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Chapter 1: Problem 87

Write the set as a single interval. $$[(-\infty,-2) \cup(4, \infty)] \cap[-5,3)$$

Short Answer

Expert verified
[-5, -2)

Step by step solution

01

Identify and graph the intervals

First, identify the given intervals and graph them on a number line. The intervals are \[(-\infty, -2) \cup (4, \infty)\]and\[-5, 3)\].
02

Find the intersection of the intervals

To find the intersection, look for the common points between the intervals \[(-\infty, -2)\cup(4, \infty)\] and \[-5, 3)\]. This means identifying where these intervals overlap.
03

Determine the overlapping section

The interval \[(-\infty,-2)\] intersects with \[-5, 3)\] in the section \[-5, -2)\]. The interval \[(4, \infty)\] does not intersect with \[-5, 3)\].
04

Write the final interval

Since the only overlap is \[-5, -2)\], the intersection as a single interval is \[-5, -2)\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

set intersection
In mathematics, a 'set intersection' refers to the common elements shared between two or more sets. When dealing with intervals, finding the intersection means identifying the overlapping regions where the intervals share common points. To illustrate, consider two intervals: [-5, 3) and (-鈭�, -2) U (4, 鈭�). The 'set intersection' operation involves looking for all points that lie within both intervals simultaneously.
number line
A number line is a visual representation that shows real numbers in order. Using a number line can help you easily understand where intervals start, end, and overlap. In our problem, let's mark the intervals (-鈭�, -2), (4, 鈭�), and [-5, 3) on the number line. This will visually show which parts of these intervals intersect.
interval overlap
Interval overlap occurs when two or more intervals have common points. To determine this, look for shared sections on the number line. For example, in the intervals (-鈭�, -2) and [-5, 3), the overlap is the section [-5, -2). This is because -5, -4, -3, and -2 are part of both intervals. For the intervals (4, 鈭�) and [-5, 3), there is no overlap because they do not share any common points.
inequalities in algebra
Inequalities in algebra involve expressions showing that one value is less than, greater than, or not equal to another. They are often used to define intervals on a number line. For example, the interval (-鈭�, -2) can be written as the inequality x < -2. Understanding and solving these inequalities is essential for finding interval intersections and overlaps.

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Most popular questions from this chapter

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