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Chapter 8: Problem 49

Concept Check Express each set in simplest interval form. \((-\infty,-6] \cap[-9, \infty)\)

Short Answer

Expert verified
The intersection is \([-9, -6]\).

Step by step solution

01

Understand the intersection

Given the two sets \((-\infty, -6]\) and \([-9, \infty)\), need to find their intersection. The intersection of two sets consists of elements present in both sets.
02

Identify the range of each set

Set \((-\infty, -6]\) includes all values less than or equal to -6. Set \([-9, \infty)\) includes all values greater than or equal to -9.
03

Find the overlapping range

The overlapping range, or intersection, is where these two sets share common values. This occurs for values between -9 and -6, inclusive of both -9 and -6 since both endpoints are included in at least one of the sets.
04

Write the final interval in simplest form

The simplest interval form that represents values between -9 and -6, inclusive, is \([-9, -6]\).

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Key Concepts

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

set intersection
In mathematics, the concept of 'set intersection' is vital. It helps you find common elements between two sets. Imagine two overlapping circles in a Venn diagram. The area where both circles overlap is the intersection. For example, if you have \((-\infty, -6]\) and \([-9, \infty)\), you find values that are in both sets. This will form the 'intersection' of these intervals. Lean on the idea that you're looking for common ground!
interval notation
Interval notation is a streamlined way to represent a continuous range of numbers. Instead of listing all numbers, you can use start and end points. Brackets express these limits. For example, \([-9, -6]\) means that numbers -9 and -6, as well as every number in between, are included. A parenthesis (like in \((-\beginfity, -6]\)) means the number next to it (on that side) is not included. This notation is compact and clear, making it easier to communicate ranges.
overlapping range
An 'overlapping range' refers to numbers or values shared between two sets or intervals. Imagine two sections on a number line that overlap. This common part is the overlapping range. In our example, \((-\beginfinity, -6]\) and \([-9, \begininty)\) share values between -9 and -6, inclusive. It's essential to visually or logically identify where these intervals intersect. This helps in various fields, from math to computer programming, to find a shared space or common ground.
simplifying intervals
Once you've identified the 'set intersection' and 'overlapping range,' the intersection should be simplified to its simplest interval form. This makes it concise and easy to understand. When we simplified \((-\infty, -6]\) \cap \([-9, \infty)\), we found \([-9, -6]\). Simplifying intervals involves combining the insights from the intersections. The goal is a straightforward way to represent the common range. Whether you're solving math problems or dealing with real-life data, a simplified interval always helps you grasp the concept faster.

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