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Chapter 8: Problem 53

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

Short Answer

Expert verified
[3,9]

Step by step solution

01

Understand the Union of Intervals

The union of two intervals is all the numbers that are in either interval. This means combining all numbers from both intervals into one.
02

Identify the Intervals and their Overlap

We have two intervals: \[3,6\] and \(4,9\). The interval \[3,6\] represents all numbers from 3 to 6, inclusive. The interval \(4,9\) represents all numbers greater than 4 and less than 9.
03

Combine the Intervals

Since \(4,9\) overlaps with \[3,6\] from 4 to 6, the union of these two intervals will start at the lower bound of the first interval, which is 3, and end at the upper bound of the second interval, which is 9.
04

Write the Simplified Interval Form

The simplest interval form considering the union of the overlapping intervals \[3,6\] and \(4,9\) is \[3,9\].

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Key Concepts

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

Union of Intervals
The union of intervals combines all the numbers that are in either interval. Imagine you have two intervals on the number line. To find the union, you need to include every number that appears in at least one of the intervals. For instance, with the intervals \[3, 6\] and \(4, 9\), the union includes all numbers between the smallest start point and the largest end point. This method helps in creating a combined interval that contains all possible values from both intervals.
Overlapping Intervals
Overlapping intervals mean some portions of two intervals cover the same numbers. For example, with \[3, 6\] and \(4, 9\), the interval from 4 to 6 is common to both. In mathematical terms, overlapping means the two intervals share some of their ranges. When intervals overlap, it affects how you compute their union. Instead of listing each interval separately, you recognize the overlap and merge the intervals efficiently. This creates a more compact representation of the union, simplifying the visualization.
Simplifying Intervals
Simplifying intervals involves combining overlapping intervals into a single, continuous interval. This process makes them easier to understand and work with. Take the example of \[3, 6\] and \(4, 9\). Since these intervals overlap from 4 to 6, you can simplify them into the interval \[3, 9\]. This new interval covers all numbers from 3 to 9, without any gaps. Simplifying intervals helps in minimizing the number of separate intervals, thus making future calculations or interpretations more straightforward.

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Most popular questions from this chapter

Graph the solution set of each system of linear inequalities. $$ \begin{aligned} 4 x+5 y & \geq 20 \\ x-2 y & \leq 5 \end{aligned} $$

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Graph each linear inequality. $$ x+3 y \geq-2 $$

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