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Chapter 2: Problem 53

Express each set in the simplest interval form. $$ [3,6] \cup(4,9) $$

Short Answer

Expert verified
[3, 9)

Step by step solution

01

- Identify the Sets

The given sets are \( [3, 6] \) and \( (4, 9) \). The first is a closed interval from 3 to 6, and the second is an open interval from 4 to 9.
02

- Understand Set Union

The union of two sets includes all points that are in either set. Therefore, it combines all elements from \( [3, 6] \) and \( (4, 9) \).
03

- Find the Overlapping Interval

The intervals \( [3, 6] \) and \( (4, 9) \) overlap between 4 and 6. Specifically, from \( [4, 6] \) part of the closed interval \( [3, 6] \) and part of the open interval \( (4, 9) \).
04

- Determine the Combined Set

Combine the intervals into one continuous interval. Starting from 3 (inclusive), through the overlapping section from 4 to 6, and continuing until 9 (not inclusive).
05

- Express in Simplified Interval Form

The simplest interval form for the union of the given sets is \( [3, 9) \).

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Key Concepts

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

Set Union
In math, a union merges different sets into a single set containing all unique elements from the original sets.
This is represented by the symbol \( \cup \), meaning 'union.' For sets A and B, the union includes all elements in A, all in B, and those in both.
For example, with sets \( [3, 6] \) and \( (4, 9) \), the union is written as \( [3, 6] \cup (4, 9) \). The result contains:
  • Elements 3 to 6 (inclusive) from \( [3, 6] \)
  • Elements greater than 4 and less than 9 from \( (4, 9) \)
This union takes every element in either set and merges them.
Closed Interval
A closed interval includes its endpoints, represented with square brackets. For instance, the interval \( [3, 6] \) is closed because it includes 3, 4, 5, and 6.
This means the values at the boundaries are part of the set. To visualize, imagine a line segment from 3 to 6. Both 3 and 6 are part of this segment.
Closed intervals are useful when you need to consider every point within a given range, including the borders.

Example: The interval \( [a, b] \) means:
  • It includes all values from \(\text{a}\) to \(\text{b}\)
  • It contains both \(\text{a}\) and \(\text{b}\)
Open Interval
An open interval does not include its endpoints, represented with parenthesis. For instance, \( (4, 9) \) is open because it includes values greater than 4 and less than 9, but not 4 and 9 themselves.
This means you never reach the edges, only points close to them. Think of an open interval as a line segment from which both boundary points are removed.

Example: The interval \( (a, b) \) means:
  • It contains all values strictly between \(\text{a}\) and \(\text{b}\)
  • It excludes \(\text{a}\) and \(\text{b}\)
Open intervals are used when the boundary values themselves should not be part of the set.
Overlapping Intervals
Overlapping intervals share at least one common point. When this happens, you need to find and merge the shared parts.
The two intervals \( [3, 6] \) and \( (4, 9) \) overlap between 4 and 6. That means every number in this range belongs to both intervals.
In our example:
  • The common part is \( [4, 6] \).
  • When we take the union, this helps us to determine the full span: from 3 (inclusive) to 9 (exclusive).
Merging overlapping intervals into one gives a single, simplified representation. From our example, the union would be \( [3,9) \).

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Most popular questions from this chapter

Solve each inequality. Give the solution set in both interval and graph form. See Examples 1鈥�5. $$ 8\left(\frac{1}{2} x+3\right)<8\left(\frac{1}{2} x-1\right) $$

Let \(A=\\{1,2,3,4,5,6\\}, B=\\{1,3,5\\}, C=\\{1,6\\}\) and \(D=\\{4\\} .\) Specify each set. See Examples I and 5. $$ B \cup D $$

Why is it impossible to mix candy worth \(\$ 4\) per lb and candy worth \(\$ 5\) per lb to obtain a final mixture worth \(\$ 6\) per Ib \(?\)

The recommended daily intake (RDI) of calcium for females aged \(19-50\) is \(1000 \mathrm{mg}\). Actual needs vary from person to person. Write this statement as an absolute value inequality, with \(x\) representing the RDI, to express the RDI plus or minus \(100 \mathrm{mg}\), and solve the inequality. (Source: National Academy of Sciences- -Institute of Medicine.)

How many liters of a \(10 \%\) alcohol solution must be mixed with 40 L of a \(50 \%\) solution to get a \(40 \%\) solution?
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