is a bracket on the left side and a parenthesis on the right side of the set. A square bracket next to a number in a set indicates that that number is also included. It would be equivalent to the inequality signs greater than or equal to or less than or equal to. When round parentheses are used in set notation, that indicates that the number next to it is not included. The inequality equivalents are greater than or less than.

Let’s see how we can represent this problem on a number line. If we draw a circle over negative 35, we first need to establish if that would be filled in or not. Well, since the square bracket next to the negative 35 indicates that the negative 35 is also included, then we fill in the circle. Next, we need to represent the fact that our set goes from negative 35 to ∞. So if we were to draw a line, which way should it go? The answer is that it should point to the right. Since that’s the direction in which our numbers increased to positive ∞, to the left the numbers decreased to negative ∞.

As an aside, we might wonder why there wasn’t a bracket next to ∞ in our set notation. And the answer is simply that convention says, we always use a parenthesis next to an ∞ or negative ∞ sign. So now, we have shown the inequality on a number line. Let’s see if we can write it as a statement. We have established that our value of π‘₯ can be equal to negative 35 and it includes all values greater than that as well. So we use the symbol greater than or equal to. So our answer will be that in option C π‘₯ is greater than or equal to negative 35.

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